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Planning horizon affects prophylactic decision-making and epidemic dynamics
Abstract
Human behavior can change the spread of infectious disease. There is limited understanding of 1 how the time in the future over which individuals make a behavioral decision, their planning 2 horizon, affects epidemic dynamics. We developed an agent-based model (along with an ODE 3 analog) to explore the decision-making of self-interested individuals on adopting prophylactic 4 behavior. The decision-making process incorporates prophylaxis efficacy and disease prevalence 5 with individuals' payoffs and planning horizon. Our results show that for short and long 6 planning horizons individuals do not consider engaging in prophylactic behavior. In contrast, 7 individuals adopt prophylactic behavior when considering intermediate planning horizons. Such 8 adoption, however, is not always monotonically associated with the prevalence of the disease, 9 depending on the perceived protection efficacy and the disease parameters. Adoption of 10 prophylactic behavior reduces the peak size while prolonging the epidemic and potentially 11 generates secondary waves of infection. These effects can be made stronger by increasing the 12 behavioral decision frequency or distorting an individual's perceived risk of infection.
Human behavior plays a significant role in the dynamics of infectious disease [7, 11] . However, 15 the inclusion of behavior in epidemiological modeling introduces numerous complications and 16 involves fields of research outside the biological sciences, including psychology, philosophy, 17 sociology, and economics. Areas of research that incorporates human behavior into 18 epidemiological models are loosely referred to as social epidemiology, behavioral epidemiology, 19 or economic epidemiology [14, 17] . We use the term 'behavioral epidemiology' to broadly refer 20 to all epidemiological approaches that incorporate human behavior. While the incorporation of 21 behavior faces many challenges [9], one of the goals of behavioral epidemiology is to 22 understand how social and behavioral factors affect the dynamics of infectious disease 23 epidemics. This goal is usually accomplished by coupling models of social behavior and 24 decision making with biological models of contagion [16, 11] .
Many social and behavioral aspects can be incorporated into a model of infectious disease.
One example is the effect of either awareness or fear spreading through a population [10, 5] . In 27 these types of models, the spread of beliefs or information is treated as a contagion much like an 28 infectious disease, though the network for the spread of information may differ from the 29 biological network [2]. Other models focus on how individuals adapt their behavior by 30 weighting the risk of infection with the cost of social distancing [6, 19] or other disincentives [1]. 31 Still others model public health interventions (e.g. isolation, vaccination, surveillance, etc.) and 32 individual responses to them [3]. Many of these models sit at the population level, incorporating 33 the effects of social factors and abstracting away details about the individuals themselves.
34 The SPIR model (Susceptible, Prophylactic, Infectious, Recovered) is an epidemiological 35 agent-based model that couples individual behavioral decisions with an extension of the SIR 36 model [13]. In this model agents that are vulnerable to infection may be in one of two states 37 which are determined by their behavior. Agents in the susceptible state engage in the status quo 38 behavior while agents in the prophylactic state employ preventative behaviors that reduce their 39 chance of infection. We use a rational choice model to represent individual behavioral decisions, 40
where individuals select the largest utility between engaging in prophylactic behavior (e.g.
hand-washing or wearing a face mask) or non-prophylactic behavior (akin to the status quo). We 42 also allow for the fact that individuals may not perceive the risk of getting infected accurately, 43 but rather receive some distorted information, for example, through the media.
