CORD-19-Sentences  

CORD-19:ad147a4279222c5e480a5636fb879540eded760e JSONTXT 8 Projects

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TextSentencer_T1 0-48 Sentence denotes Estimating the distance to an epidemic threshold
TextSentencer_T1 0-48 Sentence denotes Estimating the distance to an epidemic threshold
TextSentencer_T2 50-58 Sentence denotes Abstract
TextSentencer_T2 50-58 Sentence denotes Abstract
TextSentencer_T3 59-207 Sentence denotes The epidemic threshold of the susceptible-infected-recovered model is a boundary separating parameters that permit epidemics from those that do not.
TextSentencer_T3 59-207 Sentence denotes The epidemic threshold of the susceptible-infected-recovered model is a boundary separating parameters that permit epidemics from those that do not.
TextSentencer_T4 208-297 Sentence denotes This threshold corresponds to parameters where the system's equilibrium becomes unstable.
TextSentencer_T4 208-297 Sentence denotes This threshold corresponds to parameters where the system's equilibrium becomes unstable.
TextSentencer_T5 298-423 Sentence denotes Consequently, we use the average rate at which deviations from the equilibrium shrink to define a distance to this threshold.
TextSentencer_T5 298-423 Sentence denotes Consequently, we use the average rate at which deviations from the equilibrium shrink to define a distance to this threshold.
TextSentencer_T6 424-544 Sentence denotes However, the vital dynamics of the host population may occur slowly even when transmission is far from threshold levels.
TextSentencer_T6 424-544 Sentence denotes However, the vital dynamics of the host population may occur slowly even when transmission is far from threshold levels.
TextSentencer_T7 545-699 Sentence denotes Here, we show analytically how such slow dynamics can prevent estimation of the distance to the threshold from fluctuations in the susceptible population.
TextSentencer_T7 545-699 Sentence denotes Here, we show analytically how such slow dynamics can prevent estimation of the distance to the threshold from fluctuations in the susceptible population.
TextSentencer_T8 700-942 Sentence denotes Although these results are exact only in the limit of long-term observation of a large system, simulations show that they still provide useful insight into systems with a range of population sizes, environmental noise and observation schemes.
TextSentencer_T8 700-942 Sentence denotes Although these results are exact only in the limit of long-term observation of a large system, simulations show that they still provide useful insight into systems with a range of population sizes, environmental noise and observation schemes.
TextSentencer_T9 943-1126 Sentence denotes Having established some guidelines about when estimates are accurate, we then illustrate how multiple distance estimates can be used to estimate the rate of approach to the threshold.
TextSentencer_T9 943-1126 Sentence denotes Having established some guidelines about when estimates are accurate, we then illustrate how multiple distance estimates can be used to estimate the rate of approach to the threshold.
TextSentencer_T10 1127-1331 Sentence denotes The estimation approach is general and may be applicable to zoonotic pathogens such as Middle East respiratory syndrome-related coronavirus (MERS-CoV) as well as vaccine-preventable diseases like measles.
TextSentencer_T10 1127-1331 Sentence denotes The estimation approach is general and may be applicable to zoonotic pathogens such as Middle East respiratory syndrome-related coronavirus (MERS-CoV) as well as vaccine-preventable diseases like measles.
TextSentencer_T11 1333-1443 Sentence denotes Many infectious disease epidemics occur with sufficient regularity that their anticipation is straightforward.
TextSentencer_T11 1333-1443 Sentence denotes Many infectious disease epidemics occur with sufficient regularity that their anticipation is straightforward.
TextSentencer_T12 1444-1561 Sentence denotes For example, seasonal influenza has a pronounced winter seasonality in most of the world, with annual outbreaks [1] .
TextSentencer_T12 1444-1561 Sentence denotes For example, seasonal influenza has a pronounced winter seasonality in most of the world, with annual outbreaks [1] .
TextSentencer_T13 1562-1758 Sentence denotes Some systems are more episodic but still well understood, such as measles in sub-Saharan Africa where regional inter-epidemic periods between 1 and 4 years have been observed in recent times [2] .
TextSentencer_T13 1562-1758 Sentence denotes Some systems are more episodic but still well understood, such as measles in sub-Saharan Africa where regional inter-epidemic periods between 1 and 4 years have been observed in recent times [2] .
TextSentencer_T14 1759-1910 Sentence denotes By contrast, emerging and re-emerging infectious diseases are rarely anticipated, even though the root causes are often discerned soon after the event.
TextSentencer_T14 1759-1910 Sentence denotes By contrast, emerging and re-emerging infectious diseases are rarely anticipated, even though the root causes are often discerned soon after the event.
TextSentencer_T15 1911-2018 Sentence denotes Many childhood infectious diseases naturally spread effectively, including measles, chickenpox and rubella.
TextSentencer_T15 1911-2018 Sentence denotes Many childhood infectious diseases naturally spread effectively, including measles, chickenpox and rubella.
TextSentencer_T16 2019-2178 Sentence denotes This means that, in unvaccinated populations, one infectious individual may infect many others, measured by the pathogen's basic reproduction number, R 0 [3] .
TextSentencer_T16 2019-2178 Sentence denotes This means that, in unvaccinated populations, one infectious individual may infect many others, measured by the pathogen's basic reproduction number, R 0 [3] .
TextSentencer_T17 2179-2424 Sentence denotes Outbreaks are prevented in these cases by maintaining a very high proportion of vaccinated individuals, generating herd immunity in which the effective reproduction number is below 1, meaning small chains of transmission are quickly broken [4] .
TextSentencer_T17 2179-2424 Sentence denotes Outbreaks are prevented in these cases by maintaining a very high proportion of vaccinated individuals, generating herd immunity in which the effective reproduction number is below 1, meaning small chains of transmission are quickly broken [4] .
TextSentencer_T18 2425-2597 Sentence denotes Reduced vaccine uptake rates can move the infectious disease system from controlled (sub-critical, with effective R 0 , 1) to super-critical, when outbreaks may occur [5] .
TextSentencer_T18 2425-2597 Sentence denotes Reduced vaccine uptake rates can move the infectious disease system from controlled (sub-critical, with effective R 0 , 1) to super-critical, when outbreaks may occur [5] .
TextSentencer_T19 2598-2719 Sentence denotes Alternatively, other features of the system may be slowly changing, similarly enhancing the transmission of the pathogen.
TextSentencer_T19 2598-2719 Sentence denotes Alternatively, other features of the system may be slowly changing, similarly enhancing the transmission of the pathogen.
TextSentencer_T20 2720-2967 Sentence denotes Host demographic changes, particularly rising birth rates, can increase the supply of susceptible individuals to the population, and pathogens frequently evolve at high rates, whereby fitter strains (higher R 0 ) may be favoured by selection [6] .
TextSentencer_T20 2720-2967 Sentence denotes Host demographic changes, particularly rising birth rates, can increase the supply of susceptible individuals to the population, and pathogens frequently evolve at high rates, whereby fitter strains (higher R 0 ) may be favoured by selection [6] .
TextSentencer_T21 2968-3301 Sentence denotes Predicting a dynamical system's movement from sub-to super-critical before it happens has enormous potential to remove the element of surprise associated with emerging infectious diseases, to prioritize mitigation strategies to reverse, stop or slow the transition, and in worst cases to simply be better prepared for the inevitable.
TextSentencer_T21 2968-3301 Sentence denotes Predicting a dynamical system's movement from sub-to super-critical before it happens has enormous potential to remove the element of surprise associated with emerging infectious diseases, to prioritize mitigation strategies to reverse, stop or slow the transition, and in worst cases to simply be better prepared for the inevitable.
TextSentencer_T22 3302-3530 Sentence denotes Recent work has also illustrated that following a transition from sub-to super-critical there is a characterizable bifurcation delay-a waiting time until the outbreak actually occurs following suitable conditions being met [7] .
TextSentencer_T22 3302-3530 Sentence denotes Recent work has also illustrated that following a transition from sub-to super-critical there is a characterizable bifurcation delay-a waiting time until the outbreak actually occurs following suitable conditions being met [7] .
TextSentencer_T23 3531-3972 Sentence denotes Consequently, estimates of how far a system is from the epidemic threshold could help public health officials make judgements about policy, infer on which side of the threshold the population lies, and track the movement of a system towards a threshold ( providing early warnings) and even away from a threshold as a way of evaluating the effectiveness of any external changes to the system aimed at controlling infectious disease outbreaks.
TextSentencer_T23 3531-3972 Sentence denotes Consequently, estimates of how far a system is from the epidemic threshold could help public health officials make judgements about policy, infer on which side of the threshold the population lies, and track the movement of a system towards a threshold ( providing early warnings) and even away from a threshold as a way of evaluating the effectiveness of any external changes to the system aimed at controlling infectious disease outbreaks.
TextSentencer_T24 3973-4123 Sentence denotes A potentially robust basis for estimating the distance to a threshold is the general slowing down of a system's dynamics as a threshold is approached.
TextSentencer_T24 3973-4123 Sentence denotes A potentially robust basis for estimating the distance to a threshold is the general slowing down of a system's dynamics as a threshold is approached.
TextSentencer_T25 4124-4337 Sentence denotes To be more precise, the average decay rate of deviations from a fixed point of the system becomes increasingly smaller as the parameters of the system approach the point at which that fixed point becomes unstable.
TextSentencer_T25 4124-4337 Sentence denotes To be more precise, the average decay rate of deviations from a fixed point of the system becomes increasingly smaller as the parameters of the system approach the point at which that fixed point becomes unstable.
TextSentencer_T26 4338-4630 Sentence denotes Wissel [8] pointed out that this phenomenon, known as critical slowing down or sometimes simply as slowing down, could be used to determine whether the parameters of a system were approaching a threshold that, when crossed, could result in the system changing in an abrupt and drastic manner.
TextSentencer_T26 4338-4630 Sentence denotes Wissel [8] pointed out that this phenomenon, known as critical slowing down or sometimes simply as slowing down, could be used to determine whether the parameters of a system were approaching a threshold that, when crossed, could result in the system changing in an abrupt and drastic manner.
TextSentencer_T27 4631-4693 Sentence denotes Such changes have come to be called critical transitions [9] .
TextSentencer_T27 4631-4693 Sentence denotes Such changes have come to be called critical transitions [9] .
TextSentencer_T28 4694-4878 Sentence denotes A great deal of interest has developed in the possibility of devising model-independent methods to anticipate critical transitions in complex systems using early warning signals [10] .
TextSentencer_T28 4694-4878 Sentence denotes A great deal of interest has developed in the possibility of devising model-independent methods to anticipate critical transitions in complex systems using early warning signals [10] .
TextSentencer_T29 4879-5050 Sentence denotes In general, early warning signals are statistical properties of observations of systems that can be expected to change in characteristic ways as a threshold is approached.
TextSentencer_T29 4879-5050 Sentence denotes In general, early warning signals are statistical properties of observations of systems that can be expected to change in characteristic ways as a threshold is approached.
TextSentencer_T30 5051-5147 Sentence denotes Perhaps the most common examples are increasing autocorrelation and variance of model variables.
TextSentencer_T30 5051-5147 Sentence denotes Perhaps the most common examples are increasing autocorrelation and variance of model variables.
TextSentencer_T31 5148-5349 Sentence denotes These signals can often be derived from the increasingly slow decay of perturbations due to slowing down, and many other early warning signals are in one way or another quantifications of slowing down.
TextSentencer_T31 5148-5349 Sentence denotes These signals can often be derived from the increasingly slow decay of perturbations due to slowing down, and many other early warning signals are in one way or another quantifications of slowing down.
TextSentencer_T32 5350-5545 Sentence denotes The beauty of early warning signals is that their basis in generic properties of dynamical systems means they have the potential to be reliable even when the system is complex and unidentifiable.
TextSentencer_T32 5350-5545 Sentence denotes The beauty of early warning signals is that their basis in generic properties of dynamical systems means they have the potential to be reliable even when the system is complex and unidentifiable.
TextSentencer_T33 5546-5631 Sentence denotes Examples of complex and poorly identified systems abound in ecology and epidemiology.
TextSentencer_T33 5546-5631 Sentence denotes Examples of complex and poorly identified systems abound in ecology and epidemiology.
TextSentencer_T34 5632-5841 Sentence denotes With application to such systems in mind, several authors [11] [12] [13] demonstrated the application of early warning signals based on slowing down to forecasting infectious disease emergence and eradication.
TextSentencer_T34 5632-5841 Sentence denotes With application to such systems in mind, several authors [11] [12] [13] demonstrated the application of early warning signals based on slowing down to forecasting infectious disease emergence and eradication.
TextSentencer_T35 5842-6060 Sentence denotes Further development and integration of these methods into surveillance systems may provide a novel and broadly applicable method of evaluating the control of infectious diseases from existing surveillance data streams.
TextSentencer_T35 5842-6060 Sentence denotes Further development and integration of these methods into surveillance systems may provide a novel and broadly applicable method of evaluating the control of infectious diseases from existing surveillance data streams.
TextSentencer_T36 6061-6249 Sentence denotes To explain some of the current challenges in further developing approaches to estimating the distance to the threshold, we will make reference to some elements of dynamical systems theory.
TextSentencer_T36 6061-6249 Sentence denotes To explain some of the current challenges in further developing approaches to estimating the distance to the threshold, we will make reference to some elements of dynamical systems theory.
TextSentencer_T37 6250-6596 Sentence denotes Following Wiggins [14] , a general dynamical system may be written as a system of equations for a vector field _ x ¼ f(x, u), where the overdot indicates a derivative with respect to time, x is a vector of real numbers that determine the point of the system in its phase space, and u is a vector of real numbers that are parameters of the system.
TextSentencer_T37 6250-6596 Sentence denotes Following Wiggins [14] , a general dynamical system may be written as a system of equations for a vector field _ x ¼ f(x, u), where the overdot indicates a derivative with respect to time, x is a vector of real numbers that determine the point of the system in its phase space, and u is a vector of real numbers that are parameters of the system.
TextSentencer_T38 6597-6686 Sentence denotes A solution to the system is a function x of time that over some time interval satisfies _
TextSentencer_T38 6597-6686 Sentence denotes A solution to the system is a function x of time that over some time interval satisfies _
TextSentencer_T39 6687-6702 Sentence denotes x ¼ f(x(t), u).
TextSentencer_T39 6687-6702 Sentence denotes x ¼ f(x(t), u).
TextSentencer_T40 6703-6812 Sentence denotes A fixed point x* of the system is a solution that does not change with time (i.e. it satisfies 0 ¼ f(x*, u)).
TextSentencer_T40 6703-6812 Sentence denotes A fixed point x* of the system is a solution that does not change with time (i.e. it satisfies 0 ¼ f(x*, u)).
TextSentencer_T41 6813-6896 Sentence denotes Such a point is also referred to as a steady state or an equilibrium of the system.
TextSentencer_T41 6813-6896 Sentence denotes Such a point is also referred to as a steady state or an equilibrium of the system.
TextSentencer_T42 6897-7026 Sentence denotes A fixed point is called asymptotically stable if solutions that start at points near the fixed point move closer to it over time.
TextSentencer_T42 6897-7026 Sentence denotes A fixed point is called asymptotically stable if solutions that start at points near the fixed point move closer to it over time.
TextSentencer_T43 7027-7272 Sentence denotes Because the starting points are nearby, deviations z ¼ x 2 x* are small and can be accurately modelled by solutions to the linear system _ z ¼ Fz, where F denotes the matrix of first derivatives of f with respect to x (i.e. the Jacobian matrix).
TextSentencer_T43 7027-7272 Sentence denotes Because the starting points are nearby, deviations z ¼ x 2 x* are small and can be accurately modelled by solutions to the linear system _ z ¼ Fz, where F denotes the matrix of first derivatives of f with respect to x (i.e. the Jacobian matrix).
TextSentencer_T44 7273-7334 Sentence denotes The general solution of such a system is z(t) ¼ exp (Ft)z(0).
TextSentencer_T44 7273-7334 Sentence denotes The general solution of such a system is z(t) ¼ exp (Ft)z(0).
TextSentencer_T45 7335-7479 Sentence denotes If the real parts of all of the eigenvalues of F are negative, this solution will shrink to zero and it follows that x* is asympotically stable.
TextSentencer_T45 7335-7479 Sentence denotes If the real parts of all of the eigenvalues of F are negative, this solution will shrink to zero and it follows that x* is asympotically stable.
TextSentencer_T46 7480-7611 Sentence denotes If the real parts of any of the eigenvalues are positive, the solution will not shrink to zero and x* is not asymptotically stable.
TextSentencer_T46 7480-7611 Sentence denotes If the real parts of any of the eigenvalues are positive, the solution will not shrink to zero and x* is not asymptotically stable.
TextSentencer_T47 7612-7743 Sentence denotes Thus, as long as the real parts of the eigenvalues of F are not zero, their signs tell us whether or not any fixed point is stable.
TextSentencer_T47 7612-7743 Sentence denotes Thus, as long as the real parts of the eigenvalues of F are not zero, their signs tell us whether or not any fixed point is stable.
TextSentencer_T48 7744-7996 Sentence denotes The relationship between the speed of a system's dynamics and the distance to the threshold arises in the common case that the eigenvalues of F are continuous functions of the parameters u of the system and none of the eigenvalues have zero real parts.
TextSentencer_T48 7744-7996 Sentence denotes The relationship between the speed of a system's dynamics and the distance to the threshold arises in the common case that the eigenvalues of F are continuous functions of the parameters u of the system and none of the eigenvalues have zero real parts.
TextSentencer_T49 7997-8094 Sentence denotes In this case for a stable fixed point to become unstable, one of the eigenvalues must cross zero.
TextSentencer_T49 7997-8094 Sentence denotes In this case for a stable fixed point to become unstable, one of the eigenvalues must cross zero.
TextSentencer_T50 8095-8225 Sentence denotes Thus as the parameters approach the threshold where stability is lost, one of the eigenvalues must approach zero in its real part.
TextSentencer_T50 8095-8225 Sentence denotes Thus as the parameters approach the threshold where stability is lost, one of the eigenvalues must approach zero in its real part.
TextSentencer_T51 8226-8368 Sentence denotes We call such an eigenvalue an informative eigenvalue because its value is informative of how far the system's parameters are from a threshold.
TextSentencer_T51 8226-8368 Sentence denotes We call such an eigenvalue an informative eigenvalue because its value is informative of how far the system's parameters are from a threshold.
TextSentencer_T52 8369-8441 Sentence denotes We call the magnitude of such an eigenvalue a distance to the threshold.
TextSentencer_T52 8369-8441 Sentence denotes We call the magnitude of such an eigenvalue a distance to the threshold.
TextSentencer_T53 8442-8633 Sentence denotes If an informative eigenvalue can be monitored over time, one can determine whether the system is approaching a threshold or not and even make a forecast of when the threshold will be crossed.
TextSentencer_T53 8442-8633 Sentence denotes If an informative eigenvalue can be monitored over time, one can determine whether the system is approaching a threshold or not and even make a forecast of when the threshold will be crossed.
TextSentencer_T54 8634-8804 Sentence denotes An informative eigenvalue can be measured by monitoring the decay of small perturbations away from the fixed point along the eigendirection of the informative eigenvalue.
TextSentencer_T54 8634-8804 Sentence denotes An informative eigenvalue can be measured by monitoring the decay of small perturbations away from the fixed point along the eigendirection of the informative eigenvalue.
TextSentencer_T55 8805-8904 Sentence denotes Identifying trends in such a decay rate is the goal of early warning signals based on slowing down.
TextSentencer_T55 8805-8904 Sentence denotes Identifying trends in such a decay rate is the goal of early warning signals based on slowing down.
TextSentencer_T56 8905-9046 Sentence denotes Despite the simplicity of this goal, it is currently not clear exactly how it can be achieved when systems have multidimensional phase space.
TextSentencer_T56 8905-9046 Sentence denotes Despite the simplicity of this goal, it is currently not clear exactly how it can be achieved when systems have multidimensional phase space.
TextSentencer_T57 9047-9200 Sentence denotes When one of the eigenvalues of F gets closer to zero, only a small number of the model's observable variables may become less resilient to perturbations.
TextSentencer_T57 9047-9200 Sentence denotes When one of the eigenvalues of F gets closer to zero, only a small number of the model's observable variables may become less resilient to perturbations.
TextSentencer_T58 9201-9347 Sentence denotes The implication is that early warning signals such as increasing variance and autocorrelation will not be present in all of the model's variables.
TextSentencer_T58 9201-9347 Sentence denotes The implication is that early warning signals such as increasing variance and autocorrelation will not be present in all of the model's variables.
TextSentencer_T59 9348-9402 Sentence denotes Several authors have provided examples of such a case.
TextSentencer_T59 9348-9402 Sentence denotes Several authors have provided examples of such a case.
TextSentencer_T60 9403-9631 Sentence denotes Kuehn [15] showed that in a susceptibleinfected-susceptible (SIS) model of an epidemic on an adaptive contact network, only one of the three model variables had a clear increase in variance as the epidemic threshold was crossed.
TextSentencer_T60 9403-9631 Sentence denotes Kuehn [15] showed that in a susceptibleinfected-susceptible (SIS) model of an epidemic on an adaptive contact network, only one of the three model variables had a clear increase in variance as the epidemic threshold was crossed.
TextSentencer_T61 9632-9833 Sentence denotes Boerlijst et al. [16] even showed that, depending on the types of perturbations a system experiences, the autocorrelation of some variables may either increase or decrease as a threshold is approached.
TextSentencer_T61 9632-9833 Sentence denotes Boerlijst et al. [16] even showed that, depending on the types of perturbations a system experiences, the autocorrelation of some variables may either increase or decrease as a threshold is approached.
TextSentencer_T62 9834-10010 Sentence denotes Consequently, a review [17] identified the selection of appropriate variables in multivariate systems for detection of slowing down as an important problem in need of solution.
TextSentencer_T62 9834-10010 Sentence denotes Consequently, a review [17] identified the selection of appropriate variables in multivariate systems for detection of slowing down as an important problem in need of solution.
TextSentencer_T63 10011-10191 Sentence denotes Dakos [18] has recently used an eigendecomposition of F to derive a simple rule about which state variables have a decay rate that is most affected by the dominant eigenvalue of F.
TextSentencer_T63 10011-10191 Sentence denotes Dakos [18] has recently used an eigendecomposition of F to derive a simple rule about which state variables have a decay rate that is most affected by the dominant eigenvalue of F.
TextSentencer_T64 10192-10457 Sentence denotes However, this approach only provides a partial answer to the question of variable selection, because it does not account for the covariance of the perturbations to the system, which can be as important as the eigenvectors of F on the decay rate of a state variable.
TextSentencer_T64 10192-10457 Sentence denotes However, this approach only provides a partial answer to the question of variable selection, because it does not account for the covariance of the perturbations to the system, which can be as important as the eigenvectors of F on the decay rate of a state variable.
TextSentencer_T65 10458-10610 Sentence denotes Furthermore, another consequence of models having multiple dimensions is that the informative eigenvalue may not necessarily be the dominant eigenvalue.
TextSentencer_T65 10458-10610 Sentence denotes Furthermore, another consequence of models having multiple dimensions is that the informative eigenvalue may not necessarily be the dominant eigenvalue.
TextSentencer_T66 10611-10788 Sentence denotes When its real part gets close enough to zero, the informative eigenvalue will of course become dominant but, as we shall demonstrate, that may not happen until it is very small.