We are interested in understanding how an individual's planning horizon-the time in the 45 future over which individuals calculate their utilities to make a behavioral decision-affects 46 behavioral change and how that in turn influences the dynamics of an epidemic. 47 We introduce the model through the lens of individuals rather than at the population level, 48 because we find that the individualistic perspective gives a more natural interpretation of the 49 behavioral decision analysis we discuss. Under the assumption that the population is large, well 50 mixed, and homogeneous, we can also express the model as a system of ordinary differential 51 equations (ODEs). We work with both versions of the model, using the ODE version for 52 calculations, and the individual-based or agent-based model (ABM) version for thinking about 53 the psychological features of decision making that may affect the spread of infectious diseases. 54
One of our key findings is that individuals choose to engage in prophylactic behavior only 55 when the planning horizon is "just right." If the planning horizon is set too far into the future, it 56 is in an individual's best interest to become infected (i.e. get it over with); if the horizon is too 57 short, individuals dismiss the future risk of infection (i.e. live for the moment). What counts as 58 "just right" depends on the disease in question, and we explore two hypothetical contrasting 59 diseases, one with long recovery time and acute severity, and another with short recovery time 60 and mild severity. 61 Methods 62 The SPIR Model 63 The SPIR model couples two sub-models: one reproducing the dynamics of the infectious 64 disease, the Disease Dynamics Model, and another that determines how agents make the 65 decision to engage in prophylactic or non-prophylactic behavior, the Behavioral Decision Model. 66 Disease Dynamics Model 67 The disease dynamics model reproduces the dynamics of the infectious disease in a constant 68 population of N agents. Each agent can be in one of four states: Susceptible (S), Prophylactic 69 (P), Infectious (I), or Recovered (R). The difference between agents in states S and P is that the 70 former engage in non-prophylactic behavior and do not implement any measure to prevent 71 infection, while the latter adopt prophylactic behavior which decreases their probability of being 72 3/23 . CC-BY-NC 4.0 International license is made available under a
The copyright holder for this preprint (which was not peer-reviewed) is the author/funder. It . https://doi.org/10.1101/069013 doi: bioRxiv preprint proportion of infectious agents i represents the probability that an agent is paired with an 87 infectious agent and the per time step probability of an agent in either state S or P being infected 88 is either ib S or ib P , respectively. In addition to interacting, susceptible and prophylactic agents 89 4/23 . CC-BY-NC 4.0 International license is made available under a The copyright holder for this preprint (which was not peer-reviewed) is the author/funder. It . https://doi.org/10.1101/069013 doi: bioRxiv preprint 109 model is a rational choice model that assumes agents are self-interested and rational; thus they 110 adopt the behavior with the largest utility over the planning horizon, H. Note that the planning 111 horizon is a construct used to calculate utilities and it does not affect the time until an agent has 112 an opportunity to make another decision within the disease dynamics model.
The planning horizon is the time in the future over which agents calculate their utilities. In 114 order for the agent to make these calculations, we make the following assumptions about agents 115 6/23
Introduction infected (e.g. wearing a mask, washing hands, etc.). Agents in the infectious state I are infected 73 and infective, while those in the recovered state R are immune to and do not transmit disease. 74 The transition between states is captured with the state-transition diagram shown in Fig. 1 . For 75 reference, all the parameters and variables in the SPIR model are listed and defined in Table 1 . S, P, I, and R represent the four epidemiological states an agent can be in: Susceptible, Prophylactic, Infectious, and Recovered, respectively. The parameters over the transitions connecting the states represent the probability per time step that agents in one state move to an adjacent state: i is the proportion of infectious agents in the population; bS and bP are the respective probabilities that an agent in state S or P, encountering an infectious agent I, becomes infected; g is the recovery probability; d is the behavioral decision making probability; and W(i) is an indicator function returning value 1 when the utility of being prophylactic is greater than the utility of being susceptible and 0 otherwise (see details in Behavioral Decision Model). This sub-model assumes that in each time step, three types of events occur: (i) interactions Proportion of susceptible agents in the population. p
Proportion of prophylactic agents in the population. i
Proportion of infectious agents in the population. r
Proportion of recovered agents in the population. b S Probability that an agent in the susceptible state becomes infected upon interacting with an infectious agent. b P Probability that an agent in the prophylactic state becomes infected upon interacting with an infectious agent. ρ
Reduction in the transmission probability or rate when adopting prophylactic behavior: b P = ρb S (0 ≤ ρ ≤ 1). Note that we refer to 1 − ρ as the protection. g
Probability an infectious agent recovers. d
Probability an agent in the susceptible or prophylactic state decides which behavior to engage in. κ
Distortion of the perceived proportion of infectious agents in the population (i.e. distortion factor). If we assume that the population is well-mixed, the dynamics can be generated using a The copyright holder for this preprint (which was not peer-reviewed) is the author/funder. It . https://doi.org/10.1101/069013 doi: bioRxiv preprint system of ODEs:
where s, p, i, and r are the proportion of susceptible, prophylactic, infectious, and recovered 99 agents in the population. The parameters β, γ, and δ are transmission, recovery, and decision 100 rates, whose equivalent probabilities are, respectively, transmission (b S ), recovery (g), and 101 decision (d) ( Table 2 ). The parameter ρ refers to the reduction in transmission rate when 102 adopting prophylactic behavior. We convert between rates and probabilities using equations Table 2 . Disease parameters in rates and probabilities.
Probability Description Probability to parameter parameter rate conversion on behavior X (denoted T X|DX ),
Pr
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which simplifies to the desired expectation,
Notice that 1 fX − 1 is the expected value of an uncensored geometric with minimum value of 142 zero and that the second parenthetical term rescales the expectation to the interval [1, H] . 143 Next, we derive the expected time spent in I by conditioning on T X|DX .