TextSentencer_T66 10611-10788 Sentence denotes When its real part gets close enough to zero, the informative eigenvalue will of course become dominant but, as we shall demonstrate, that may not happen until it is very small.
TextSentencer_T67 10789-10903 Sentence denotes So although slowing down is often explained to be a consequence of the dominant rsif.royalsocietypublishing.org J.
TextSentencer_T67 10789-10903 Sentence denotes So although slowing down is often explained to be a consequence of the dominant rsif.royalsocietypublishing.org J.
TextSentencer_T68 10904-10906 Sentence denotes R.
TextSentencer_T68 10904-10906 Sentence denotes R.
TextSentencer_T69 10907-10911 Sentence denotes Soc.
TextSentencer_T69 10907-10911 Sentence denotes Soc.
TextSentencer_T70 10912-10925 Sentence denotes Interface 15:
TextSentencer_T70 10912-10925 Sentence denotes Interface 15:
TextSentencer_T71 10926-11101 Sentence denotes 20180034 eigenvalue approaching zero, methods to estimate the dominant eigenvalue of F from a multivariate time series may not reliably estimate the distance to the threshold.
TextSentencer_T71 10926-11101 Sentence denotes 20180034 eigenvalue approaching zero, methods to estimate the dominant eigenvalue of F from a multivariate time series may not reliably estimate the distance to the threshold.
TextSentencer_T72 11102-11222 Sentence denotes There does not seem to be any general approach for estimating the distance to the threshold in multidimensional systems.
TextSentencer_T72 11102-11222 Sentence denotes There does not seem to be any general approach for estimating the distance to the threshold in multidimensional systems.
TextSentencer_T73 11223-11384 Sentence denotes In this work, we derive an explicit relationship between the eigenvalues of F and the autocorrelation function of each of the variables in a multivariate system.
TextSentencer_T73 11223-11384 Sentence denotes In this work, we derive an explicit relationship between the eigenvalues of F and the autocorrelation function of each of the variables in a multivariate system.
TextSentencer_T74 11385-11613 Sentence denotes The resulting equations lead us to a simple condition for determining the types of perturbations under which estimation of a variable's autocovariance function can be translated into an estimate of the distance to the threshold.
TextSentencer_T74 11385-11613 Sentence denotes The resulting equations lead us to a simple condition for determining the types of perturbations under which estimation of a variable's autocovariance function can be translated into an estimate of the distance to the threshold.
TextSentencer_T75 11614-11753 Sentence denotes We demonstrate the application of this method to the susceptible-infected-removed (SIR) model for directly transmitted infectious diseases.
TextSentencer_T75 11614-11753 Sentence denotes We demonstrate the application of this method to the susceptible-infected-removed (SIR) model for directly transmitted infectious diseases.
TextSentencer_T76 11754-12003 Sentence denotes We find that, for parameters relevant to many vaccine-preventable diseases, the autocorrelation of the number infected almost always is indicative of the distance to the epidemic threshold, while the autocorrelation of the number susceptible is not.
TextSentencer_T76 11754-12003 Sentence denotes We find that, for parameters relevant to many vaccine-preventable diseases, the autocorrelation of the number infected almost always is indicative of the distance to the epidemic threshold, while the autocorrelation of the number susceptible is not.
TextSentencer_T77 12004-12212 Sentence denotes We examine the sensitivity of the accuracy of these estimates to environmental noise, small population size, the frequency of observation and observation of case reports instead of the actual number infected.
TextSentencer_T77 12004-12212 Sentence denotes We examine the sensitivity of the accuracy of these estimates to environmental noise, small population size, the frequency of observation and observation of case reports instead of the actual number infected.
TextSentencer_T78 12213-12334 Sentence denotes We also show a simple example of estimating the change in the distance to the threshold over the length of a time series.
TextSentencer_T78 12213-12334 Sentence denotes We also show a simple example of estimating the change in the distance to the threshold over the length of a time series.
TextSentencer_T79 12335-12497 Sentence denotes These results demonstrate the general feasibility of developing statistical systems for forecasting disease emergence and documenting the approach to elimination.
TextSentencer_T79 12335-12497 Sentence denotes These results demonstrate the general feasibility of developing statistical systems for forecasting disease emergence and documenting the approach to elimination.
TextSentencer_T80 12498-12597 Sentence denotes The model that motivated the development of the following methods is the SIR model with demography.
TextSentencer_T80 12498-12597 Sentence denotes The model that motivated the development of the following methods is the SIR model with demography.
TextSentencer_T81 12598-12845 Sentence denotes We let X(t) denote the number of susceptible individuals, Y(t) the number of infected (and infectious) individuals, Z(t) the number of removed individuals (recovered or vaccinated) and N(t) ¼ X(t) þ Y(t) þ Z(t) the total population size at time t.
TextSentencer_T81 12598-12845 Sentence denotes We let X(t) denote the number of susceptible individuals, Y(t) the number of infected (and infectious) individuals, Z(t) the number of removed individuals (recovered or vaccinated) and N(t) ¼ X(t) þ Y(t) þ Z(t) the total population size at time t.
TextSentencer_T82 12846-13026 Sentence denotes Typically, we assume that these numbers are the integer-valued random variables of a Markov process having the parameters defined in table 1 and the transitions defined in table 2.
TextSentencer_T82 12846-13026 Sentence denotes Typically, we assume that these numbers are the integer-valued random variables of a Markov process having the parameters defined in table 1 and the transitions defined in table 2.
TextSentencer_T83 13027-13124 Sentence denotes In the following, we often omit explicit notation of time dependence for the sake of conciseness.
TextSentencer_T83 13027-13124 Sentence denotes In the following, we often omit explicit notation of time dependence for the sake of conciseness.
TextSentencer_T84 13125-13344 Sentence denotes We also consider models where the death rate or the force of infection (i.e. the per capita rate at which susceptibles become infected) is subject to variation over time due to fluctuations in the environment over time.
TextSentencer_T84 13125-13344 Sentence denotes We also consider models where the death rate or the force of infection (i.e. the per capita rate at which susceptibles become infected) is subject to variation over time due to fluctuations in the environment over time.
TextSentencer_T85 13345-13466 Sentence denotes We follow Bretó & Ionides [19] in modelling such variation as multiplicative gamma white (temporally uncorrelated) noise.
TextSentencer_T85 13345-13466 Sentence denotes We follow Bretó & Ionides [19] in modelling such variation as multiplicative gamma white (temporally uncorrelated) noise.
TextSentencer_T86 13467-13659 Sentence denotes This noise could represent changes in rates due to weather conditions or social mixing [20] or even model errors in model specification, such as a failure to model spatial heterogeneity [21] .
TextSentencer_T86 13467-13659 Sentence denotes This noise could represent changes in rates due to weather conditions or social mixing [20] or even model errors in model specification, such as a failure to model spatial heterogeneity [21] .
TextSentencer_T87 13660-13840 Sentence denotes Bretó & Ionides [19] show that the model remains Markovian with such noise with the modified propensities for the death and transmission events given by the expressions in table 3.
TextSentencer_T87 13660-13840 Sentence denotes Bretó & Ionides [19] show that the model remains Markovian with such noise with the modified propensities for the death and transmission events given by the expressions in table 3.
TextSentencer_T88 13841-14141 Sentence denotes Inclusion of multiplicative gamma noise leads to the possibility that more than one individual becomes infected or dies in a single event (i.e. the associated counting processes are compound) and table 3 gives the propensity of birth and death events for all positive integers k, k 1 , k 2 and k 3 .
TextSentencer_T88 13841-14141 Sentence denotes Inclusion of multiplicative gamma noise leads to the possibility that more than one individual becomes infected or dies in a single event (i.e. the associated counting processes are compound) and table 3 gives the propensity of birth and death events for all positive integers k, k 1 , k 2 and k 3 .
TextSentencer_T89 14142-14205 Sentence denotes There are several biological assumptions implicit in our model.
TextSentencer_T89 14142-14205 Sentence denotes There are several biological assumptions implicit in our model.
TextSentencer_T90 14206-14441 Sentence denotes We use the standard assumption of frequency-dependent transmission, which has been shown to be a more appropriate model than the common alternative assumption of density-dependent transmission for a number of infectious diseases [22] .
TextSentencer_T90 14206-14441 Sentence denotes We use the standard assumption of frequency-dependent transmission, which has been shown to be a more appropriate model than the common alternative assumption of density-dependent transmission for a number of infectious diseases [22] .
TextSentencer_T91 14442-14514 Sentence denotes However, we calculate frequency using the parameter N 0 instead of N(t).
TextSentencer_T91 14442-14514 Sentence denotes However, we calculate frequency using the parameter N 0 instead of N(t).
TextSentencer_T92 14515-14647 Sentence denotes The initial population size N 0 is also the expected value of the population size because we set the birth rate in table 2 as N 0 m.
TextSentencer_T92 14515-14647 Sentence denotes The initial population size N 0 is also the expected value of the population size because we set the birth rate in table 2 as N 0 m.
TextSentencer_T93 14648-14752 Sentence denotes Another assumption is that the average death rate of individuals is constant throughout their lifetimes.
TextSentencer_T93 14648-14752 Sentence denotes Another assumption is that the average death rate of individuals is constant throughout their lifetimes.
TextSentencer_T94 14753-14878 Sentence denotes According to Anderson & May [23] , this is a common assumption among the traditional literature in mathematical epidemiology.
TextSentencer_T94 14753-14878 Sentence denotes According to Anderson & May [23] , this is a common assumption among the traditional literature in mathematical epidemiology.
TextSentencer_T95 14879-15009 Sentence denotes It is biologically accurate in that humans are subject to a small and relatively constant mortality rate until they reach old age.
TextSentencer_T95 14879-15009 Sentence denotes It is biologically accurate in that humans are subject to a small and relatively constant mortality rate until they reach old age.
TextSentencer_T96 15010-15129 Sentence denotes A more realistic model would include much higher mortality at old age, but such realism is not necessary for our study.
TextSentencer_T96 15010-15129 Sentence denotes A more realistic model would include much higher mortality at old age, but such realism is not necessary for our study.
TextSentencer_T97 15130-15361 Sentence denotes Another key feature of our model is the inclusion of the h term in the force of infection (tables 2 and 3), which relaxes the assumption that the population is closed to infection from other populations or environmental reservoirs.
TextSentencer_T97 15130-15361 Sentence denotes Another key feature of our model is the inclusion of the h term in the force of infection (tables 2 and 3), which relaxes the assumption that the population is closed to infection from other populations or environmental reservoirs.
TextSentencer_T98 15362-15529 Sentence denotes We include such a term to allow our model to represent populations in which an infectious disease is repeatedly introduced but unable to persist within the population.
TextSentencer_T98 15362-15529 Sentence denotes We include such a term to allow our model to represent populations in which an infectious disease is repeatedly introduced but unable to persist within the population.
TextSentencer_T99 15530-15625 Sentence denotes Although the model is stochastic, the expected value of the model's variables is deterministic.
TextSentencer_T99 15530-15625 Sentence denotes Although the model is stochastic, the expected value of the model's variables is deterministic.
TextSentencer_T100 15626-15828 Sentence denotes The rate of change in the expected value when the system is in a given state can be approximated by summing over all possible updates in tables 2 and 3 and weighting each update by its propensity [24] .
TextSentencer_T100 15626-15828 Sentence denotes The rate of change in the expected value when the system is in a given state can be approximated by summing over all possible updates in tables 2 and 3 and weighting each update by its propensity [24] .
TextSentencer_T101 15829-15964 Sentence denotes Calculating the rate of change in the expected value of X, Y and Z in this way leads to the following system of differential equations:
TextSentencer_T101 15829-15964 Sentence denotes Calculating the rate of change in the expected value of X, Y and Z in this way leads to the following system of differential equations:
TextSentencer_T102 15965-15968 Sentence denotes and
TextSentencer_T102 15965-15968 Sentence denotes and
TextSentencer_T103 15969-16028 Sentence denotes where the overdot indicates a time derivative and where l ¼
TextSentencer_T103 15969-16028 Sentence denotes where the overdot indicates a time derivative and where l ¼
TextSentencer_T104 16030-16161 Sentence denotes The equations for l and m in the case of non-zero environmental noise are the infinitesimal means derived in Bretó & Ionides [19] .
TextSentencer_T104 16030-16161 Sentence denotes The equations for l and m in the case of non-zero environmental noise are the infinitesimal means derived in Bretó & Ionides [19] .
TextSentencer_T105 16162-16333 Sentence denotes By setting the differential equations equal to zero and solving for kXl and kYl, we can find the approximate fixed point of the system for a given set of model parameters.
TextSentencer_T105 16162-16333 Sentence denotes By setting the differential equations equal to zero and solving for kXl and kYl, we can find the approximate fixed point of the system for a given set of model parameters.
TextSentencer_T106 16334-16453 Sentence denotes The equations for the fixed point of the differential equations allow us to explain what we mean by epidemic threshold.
TextSentencer_T106 16334-16453 Sentence denotes The equations for the fixed point of the differential equations allow us to explain what we mean by epidemic threshold.
TextSentencer_T107 16454-16536 Sentence denotes For the sake of clarity, we consider the equations only when t d and t f are zero.
TextSentencer_T107 16454-16536 Sentence denotes For the sake of clarity, we consider the equations only when t d and t f are zero.
TextSentencer_T108 16537-16634 Sentence denotes In that case the exact equation for the Y-coordinate of the fixed point, which we denote Y * , is
TextSentencer_T108 16537-16634 Sentence denotes In that case the exact equation for the Y-coordinate of the fixed point, which we denote Y * , is
TextSentencer_T109 16635-16636 Sentence denotes .
TextSentencer_T109 16635-16636 Sentence denotes .
TextSentencer_T110 16637-16801 Sentence denotes R 0 is known as the basic reproduction number and we consider the epidemic threshold for the SIR model to be the surface in parameter space where R 0 ¼ 1 and h ¼ 0.
TextSentencer_T110 16637-16801 Sentence denotes R 0 is known as the basic reproduction number and we consider the epidemic threshold for the SIR model to be the surface in parameter space where R 0 ¼ 1 and h ¼ 0.
TextSentencer_T111 16802-16888 Sentence denotes To see why, note that, when h ¼ 0, equation (2.6) has a non-zero value only when R 0 .
TextSentencer_T111 16802-16888 Sentence denotes To see why, note that, when h ¼ 0, equation (2.6) has a non-zero value only when R 0 .
TextSentencer_T112 16889-16907 Sentence denotes 1; only when R 0 .
TextSentencer_T112 16889-16907 Sentence denotes 1; only when R 0 .
TextSentencer_T113 16908-17048 Sentence denotes 1 will the introduction of an infection into a susceptible population lead to an epidemic according to the system of differential equations.
TextSentencer_T113 16908-17048 Sentence denotes 1 will the introduction of an infection into a susceptible population lead to an epidemic according to the system of differential equations.
TextSentencer_T114 17049-17185 Sentence denotes Accordingly, one can interpret R 0 as the average number of new infections caused by an infected individual in a susceptible population.
TextSentencer_T114 17049-17185 Sentence denotes Accordingly, one can interpret R 0 as the average number of new infections caused by an infected individual in a susceptible population.
TextSentencer_T115 17186-17416 Sentence denotes From the point of view of fixed points, the epidemic threshold separates the region of parameter space where a fixed point occurs with Y * ¼ 0, a disease-free equilibrium, from the region where a fixed point occurs with some Y * .
TextSentencer_T115 17186-17416 Sentence denotes From the point of view of fixed points, the epidemic threshold separates the region of parameter space where a fixed point occurs with Y * ¼ 0, a disease-free equilibrium, from the region where a fixed point occurs with some Y * .
TextSentencer_T116 17417-17443 Sentence denotes 0, an endemic equilibrium.
TextSentencer_T116 17417-17443 Sentence denotes 0, an endemic equilibrium.
TextSentencer_T117 17444-17548 Sentence denotes In short, we define the epidemic threshold for the SIR model as the location of the model's bifurcation.
TextSentencer_T117 17444-17548 Sentence denotes In short, we define the epidemic threshold for the SIR model as the location of the model's bifurcation.
TextSentencer_T118 17549-17654 Sentence denotes In the electronic supplementary material, we show how this definition remains useful in the case that h .
TextSentencer_T118 17549-17654 Sentence denotes In the electronic supplementary material, we show how this definition remains useful in the case that h .
TextSentencer_T119 17655-17690 Sentence denotes 0 and the bifurcation is imperfect.
TextSentencer_T119 17655-17690 Sentence denotes 0 and the bifurcation is imperfect.
TextSentencer_T120 17691-17858 Sentence denotes For other more realistic models, such as the structured immunity model of Reluga et al. [25] , there may be multiple stable states for a given value of the parameters.
TextSentencer_T120 17691-17858 Sentence denotes For other more realistic models, such as the structured immunity model of Reluga et al. [25] , there may be multiple stable states for a given value of the parameters.
TextSentencer_T121 17859-18008 Sentence denotes For such models, there may not be a single bifurcation occurring at R 0 ¼ 1 but instead multiple bifurcations at different points in parameter space.
TextSentencer_T121 17859-18008 Sentence denotes For such models, there may not be a single bifurcation occurring at R 0 ¼ 1 but instead multiple bifurcations at different points in parameter space.
TextSentencer_T122 18009-18270 Sentence denotes The term epidemic threshold is ambiguous for such models because, for example, the fraction of the population infected may jump up as the parameters cross one threshold but not jump down until the parameters move much further backward in the opposite direction.
TextSentencer_T122 18009-18270 Sentence denotes The term epidemic threshold is ambiguous for such models because, for example, the fraction of the population infected may jump up as the parameters cross one threshold but not jump down until the parameters move much further backward in the opposite direction.
TextSentencer_T123 18271-18427 Sentence denotes However, our methods are applicable to estimating the distance to any threshold parameter values that corresponds to the loss of an equilibrium's stability.
TextSentencer_T123 18271-18427 Sentence denotes However, our methods are applicable to estimating the distance to any threshold parameter values that corresponds to the loss of an equilibrium's stability.
TextSentencer_T124 18428-18611 Sentence denotes When the dynamics are characterized by small fluctuations around a fixed point, the degree of autocorrelation of these fluctuations may be indicative of the distance to the threshold.
TextSentencer_T124 18428-18611 Sentence denotes When the dynamics are characterized by small fluctuations around a fixed point, the degree of autocorrelation of these fluctuations may be indicative of the distance to the threshold.
TextSentencer_T125 18612-18725 Sentence denotes A first step in demonstrating this relationship is to derive a probability density function for the fluctuations.
TextSentencer_T125 18612-18725 Sentence denotes A first step in demonstrating this relationship is to derive a probability density function for the fluctuations.
TextSentencer_T126 18726-18843 Sentence denotes Let z(t) denote a vector of deviations from the fixed point that is in units of the square root of the system's size.
TextSentencer_T126 18726-18843 Sentence denotes Let z(t) denote a vector of deviations from the fixed point that is in units of the square root of the system's size.
TextSentencer_T127 18844-18909 Sentence denotes Let p(z) be the probability density function of these deviations.
TextSentencer_T127 18844-18909 Sentence denotes Let p(z) be the probability density function of these deviations.
TextSentencer_T128 18910-19027 Sentence denotes In the limit of a large system size, this function may be approximated as the solution to the Fokker -Planck equation
TextSentencer_T128 18910-19027 Sentence denotes In the limit of a large system size, this function may be approximated as the solution to the Fokker -Planck equation
TextSentencer_T129 19028-19235 Sentence denotes where the matrix F (with elements f ij ) determines the expected trajectory of z towards zero and the matrix D (with elements d ij ) describes the covariance of a Gaussian white noise process that acts on z.
TextSentencer_T129 19028-19235 Sentence denotes where the matrix F (with elements f ij ) determines the expected trajectory of z towards zero and the matrix D (with elements d ij ) describes the covariance of a Gaussian white noise process that acts on z.
TextSentencer_T130 19236-19307 Sentence denotes The matrices F and D follow directly from the transition probabilities.
TextSentencer_T130 19236-19307 Sentence denotes The matrices F and D follow directly from the transition probabilities.
TextSentencer_T131 19308-19385 Sentence denotes For the SIR model in the previous subsection, we take N 0 as the system size,
TextSentencer_T131 19308-19385 Sentence denotes For the SIR model in the previous subsection, we take N 0 as the system size,
TextSentencer_T132 19386-19429 Sentence denotes where l and d l=dY are evaluated at Y ¼ Y*.
TextSentencer_T132 19386-19429 Sentence denotes where l and d l=dY are evaluated at Y ¼ Y*.
TextSentencer_T133 19430-19491 Sentence denotes Note that we have omitted deviations from Z* in our vector z.
TextSentencer_T133 19430-19491 Sentence denotes Note that we have omitted deviations from Z* in our vector z.
TextSentencer_T134 19492-19684 Sentence denotes Including these deviations would be straightforward but the behaviour of Z is in many ways similar to that of X and the value of Z does not affect the rates at which X and Y change (table 2) .
TextSentencer_T134 19492-19684 Sentence denotes Including these deviations would be straightforward but the behaviour of Z is in many ways similar to that of X and the value of Z does not affect the rates at which X and Y change (table 2) .
TextSentencer_T135 19685-19784 Sentence denotes Therefore, we have omitted the fluctuations of Z in the following to make our results more concise.
TextSentencer_T135 19685-19784 Sentence denotes Therefore, we have omitted the fluctuations of Z in the following to make our results more concise.
TextSentencer_T136 19785-19821 Sentence denotes For the covariance matrix, we obtain
TextSentencer_T136 19785-19821 Sentence denotes For the covariance matrix, we obtain
TextSentencer_T137 19822-19841 Sentence denotes ð2:12Þ and m XY,; ¼
TextSentencer_T137 19822-19841 Sentence denotes ð2:12Þ and m XY,; ¼
TextSentencer_T138 19842-19848 Sentence denotes ð2:13Þ
TextSentencer_T138 19842-19848 Sentence denotes ð2:13Þ
TextSentencer_T139 19849-20068 Sentence denotes A solution to equation (2.7) is a Gaussian density function with a mean of zero and a covariance matrix S (with elements s ij ) that depends on F and D. van Kampen [24] provides a detailed introduction to these methods.
TextSentencer_T139 19849-20068 Sentence denotes A solution to equation (2.7) is a Gaussian density function with a mean of zero and a covariance matrix S (with elements s ij ) that depends on F and D. van Kampen [24] provides a detailed introduction to these methods.
TextSentencer_T140 20069-20193 Sentence denotes For these Gaussian solutions, the autocovariance function of the deviations may be written in terms of the eigenvalues of F.
TextSentencer_T140 20069-20193 Sentence denotes For these Gaussian solutions, the autocovariance function of the deviations may be written in terms of the eigenvalues of F.
TextSentencer_T141 20194-20258 Sentence denotes The relationship is particularly simple when the eigenvectors of
TextSentencer_T141 20194-20258 Sentence denotes The relationship is particularly simple when the eigenvectors of
TextSentencer_T142 20260-20294 Sentence denotes rsif.royalsocietypublishing.org J.
TextSentencer_T142 20260-20294 Sentence denotes rsif.royalsocietypublishing.org J.
TextSentencer_T143 20295-20297 Sentence denotes R.
TextSentencer_T143 20295-20297 Sentence denotes R.
TextSentencer_T144 20298-20302 Sentence denotes Soc.
TextSentencer_T144 20298-20302 Sentence denotes Soc.
TextSentencer_T145 20303-20316 Sentence denotes Interface 15:
TextSentencer_T145 20303-20316 Sentence denotes Interface 15:
TextSentencer_T146 20317-20369 Sentence denotes 20180034 F are used as the basis of the coordinates.
TextSentencer_T146 20317-20369 Sentence denotes 20180034 F are used as the basis of the coordinates.
TextSentencer_T147 20370-20479 Sentence denotes Thus, letz ¼ W À1 z, where W is a matrix of the eigenvectors of F, and letS denote the covariance matrix ofz.
TextSentencer_T147 20370-20479 Sentence denotes Thus, letz ¼ W À1 z, where W is a matrix of the eigenvectors of F, and letS denote the covariance matrix ofz.
TextSentencer_T148 20480-20547 Sentence denotes Then, using the decomposition of Kwon et al. [26] , it follows that
TextSentencer_T148 20480-20547 Sentence denotes Then, using the decomposition of Kwon et al. [26] , it follows that
TextSentencer_T149 20548-20642 Sentence denotes where l i denotes an eigenvalue of F and we assume that all of these eigenvalues are distinct.
TextSentencer_T149 20548-20642 Sentence denotes where l i denotes an eigenvalue of F and we assume that all of these eigenvalues are distinct.
TextSentencer_T150 20643-20801 Sentence denotes The autocovariance matrix is defined as S t ¼ kz(t À t)z(t) T l, where the angular brackets denote the expected value over time or realizations of the system.
TextSentencer_T150 20643-20801 Sentence denotes The autocovariance matrix is defined as S t ¼ kz(t À t)z(t) T l, where the angular brackets denote the expected value over time or realizations of the system.
TextSentencer_T151 20802-20872 Sentence denotes It follows from the stationarity of the solution that S t ¼ exp (Ft)S.
TextSentencer_T151 20802-20872 Sentence denotes It follows from the stationarity of the solution that S t ¼ exp (Ft)S.
TextSentencer_T152 20873-20932 Sentence denotes In the eigenvector basis, we haves t,ij ¼ exp (l i t)s ij .
TextSentencer_T152 20873-20932 Sentence denotes In the eigenvector basis, we haves t,ij ¼ exp (l i t)s ij .
TextSentencer_T153 20933-21096 Sentence denotes Thus, the behaviour of the autocovariance along an eigendirection as a function of the lag t is a simple and identifiable function of the corresponding eigenvalue.
TextSentencer_T153 20933-21096 Sentence denotes Thus, the behaviour of the autocovariance along an eigendirection as a function of the lag t is a simple and identifiable function of the corresponding eigenvalue.
TextSentencer_T154 21097-21175 Sentence denotes If l i is real, thens t,ii decays exponentially towards zero at the rate l i .
TextSentencer_T154 21097-21175 Sentence denotes If l i is real, thens t,ii decays exponentially towards zero at the rate l i .
TextSentencer_T155 21176-21424 Sentence denotes If l i has an imaginary component, then the real and imaginary parts ofs t,ii oscillate around zero with a frequency given by the imaginary component of l i and an amplitude that decays exponentially at the rate given by the real component of l i .
TextSentencer_T155 21176-21424 Sentence denotes If l i has an imaginary component, then the real and imaginary parts ofs t,ii oscillate around zero with a frequency given by the imaginary component of l i and an amplitude that decays exponentially at the rate given by the real component of l i .
TextSentencer_T156 21425-21560 Sentence denotes Since S t ¼ WS t W, s t,ii will be a linear combination of the elements ofS t (see the electronic supplementary material, equation S4).
TextSentencer_T156 21425-21560 Sentence denotes Since S t ¼ WS t W, s t,ii will be a linear combination of the elements ofS t (see the electronic supplementary material, equation S4).
TextSentencer_T157 21561-21704 Sentence denotes Therefore, the elements of the autocovariance matrix S t are linear combinations of functions from which the eigenvalues of F are identifiable.
TextSentencer_T157 21561-21704 Sentence denotes Therefore, the elements of the autocovariance matrix S t are linear combinations of functions from which the eigenvalues of F are identifiable.
TextSentencer_T158 21705-21944 Sentence denotes The relationship between the eigenvalues and the autocovariance established in the previous subsection clarifies the question of when the autocovariance of a variable contains sufficient information to estimate the distance to a threshold.
TextSentencer_T158 21705-21944 Sentence denotes The relationship between the eigenvalues and the autocovariance established in the previous subsection clarifies the question of when the autocovariance of a variable contains sufficient information to estimate the distance to a threshold.
TextSentencer_T159 21945-22002 Sentence denotes Any threshold corresponds to an eigenvalue crossing zero.
TextSentencer_T159 21945-22002 Sentence denotes Any threshold corresponds to an eigenvalue crossing zero.
TextSentencer_T160 22003-22156 Sentence denotes Recall that we call such an eigenvalue an informative eigenvalue and that the magnitude of its real part can be considered the distance to the threshold.
TextSentencer_T160 22003-22156 Sentence denotes Recall that we call such an eigenvalue an informative eigenvalue and that the magnitude of its real part can be considered the distance to the threshold.
TextSentencer_T161 22157-22344 Sentence denotes If it is known that the imaginary part of the eigenvalue will also be zero at the threshold, then the magnitude of the imaginary part can be considered a second component of the distance.
TextSentencer_T161 22157-22344 Sentence denotes If it is known that the imaginary part of the eigenvalue will also be zero at the threshold, then the magnitude of the imaginary part can be considered a second component of the distance.
TextSentencer_T162 22345-22447 Sentence denotes Note that in the case that an informative eigenvalue is complex it will be a part of a conjugate pair.
TextSentencer_T162 22345-22447 Sentence denotes Note that in the case that an informative eigenvalue is complex it will be a part of a conjugate pair.
TextSentencer_T163 22448-22684 Sentence denotes Estimation of the decay rate and frequency of oscillation of a variable's autocovariance function can provide an estimate of the distance to the threshold when they are close to the real and imaginary parts of an informative eigenvalue.
TextSentencer_T163 22448-22684 Sentence denotes Estimation of the decay rate and frequency of oscillation of a variable's autocovariance function can provide an estimate of the distance to the threshold when they are close to the real and imaginary parts of an informative eigenvalue.
TextSentencer_T164 22685-22932 Sentence denotes This condition on the autocovariance function for an estimate to be accurate, together with equation S t ¼ WS t W T and equation (2.12), can be translated into conditions on the eigenvectors W of F and the covariance matrix D of the perturbations.
TextSentencer_T164 22685-22932 Sentence denotes This condition on the autocovariance function for an estimate to be accurate, together with equation S t ¼ WS t W T and equation (2.12), can be translated into conditions on the eigenvectors W of F and the covariance matrix D of the perturbations.
TextSentencer_T165 22933-23094 Sentence denotes Thus, we now have a general link between the parameters of models and the potential for a model variable to provide an estimate of the distance to the threshold.
TextSentencer_T165 22933-23094 Sentence denotes Thus, we now have a general link between the parameters of models and the potential for a model variable to provide an estimate of the distance to the threshold.
TextSentencer_T166 23095-23241 Sentence denotes In the electronic supplementary material, we provide an explicit calculation of the values of D that permit a distance estimate for each variable.
TextSentencer_T166 23095-23241 Sentence denotes In the electronic supplementary material, we provide an explicit calculation of the values of D that permit a distance estimate for each variable.
TextSentencer_T167 23242-23314 Sentence denotes We estimate the distance to the threshold from a time series as follows.
TextSentencer_T167 23242-23314 Sentence denotes We estimate the distance to the threshold from a time series as follows.
TextSentencer_T168 23315-23623 Sentence denotes The main idea is to suppose that the autocorrelation will exponentially decay with increasing lags at a rate equal to the real part of the informative eigenvalue and that any oscillations in the autocorrelation function have a frequency equal in magnitude to the imaginary part of the informative eigenvalue.
TextSentencer_T168 23315-23623 Sentence denotes The main idea is to suppose that the autocorrelation will exponentially decay with increasing lags at a rate equal to the real part of the informative eigenvalue and that any oscillations in the autocorrelation function have a frequency equal in magnitude to the imaginary part of the informative eigenvalue.
TextSentencer_T169 23624-23761 Sentence denotes The first step is then to estimate the autocorrelation of the time series for a series of lags, which we did using the acf function in R.
TextSentencer_T169 23624-23761 Sentence denotes The first step is then to estimate the autocorrelation of the time series for a series of lags, which we did using the acf function in R.
TextSentencer_T170 23762-23927 Sentence denotes Because sometimes the autocorrelation can have cycles with a period of several years, we used lags from 0 to 30 observations less than the length of the time series.
TextSentencer_T170 23762-23927 Sentence denotes Because sometimes the autocorrelation can have cycles with a period of several years, we used lags from 0 to 30 observations less than the length of the time series.
TextSentencer_T171 23928-24071 Sentence denotes Next, we use a nonlinear least-squares optimizer to fit two models for the estimated autocorrelationŝ ii,t =ŝ ii , atan2(1, a) ) þ e t , ð2:16Þ
TextSentencer_T171 23928-24071 Sentence denotes Next, we use a nonlinear least-squares optimizer to fit two models for the estimated autocorrelationŝ ii,t =ŝ ii , atan2(1, a) ) þ e t , ð2:16Þ
TextSentencer_T172 24072-24273 Sentence denotes where e t is an error term, g is the decay rate parameter, v is the frequency parameter and a is a phase angle parameter, and atan 2 is the inverse tangent function with arguments in the order of y, x.
TextSentencer_T172 24072-24273 Sentence denotes where e t is an error term, g is the decay rate parameter, v is the frequency parameter and a is a phase angle parameter, and atan 2 is the inverse tangent function with arguments in the order of y, x.
TextSentencer_T173 24274-24320 Sentence denotes We use the nlsLM function to fit these models.
TextSentencer_T173 24274-24320 Sentence denotes We use the nlsLM function to fit these models.
TextSentencer_T174 24321-24472 Sentence denotes This function is available in the minpack.lm package [27] , and it provides an R interface to the Levenberg-Marquardt optimizer in the MINPACK library.
TextSentencer_T174 24321-24472 Sentence denotes This function is available in the minpack.lm package [27] , and it provides an R interface to the Levenberg-Marquardt optimizer in the MINPACK library.
TextSentencer_T175 24473-24651 Sentence denotes We used nlsLM instead of the nls function that comes with R because it was less sensitive to the choice of initial values of the parameters for the optimization of the model fit.
TextSentencer_T175 24473-24651 Sentence denotes We used nlsLM instead of the nls function that comes with R because it was less sensitive to the choice of initial values of the parameters for the optimization of the model fit.
TextSentencer_T176 24652-24893 Sentence denotes For initial values, we set a to zero, g to the least-squares slope of the log of the absolute value of the estimated autocorrelation versus the lag, and v to the frequency that maximized the spectral density of the estimated autocorrelation.
TextSentencer_T176 24652-24893 Sentence denotes For initial values, we set a to zero, g to the least-squares slope of the log of the absolute value of the estimated autocorrelation versus the lag, and v to the frequency that maximized the spectral density of the estimated autocorrelation.
TextSentencer_T177 24894-24983 Sentence denotes Only autocorrelations with relatively small lags were used for the initial estimate of g.
TextSentencer_T177 24894-24983 Sentence denotes Only autocorrelations with relatively small lags were used for the initial estimate of g.
TextSentencer_T178 24984-25227 Sentence denotes Specifically, all lags including and following the first lag that was less in magnitude than F(0:975)= ffiffiffi n p , where F is the cumulative distribution function of a standard normal random variable and n is the length of the time series.
TextSentencer_T178 24984-25227 Sentence denotes Specifically, all lags including and following the first lag that was less in magnitude than F(0:975)= ffiffiffi n p , where F is the cumulative distribution function of a standard normal random variable and n is the length of the time series.
TextSentencer_T179 25228-25370 Sentence denotes We fitted the data with and without an oscillation component in the model and used the following information criterion to evaluate the models:
TextSentencer_T179 25228-25370 Sentence denotes We fitted the data with and without an oscillation component in the model and used the following information criterion to evaluate the models:
TextSentencer_T180 25371-25472 Sentence denotes where RSS stands for the residual sum of squares, p ¼ 1 for model (2.15) and p ¼ 3 for model (2.16) .
TextSentencer_T180 25371-25472 Sentence denotes where RSS stands for the residual sum of squares, p ¼ 1 for model (2.15) and p ¼ 3 for model (2.16) .
TextSentencer_T181 25473-25531 Sentence denotes We used the estimates from the model with the lower score.
TextSentencer_T181 25473-25531 Sentence denotes We used the estimates from the model with the lower score.
TextSentencer_T182 25532-25691 Sentence denotes If neither model's information criterion exceeded P t (ŝ ii,t =ŝ ii ) 2 , we concluded that the data contained insufficient information to provide an estimate.
TextSentencer_T182 25532-25691 Sentence denotes If neither model's information criterion exceeded P t (ŝ ii,t =ŝ ii ) 2 , we concluded that the data contained insufficient information to provide an estimate.
TextSentencer_T183 25692-25833 Sentence denotes When estimates were available, we calculated the distance to the threshold as ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi v 2 þ g 2 p .
TextSentencer_T183 25692-25833 Sentence denotes When estimates were available, we calculated the distance to the threshold as ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi v 2 þ g 2 p .
TextSentencer_T184 25834-25938 Sentence denotes If the input time series consisted of aggregated counts, we modified the model having no oscillations tô
TextSentencer_T184 25834-25938 Sentence denotes If the input time series consisted of aggregated counts, we modified the model having no oscillations tô
TextSentencer_T185 25939-26008 Sentence denotes and we excluded the lag-0 autocorrelation from our data to be fitted.
TextSentencer_T185 25939-26008 Sentence denotes and we excluded the lag-0 autocorrelation from our data to be fitted.
TextSentencer_T186 26009-26211 Sentence denotes This model, which has a new parameter K that determines the autocorrelation at a lag of 1, matches the form of the autocorrelation function for counts of deaths in a birth -deathimmigration model [28] .
TextSentencer_T186 26009-26211 Sentence denotes This model, which has a new parameter K that determines the autocorrelation at a lag of 1, matches the form of the autocorrelation function for counts of deaths in a birth -deathimmigration model [28] .
TextSentencer_T187 26212-26311 Sentence denotes The birth -death -immigration model can provide a good approximation of the SIR model when R 0 , 1.
TextSentencer_T187 26212-26311 Sentence denotes The birth -death -immigration model can provide a good approximation of the SIR model when R 0 , 1.
TextSentencer_T188 26312-26379 Sentence denotes The information criterion for this model was calculated with p ¼ 2.
TextSentencer_T188 26312-26379 Sentence denotes The information criterion for this model was calculated with p ¼ 2.
TextSentencer_T189 26380-26458 Sentence denotes In the following, we apply our theory and estimation methods to the SIR model.
TextSentencer_T189 26380-26458 Sentence denotes In the following, we apply our theory and estimation methods to the SIR model.
TextSentencer_T190 26459-26640 Sentence denotes To generate data for estimation, we simulated time series of the number of individuals in each state according to our Markov process model using the Euler scheme of He et al. [20] .
TextSentencer_T190 26459-26640 Sentence denotes To generate data for estimation, we simulated time series of the number of individuals in each state according to our Markov process model using the Euler scheme of He et al. [20] .
TextSentencer_T191 26641-26701 Sentence denotes The pomp [29, 30] R package was used to implement the model.
TextSentencer_T191 26641-26701 Sentence denotes The pomp [29, 30] R package was used to implement the model.
TextSentencer_T192 26702-26857 Sentence denotes Our typical procedure was to simulate data with most of the parameters fixed at the default values in table 1 and for several choices of transmission rate.
TextSentencer_T192 26702-26857 Sentence denotes Our typical procedure was to simulate data with most of the parameters fixed at the default values in table 1 and for several choices of transmission rate.
TextSentencer_T193 26858-26948 Sentence denotes The default parameters were chosen to be typical of an acute infectious disease of humans.
TextSentencer_T193 26858-26948 Sentence denotes The default parameters were chosen to be typical of an acute infectious disease of humans.
TextSentencer_T194 26949-27142 Sentence denotes Our default infectious period of 22 days is consistent with the combined latent and infectious period in a past model of pertussis [31] , as is our default host mortality rate of 0.02 per year.
TextSentencer_T194 26949-27142 Sentence denotes Our default infectious period of 22 days is consistent with the combined latent and infectious period in a past model of pertussis [31] , as is our default host mortality rate of 0.02 per year.
TextSentencer_T195 27143-27418 Sentence denotes Our default initial population size of 10 million is chosen to be similar to that of a very large city and to be large enough for the linear noise approximation to be reasonable. to be weekly because infectious disease notification data are often available at that frequency.
TextSentencer_T195 27143-27418 Sentence denotes Our default initial population size of 10 million is chosen to be similar to that of a very large city and to be large enough for the linear noise approximation to be reasonable. to be weekly because infectious disease notification data are often available at that frequency.
TextSentencer_T196 27419-27563 Sentence denotes Simple sensitivity analyses of the distance estimates were carried out by allowing one or two of the parameters to vary from the default values.
TextSentencer_T196 27419-27563 Sentence denotes Simple sensitivity analyses of the distance estimates were carried out by allowing one or two of the parameters to vary from the default values.
TextSentencer_T197 27564-27658 Sentence denotes The full set of parameters used for each set of distance estimates is reported in the results.
TextSentencer_T197 27564-27658 Sentence denotes The full set of parameters used for each set of distance estimates is reported in the results.
TextSentencer_T198 27659-27891 Sentence denotes The initial values of the states were set to the equilibrium values and the model was run for 10 simulation years before sampling to allow the initially sampled states to vary according to the stationary distribution of the process.
TextSentencer_T198 27659-27891 Sentence denotes The initial values of the states were set to the equilibrium values and the model was run for 10 simulation years before sampling to allow the initially sampled states to vary according to the stationary distribution of the process.
TextSentencer_T199 27892-27977 Sentence denotes The sampling scheme was 1040 observations at a frequency of one observation per week.
TextSentencer_T199 27892-27977 Sentence denotes The sampling scheme was 1040 observations at a frequency of one observation per week.
TextSentencer_T200 27978-28094 Sentence denotes This corresponds to about 20 years of weekly observations, which is a realistic size for an epidemiological dataset.
TextSentencer_T200 27978-28094 Sentence denotes This corresponds to about 20 years of weekly observations, which is a realistic size for an epidemiological dataset.
TextSentencer_T201 28095-28267 Sentence denotes Sampled time series of both the number infected and the number susceptible were used to generate an estimate of the distance to the threshold by the method described above.
TextSentencer_T201 28095-28267 Sentence denotes Sampled time series of both the number infected and the number susceptible were used to generate an estimate of the distance to the threshold by the method described above.
TextSentencer_T202 28268-28442 Sentence denotes The true value for each estimate was calculated by plugging the simulation parameters into equation (2.8) , solving for the fixed points and calculating the eigenvalues of F.
TextSentencer_T202 28268-28442 Sentence denotes The true value for each estimate was calculated by plugging the simulation parameters into equation (2.8) , solving for the fixed points and calculating the eigenvalues of F.
TextSentencer_T203 28443-28644 Sentence denotes If there were two real eigenvalues, the informative eigenvalue was identified as the eigenvalue that would cross zero if the parameters were moved through the bifurcation point where R 0 ¼ 1 and h ¼ 0.
TextSentencer_T203 28443-28644 Sentence denotes If there were two real eigenvalues, the informative eigenvalue was identified as the eigenvalue that would cross zero if the parameters were moved through the bifurcation point where R 0 ¼ 1 and h ¼ 0.
TextSentencer_T204 28645-28784 Sentence denotes We also conducted simulations with a linearly increasing transmission rate to evaluate the performance of estimates of changes in distance.
TextSentencer_T204 28645-28784 Sentence denotes We also conducted simulations with a linearly increasing transmission rate to evaluate the performance of estimates of changes in distance.
TextSentencer_T205 28785-28959 Sentence denotes To ease comparison with estimates from our other simulations, we used a similar amount of data for the individual distance estimates used to calculate the change in distance.
TextSentencer_T205 28785-28959 Sentence denotes To ease comparison with estimates from our other simulations, we used a similar amount of data for the individual distance estimates used to calculate the change in distance.
TextSentencer_T206 28960-29125 Sentence denotes The simulations were sampled for twice as long, the time series were split into two windows of 1040 weekly observations and an estimate was obtained for each window.
TextSentencer_T206 28960-29125 Sentence denotes The simulations were sampled for twice as long, the time series were split into two windows of 1040 weekly observations and an estimate was obtained for each window.
TextSentencer_T207 29126-29289 Sentence denotes We first present some general considerations regarding when the distance to the epidemic threshold can be estimated from the fluctuation dynamics of the SIR model.
TextSentencer_T207 29126-29289 Sentence denotes We first present some general considerations regarding when the distance to the epidemic threshold can be estimated from the fluctuation dynamics of the SIR model.
TextSentencer_T208 29290-29421 Sentence denotes Figure 1 shows representative examples of the kinds of time series that we suppose could become available for statistical analysis.
TextSentencer_T208 29290-29421 Sentence denotes Figure 1 shows representative examples of the kinds of time series that we suppose could become available for statistical analysis.
TextSentencer_T209 29422-29620 Sentence denotes For two of the parameter values, cycles are visible in both the number of susceptibles, X, and the number of infecteds, Y; this is a consequence of the eigenvalues of the Jacobian, F, being complex.
TextSentencer_T209 29422-29620 Sentence denotes For two of the parameter values, cycles are visible in both the number of susceptibles, X, and the number of infecteds, Y; this is a consequence of the eigenvalues of the Jacobian, F, being complex.
TextSentencer_T210 29621-29676 Sentence denotes This behaviour is typical of parameters for which R 0 .
TextSentencer_T210 29621-29676 Sentence denotes This behaviour is typical of parameters for which R 0 .
TextSentencer_T211 29677-29679 Sentence denotes 1.
TextSentencer_T211 29677-29679 Sentence denotes 1.
TextSentencer_T212 29680-29842 Sentence denotes We have explained in the Methods section that, in this case, any white noise perturbation may allow for the distance estimate to be obtained from either variable.
TextSentencer_T212 29680-29842 Sentence denotes We have explained in the Methods section that, in this case, any white noise perturbation may allow for the distance estimate to be obtained from either variable.
TextSentencer_T213 29843-29898 Sentence denotes When R 0 , 1, typically there are two real eigenvalues.
TextSentencer_T213 29843-29898 Sentence denotes When R 0 , 1, typically there are two real eigenvalues.
TextSentencer_T214 29899-30040 Sentence denotes Without any knowledge about the model, we would expect that the ability to obtain an estimate depends on the covariance of the perturbations.
TextSentencer_T214 29899-30040 Sentence denotes Without any knowledge about the model, we would expect that the ability to obtain an estimate depends on the covariance of the perturbations.
TextSentencer_T215 30041-30207 Sentence denotes With the knowledge that the observations come from an SIR model, we could expect that the dynamics of the X and Y variables will be largely independent of each other.
TextSentencer_T215 30041-30207 Sentence denotes With the knowledge that the observations come from an SIR model, we could expect that the dynamics of the X and Y variables will be largely independent of each other.
TextSentencer_T216 30208-30309 Sentence denotes The number infected will generally be too small to affect the fluctuations in the number susceptible.
TextSentencer_T216 30208-30309 Sentence denotes The number infected will generally be too small to affect the fluctuations in the number susceptible.
TextSentencer_T217 30310-30567 Sentence denotes Thus, the rate at which susceptible perturbations decay will depend mostly on the per capita death rate m, whereas the rate at which infected perturbations decay will depend on the sum of the per capita rates at which Y grows and shrinks, bX * /N 0 2 g 2 m.
TextSentencer_T217 30310-30567 Sentence denotes Thus, the rate at which susceptible perturbations decay will depend mostly on the per capita death rate m, whereas the rate at which infected perturbations decay will depend on the sum of the per capita rates at which Y grows and shrinks, bX * /N 0 2 g 2 m.
TextSentencer_T218 30568-30711 Sentence denotes Thus, the variable Y is generally the one that should be observed to estimate the distance to the threshold when the disease is not widespread.
TextSentencer_T218 30568-30711 Sentence denotes Thus, the variable Y is generally the one that should be observed to estimate the distance to the threshold when the disease is not widespread.
TextSentencer_T219 30712-30837 Sentence denotes In the electronic supplementary material, we derive explicit equations for the autocorrelations that support this conclusion.
TextSentencer_T219 30712-30837 Sentence denotes In the electronic supplementary material, we derive explicit equations for the autocorrelations that support this conclusion.
TextSentencer_T220 30838-31024 Sentence denotes Having provided some general insights into why distance estimates may be obtained from Y and not X when R 0 , 1, we next consider a more specific answer for a specific set of parameters.
TextSentencer_T220 30838-31024 Sentence denotes Having provided some general insights into why distance estimates may be obtained from Y and not X when R 0 , 1, we next consider a more specific answer for a specific set of parameters.
TextSentencer_T221 31025-31212 Sentence denotes We use the approach described in the Methods section to find the set of noise parameters that allow the distance to the threshold to be estimated with a given accuracy from each variable.
TextSentencer_T221 31025-31212 Sentence denotes We use the approach described in the Methods section to find the set of noise parameters that allow the distance to the threshold to be estimated with a given accuracy from each variable.
TextSentencer_T222 31213-31264 Sentence denotes These sets appear as regions in space in figure 2 .
TextSentencer_T222 31213-31264 Sentence denotes These sets appear as regions in space in figure 2 .
TextSentencer_T223 31265-31418 Sentence denotes Consistently with the conclusions of the previous paragraph, the regions are much larger for the number infected, Y , than for the number susceptible, X.
TextSentencer_T223 31265-31418 Sentence denotes Consistently with the conclusions of the previous paragraph, the regions are much larger for the number infected, Y , than for the number susceptible, X.
TextSentencer_T224 31419-31565 Sentence denotes The regions for Y include the perturbations that result from the intrinsic noise present in simulations of the model with finite population sizes.
TextSentencer_T224 31419-31565 Sentence denotes The regions for Y include the perturbations that result from the intrinsic noise present in simulations of the model with finite population sizes.
TextSentencer_T225 31566-31680 Sentence denotes By As these examples suggest, these rates tend to increase as the parameters of the system move away from R 0 ¼ 1.
TextSentencer_T225 31566-31680 Sentence denotes By As these examples suggest, these rates tend to increase as the parameters of the system move away from R 0 ¼ 1.
TextSentencer_T226 31681-31754 Sentence denotes Parameters for the simulations are in table 1, with b set to R 0 (g þ m).
TextSentencer_T226 31681-31754 Sentence denotes Parameters for the simulations are in table 1, with b set to R 0 (g þ m).
TextSentencer_T227 31755-31923 Sentence denotes contrast, a large part of the lower-error region identified for X is in fact not feasible because the covariance matrix constraint of d 2 12 d 11 d 22 is not satisfied.
TextSentencer_T227 31755-31923 Sentence denotes contrast, a large part of the lower-error region identified for X is in fact not feasible because the covariance matrix constraint of d 2 12 d 11 d 22 is not satisfied.
TextSentencer_T228 31924-32217 Sentence denotes Having shown that, in principle, it is often possible to obtain distance estimates from the SIR model from at least one of the variables, we next turn to the question of whether estimates may be obtained in practice from a simulated time series of realistic length using our estimation method.
TextSentencer_T228 31924-32217 Sentence denotes Having shown that, in principle, it is often possible to obtain distance estimates from the SIR model from at least one of the variables, we next turn to the question of whether estimates may be obtained in practice from a simulated time series of realistic length using our estimation method.
TextSentencer_T229 32218-32380 Sentence denotes Figure 3 shows that for time series of about 1000 observations our estimation method was generally successful when the perturbations are predicted to be suitable.
TextSentencer_T229 32218-32380 Sentence denotes Figure 3 shows that for time series of about 1000 observations our estimation method was generally successful when the perturbations are predicted to be suitable.
TextSentencer_T230 32381-32527 Sentence denotes The perturbations were simply intrinsic noise, so the low accuracy of estimates based on X when R 0 ¼ (0.1, 0.5, 0.9) is consistent with figure 2.
TextSentencer_T230 32381-32527 Sentence denotes The perturbations were simply intrinsic noise, so the low accuracy of estimates based on X when R 0 ¼ (0.1, 0.5, 0.9) is consistent with figure 2.
TextSentencer_T231 32528-32551 Sentence denotes As expected, when R 0 .
TextSentencer_T231 32528-32551 Sentence denotes As expected, when R 0 .
TextSentencer_T232 32552-32606 Sentence denotes 1 estimates from both X and Y were similarly accurate.
TextSentencer_T232 32552-32606 Sentence denotes 1 estimates from both X and Y were similarly accurate.
TextSentencer_T233 32607-32671 Sentence denotes Therefore, Y permits a distance estimate for all R 0 considered.
TextSentencer_T233 32607-32671 Sentence denotes Therefore, Y permits a distance estimate for all R 0 considered.
TextSentencer_T234 32672-32744 Sentence denotes Fortunately, Y is the variable which is more often observed in practice.
TextSentencer_T234 32672-32744 Sentence denotes Fortunately, Y is the variable which is more often observed in practice.
TextSentencer_T235 32745-32911 Sentence denotes We have included estimates based on X in our results primarily to evaluate our analytical predictions about when a variable can provide an accurate distance estimate.
TextSentencer_T235 32745-32911 Sentence denotes We have included estimates based on X in our results primarily to evaluate our analytical predictions about when a variable can provide an accurate distance estimate.
TextSentencer_T236 32912-32922 Sentence denotes To better.
TextSentencer_T236 32912-32922 Sentence denotes To better.
TextSentencer_T237 32923-33095 Sentence denotes A rough guideline for accurate estimation seems to be that the duration of observation be at least as long as the period of any oscillation in the autocorrelation function.
TextSentencer_T237 32923-33095 Sentence denotes A rough guideline for accurate estimation seems to be that the duration of observation be at least as long as the period of any oscillation in the autocorrelation function.
TextSentencer_T238 33096-33219 Sentence denotes Another guideline is that the time between observations be much less than the period of oscillation of the autocorrelation.
TextSentencer_T238 33096-33219 Sentence denotes Another guideline is that the time between observations be much less than the period of oscillation of the autocorrelation.
TextSentencer_T239 33220-33361 Sentence denotes In the R 0 ¼ 16 panel of figure 4 , no distance estimates were obtained when the observation frequency went from 0.25 per week to 1 per year.
TextSentencer_T239 33220-33361 Sentence denotes In the R 0 ¼ 16 panel of figure 4 , no distance estimates were obtained when the observation frequency went from 0.25 per week to 1 per year.
TextSentencer_T240 33362-33555 Sentence denotes For these parameters, the autocorrelation had a period of about 3 years, so three observations per cycle seems much less likely to provide sufficient information than 40 observations per cycle.
TextSentencer_T240 33362-33555 Sentence denotes For these parameters, the autocorrelation had a period of about 3 years, so three observations per cycle seems much less likely to provide sufficient information than 40 observations per cycle.
TextSentencer_T241 33556-33658 Sentence denotes A similar guideline on the sampling frequency holds when the autocorrelation function is not periodic.
TextSentencer_T241 33556-33658 Sentence denotes A similar guideline on the sampling frequency holds when the autocorrelation function is not periodic.
TextSentencer_T242 33659-33772 Sentence denotes In the R 0 ¼ 0.9 panel, no estimates based on Y are available as the sampling frequency dips below 0.25 per week.
TextSentencer_T242 33659-33772 Sentence denotes In the R 0 ¼ 0.9 panel, no estimates based on Y are available as the sampling frequency dips below 0.25 per week.
TextSentencer_T243 33773-33851 Sentence denotes Here, the autocorrelation shrinks by a factor of e % 2.7 about every 23 weeks.
TextSentencer_T243 33773-33851 Sentence denotes Here, the autocorrelation shrinks by a factor of e % 2.7 about every 23 weeks.
TextSentencer_T244 33852-33968 Sentence denotes This time can be used to characterize the time scale of a decaying function and is sometimes called the return time.
TextSentencer_T244 33852-33968 Sentence denotes This time can be used to characterize the time scale of a decaying function and is sometimes called the return time.
TextSentencer_T245 33969-34063 Sentence denotes A third guideline, then, is that the time between samples should be less than the return time.
TextSentencer_T245 33969-34063 Sentence denotes A third guideline, then, is that the time between samples should be less than the return time.
TextSentencer_T246 34064-34311 Sentence denotes In summary, for accurate estimates the duration of observation should be much greater than the time scale of the autocorrelation function but the time between observations should be much smaller than the time scale of the autocorrelation function.
TextSentencer_T246 34064-34311 Sentence denotes In summary, for accurate estimates the duration of observation should be much greater than the time scale of the autocorrelation function but the time between observations should be much smaller than the time scale of the autocorrelation function.
TextSentencer_T247 34312-34497 Sentence denotes In addition to sampling frequency, another key characteristic of observations is whether they represent direct observation of the state of the system or cumulative flows between states.
TextSentencer_T247 34312-34497 Sentence denotes In addition to sampling frequency, another key characteristic of observations is whether they represent direct observation of the state of the system or cumulative flows between states.
TextSentencer_T248 34498-34620 Sentence denotes In particular, it is relatively rare for the number of infections in a population at a given point in time to be observed.
TextSentencer_T248 34498-34620 Sentence denotes In particular, it is relatively rare for the number of infections in a population at a given point in time to be observed.
TextSentencer_T249 34621-34862 Sentence denotes A more typical type of observation is the count of the number of infected individuals that moved into the removed class, for example, because these individuals were diagnosed with infection and then greatly reduced contact with others [33] .
TextSentencer_T249 34621-34862 Sentence denotes A more typical type of observation is the count of the number of infected individuals that moved into the removed class, for example, because these individuals were diagnosed with infection and then greatly reduced contact with others [33] .
TextSentencer_T250 34863-34903 Sentence denotes We refer to such counts as case reports.
TextSentencer_T250 34863-34903 Sentence denotes We refer to such counts as case reports.
TextSentencer_T251 34904-35021 Sentence denotes Figure 4 shows that the estimates based on case reports were similar to those based on direct observation of X or Y .
TextSentencer_T251 34904-35021 Sentence denotes Figure 4 shows that the estimates based on case reports were similar to those based on direct observation of X or Y .
TextSentencer_T252 35022-35178 Sentence denotes The main difference is that when the data consist of case reports we concluded more often that there was insufficient information available for an estimate.
TextSentencer_T252 35022-35178 Sentence denotes The main difference is that when the data consist of case reports we concluded more often that there was insufficient information available for an estimate.
TextSentencer_T253 35179-35372 Sentence denotes For example, many estimates based on Y are plotted when R 0 ¼ 0.9 and the observation frequency is 0.25 per week, but the same set of simulations resulted in no estimates based on case reports.
TextSentencer_T253 35179-35372 Sentence denotes For example, many estimates based on Y are plotted when R 0 ¼ 0.9 and the observation frequency is 0.25 per week, but the same set of simulations resulted in no estimates based on case reports.
TextSentencer_T254 35373-35615 Sentence denotes When the reporting of each recovery is not sure but instead occurs with a certain probability, as in the study of Gamado et al. [34] , the number of simulations that resulted in estimates went down with the reporting probability ( figure 5 ).
TextSentencer_T254 35373-35615 Sentence denotes When the reporting of each recovery is not sure but instead occurs with a certain probability, as in the study of Gamado et al. [34] , the number of simulations that resulted in estimates went down with the reporting probability ( figure 5 ).
TextSentencer_T255 35616-35796 Sentence denotes In conclusion, estimates based on case reports can be as accurate as those based on state variables, but also can be less likely to be available for a given number of observations.
TextSentencer_T255 35616-35796 Sentence denotes In conclusion, estimates based on case reports can be as accurate as those based on state variables, but also can be less likely to be available for a given number of observations.
TextSentencer_T256 35797-35998 Sentence denotes The electronic supplementary material contains the results of our sensitivity analysis of environmental noise, population size and our estimation of the rate of change of the distance to the threshold.
TextSentencer_T256 35797-35998 Sentence denotes The electronic supplementary material contains the results of our sensitivity analysis of environmental noise, population size and our estimation of the rate of change of the distance to the threshold.
TextSentencer_T257 35999-36177 Sentence denotes This work has presented a general solution to the problem of the selection of appropriate variables in multivariate systems reporting probability distance to threshold Figure 5 .
TextSentencer_T257 35999-36177 Sentence denotes This work has presented a general solution to the problem of the selection of appropriate variables in multivariate systems reporting probability distance to threshold Figure 5 .
TextSentencer_T258 36178-36354 Sentence denotes The accuracy of distance estimates did not clearly decline with the reporting probability but the number of times the data contained sufficient information for an estimate did.
TextSentencer_T258 36178-36354 Sentence denotes The accuracy of distance estimates did not clearly decline with the reporting probability but the number of times the data contained sufficient information for an estimate did.
TextSentencer_T259 36355-36395 Sentence denotes The true distance is marked with an 'x'.
TextSentencer_T259 36355-36395 Sentence denotes The true distance is marked with an 'x'.
TextSentencer_T260 36396-36648 Sentence denotes These estimates were based on a time series of binomially distributed case reports, where the accumulated number of recovery events since the last observation determined the number of trials and the reporting probability was the probability of success.
TextSentencer_T260 36396-36648 Sentence denotes These estimates were based on a time series of binomially distributed case reports, where the accumulated number of recovery events since the last observation determined the number of trials and the reporting probability was the probability of success.
TextSentencer_T261 36649-36735 Sentence denotes Parameters besides the reporting probability are in table 1, with b set to 0.9(g þ m).
TextSentencer_T261 36649-36735 Sentence denotes Parameters besides the reporting probability are in table 1, with b set to 0.9(g þ m).
TextSentencer_T262 36736-36770 Sentence denotes rsif.royalsocietypublishing.org J.
TextSentencer_T262 36736-36770 Sentence denotes rsif.royalsocietypublishing.org J.
TextSentencer_T263 36771-36773 Sentence denotes R.
TextSentencer_T263 36771-36773 Sentence denotes R.
TextSentencer_T264 36774-36778 Sentence denotes Soc.
TextSentencer_T264 36774-36778 Sentence denotes Soc.
TextSentencer_T265 36779-36792 Sentence denotes Interface 15:
TextSentencer_T265 36779-36792 Sentence denotes Interface 15:
TextSentencer_T266 36793-36801 Sentence denotes 20180034
TextSentencer_T266 36793-36801 Sentence denotes 20180034
TextSentencer_T267 36802-36861 Sentence denotes for detection of slowing down as a threshold is approached.
TextSentencer_T267 36802-36861 Sentence denotes for detection of slowing down as a threshold is approached.
TextSentencer_T268 36862-37025 Sentence denotes The solution is a method of calculating what type of white noise perturbations, if any, allow slowing down to be detected based on observation of a given variable.
TextSentencer_T268 36862-37025 Sentence denotes The solution is a method of calculating what type of white noise perturbations, if any, allow slowing down to be detected based on observation of a given variable.
TextSentencer_T269 37026-37201 Sentence denotes To provide a specific example, this general solution has been applied to the SIR model and been shown to be consistent both with a modelspecific analysis and with simulations.
TextSentencer_T269 37026-37201 Sentence denotes To provide a specific example, this general solution has been applied to the SIR model and been shown to be consistent both with a modelspecific analysis and with simulations.
TextSentencer_T270 37202-37393 Sentence denotes This application has also served to demonstrate and stress-test a method of estimating a distance to a threshold that is defined as one of the eigenvalues of the linearized model's matrix, F.
TextSentencer_T270 37202-37393 Sentence denotes This application has also served to demonstrate and stress-test a method of estimating a distance to a threshold that is defined as one of the eigenvalues of the linearized model's matrix, F.
TextSentencer_T271 37394-37473 Sentence denotes Importantly, this informative eigenvalue is not always the dominant eigenvalue.
TextSentencer_T271 37394-37473 Sentence denotes Importantly, this informative eigenvalue is not always the dominant eigenvalue.
TextSentencer_T272 37474-37738 Sentence denotes When the informative eigenvalue is not dominant it is a consequence of the vital dynamics of the host occurring on a time scale that is much longer than the dynamics of small outbreaks that occur when the infection does not spread very well in the host population.
TextSentencer_T272 37474-37738 Sentence denotes When the informative eigenvalue is not dominant it is a consequence of the vital dynamics of the host occurring on a time scale that is much longer than the dynamics of small outbreaks that occur when the infection does not spread very well in the host population.
TextSentencer_T273 37739-37846 Sentence denotes Such a difference in time scales seems likely to occur in other multivariate models of population dynamics.
TextSentencer_T273 37739-37846 Sentence denotes Such a difference in time scales seems likely to occur in other multivariate models of population dynamics.
TextSentencer_T274 37847-38033 Sentence denotes Looking beyond the SIR model, the question also arises of whether our method of identifying appropriate variables will be practical for models with many more than two degrees of freedom.
TextSentencer_T274 37847-38033 Sentence denotes Looking beyond the SIR model, the question also arises of whether our method of identifying appropriate variables will be practical for models with many more than two degrees of freedom.
TextSentencer_T275 38034-38171 Sentence denotes In the SIR model, the autocorrelation function of one of the state variables is often very similar to that of one of the eigendirections.
TextSentencer_T275 38034-38171 Sentence denotes In the SIR model, the autocorrelation function of one of the state variables is often very similar to that of one of the eigendirections.
TextSentencer_T276 38172-38359 Sentence denotes This allowed us to select variables based on the criterion of how well the autocorrelation function matched up with that of the eigendirection corresponding to the informative eigenvalue.
TextSentencer_T276 38172-38359 Sentence denotes This allowed us to select variables based on the criterion of how well the autocorrelation function matched up with that of the eigendirection corresponding to the informative eigenvalue.
TextSentencer_T277 38360-38526 Sentence denotes In general, as the number of variables grows we might expect the autocorrelation function of each variable to become more strongly influenced by multiple eigenvalues.
TextSentencer_T277 38360-38526 Sentence denotes In general, as the number of variables grows we might expect the autocorrelation function of each variable to become more strongly influenced by multiple eigenvalues.
TextSentencer_T278 38527-38750 Sentence denotes For this more challenging case, we wonder whether harmonic inversion methods [35] might be capable of estimating the values of each of the eigenvalues that strongly influence each variable from its autocorrelation function.
TextSentencer_T278 38527-38750 Sentence denotes For this more challenging case, we wonder whether harmonic inversion methods [35] might be capable of estimating the values of each of the eigenvalues that strongly influence each variable from its autocorrelation function.
TextSentencer_T279 38751-38910 Sentence denotes Variables that allow the informative eigenvalue to be estimated in this manner could then be considered appropriate for tracking the distance to the threshold.
TextSentencer_T279 38751-38910 Sentence denotes Variables that allow the informative eigenvalue to be estimated in this manner could then be considered appropriate for tracking the distance to the threshold.
TextSentencer_T280 38911-39074 Sentence denotes The distance to thresholds in systems will generally change over time, and our results concluded with a simple demonstration of how these changes might be tracked.
TextSentencer_T280 38911-39074 Sentence denotes The distance to thresholds in systems will generally change over time, and our results concluded with a simple demonstration of how these changes might be tracked.
TextSentencer_T281 39075-39353 Sentence denotes In the context of infectious disease surveillance, an exciting prospect of this approach is the possibility that surveillance programmes might be able to determine that some change in the system is moving it closer to the epidemic threshold long before the threshold is crossed.
TextSentencer_T281 39075-39353 Sentence denotes In the context of infectious disease surveillance, an exciting prospect of this approach is the possibility that surveillance programmes might be able to determine that some change in the system is moving it closer to the epidemic threshold long before the threshold is crossed.
TextSentencer_T282 39354-39554 Sentence denotes Besides increasing awareness, such measurement may allow for management of the distance to the threshold in some systems, for example, by guiding the allocation of resources to vaccination programmes.
TextSentencer_T282 39354-39554 Sentence denotes Besides increasing awareness, such measurement may allow for management of the distance to the threshold in some systems, for example, by guiding the allocation of resources to vaccination programmes.
TextSentencer_T283 39555-39711 Sentence denotes In this way, infectious disease control goals could move beyond early detection of and rapid response to epidemics towards targeted prevention of epidemics.
TextSentencer_T283 39555-39711 Sentence denotes In this way, infectious disease control goals could move beyond early detection of and rapid response to epidemics towards targeted prevention of epidemics.
TextSentencer_T284 39712-39839 Sentence denotes Furthermore, tracking has the potential to measure the relenting and reversing of system dynamics in response to control goals.
TextSentencer_T284 39712-39839 Sentence denotes Furthermore, tracking has the potential to measure the relenting and reversing of system dynamics in response to control goals.
TextSentencer_T285 39840-40250 Sentence denotes Finally, establishing the conditions under which statistical analysis of fluctuations in the number of infected individuals is more informative than similar analysis of susceptible individuals does not make a case against susceptible reconstruction methods [36] in distance-to-threshold studies because such methods estimate major trends in the susceptible population size rather than fluctuations around them.
TextSentencer_T285 39840-40250 Sentence denotes Finally, establishing the conditions under which statistical analysis of fluctuations in the number of infected individuals is more informative than similar analysis of susceptible individuals does not make a case against susceptible reconstruction methods [36] in distance-to-threshold studies because such methods estimate major trends in the susceptible population size rather than fluctuations around them.
TextSentencer_T286 40251-40445 Sentence denotes Rather, our result makes a case for analysis of the fluctuations in the number infected, whether estimated from readily available time-series incidence data or from pathogen sequence data [37] .
TextSentencer_T286 40251-40445 Sentence denotes Rather, our result makes a case for analysis of the fluctuations in the number infected, whether estimated from readily available time-series incidence data or from pathogen sequence data [37] .
TextSentencer_T287 40446-40465 Sentence denotes Data accessibility.
TextSentencer_T287 40446-40465 Sentence denotes Data accessibility.
TextSentencer_T288 40466-40495 Sentence denotes Software and reproducibility:
TextSentencer_T288 40466-40495 Sentence denotes Software and reproducibility:
TextSentencer_T289 40496-40569 Sentence denotes Code to reproduce our results is available in an online repository [38] .
TextSentencer_T289 40496-40569 Sentence denotes Code to reproduce our results is available in an online repository [38] .