After considerable algebra, we get the expectation,
Again, the expectation of the uncensored geometric is 1 g − 1 . The the second parenthetical 146 term compresses the expected time into the interval between E T X|DX and H. Notice that 147 Eq. (6) is defined only so long as f X = g. When f X = g, we instead have
Finally, the agent calculates the expected time in state R by subtracting the expectations for 149 X and I from H,
Notice that for each of the expected waiting times calculated in Eqs. (4), (6) and (7), as H goes 151 to infinity, the rescaling terms go to one so that the equations yield the familiar expected values 152
for the uncensored geometrics.
Having calculated these expected waiting times, the agent then calculates the utility for the 154
. CC-BY-NC 4.0 International license is made available under a The copyright holder for this preprint (which was not peer-reviewed) is the author/funder. It . https://doi.org/10.1101/069013 doi: bioRxiv preprint two possible behaviors using, (9) and 156
Note that when agents calculate expected times for states S and P, they need not consider the 157 possibility of alternating to the other state in the future. This is because they assume a constant i 158 which implies the best strategy now will remain the best strategy at all times during H.
Distorting knowledge of i. Recall that assumption (iii) that underlies the behavioral decision 165 model is that agents know the prevalence of the disease accurately. We relax this assumption to 166 investigate how distorting this information effects the SPIR model. To achieve this, we replace i 167 with i 1 /κ in the calculation of utilities where κ serves as a distortion factor. When κ = 1, i is not 168 distorted; when κ > 1, the agent perceives i to be above its real value and when κ < 1 the 169 opposite is true. To implement this distortion, we simply redefine f X in the expected waiting 170 time equations (i.e. Eqs. (3)- (7)) with f X = i 1 /κ b S when X = S and f X = i 1 /κ ρb S when X = P. 171
The SPIR model is suitable for helping understand the influence of human behavior for diverse 173 infectious disease epidemics. To illustrate specific characteristics of the model, however, we 174 focus here on two contrasting diseases characterized by their severity, recovery time, and harm: 175
Disease 1 is acute, has a long recovery time, and may cause chronic harm, and Disease 2 is mild, 176
. CC-BY-NC 4.0 International license is made available under a The copyright holder for this preprint (which was not peer-reviewed) is the author/funder. It . https://doi.org/10.1101/069013 doi: bioRxiv preprint has a short recovery time, and cause no lasting harm. Table 3 shows the biological and 177 behavioral parameter values used to generate the results discussed next, unless stated otherwise. 178 Table 3 . Input parameters for two hypothetical contrasting diseases.
Disease 1 Disease 2 Type Name
Behavioral Decision Analysis
Here we analyze the behavioral decision model used by the agents to decide whether or not to The copyright holder for this preprint (which was not peer-reviewed) is the author/funder. susceptible behavior and for the prophylactic behavior otherwise.
Region C corresponds to the situation in which two switch points exist instead of a single one 208 (Fig. 2C) . When the proportion of infectious agents is between these switch points, agents adopt 209
prophylactic behavior, while values outside this range drives agents to adopt non-prophylactic 210 behavior. This situation is of particular interest because it shows that the adoption of 211 prophylactic behavior is not always monotonically associated with the prevalence of the disease. 212
The utility calculations that agents use to decide whether to adopt a behavior are complex 213 (see Eqs. (9) and (10)); an exhaustive exploration of the parameter space is not undertaken here. 214 We instead investigate several paradigm cases related to the payoff ordering. We assume that the 215 payoff for the infectious state (u I ) relies upon biological parameters of the disease and always 216 corresponds to the lowest payoff, thus we need only consider the relationship between the other 217 three payoffs. In particular, we are interested in looking at situations where the recovery payoff 218 ranges from complete recovery (case 1) to less than the prophylactic state (case 4). Figure 4 . Heat maps of switch points for payoffs ordering cases of Disease 1 and Disease 2.
In (A) and (E) the payoff of being susceptible and recovered are equal, which means that agents recover completely from the disease after infection. In (B) and (F), the recovered payoff is lower than the susceptible payoff, but still greater than the prophylactic payoff meaning that it is more advantageous being recovered than in the prophylactic state. In (C) and (G), any advantage comparison between being in the prophylactic or recovered state is eliminated. In (D) and (G), the disease debilitates the agent meaning that they would be better off engaging in prophylactic behavior rather be in the prophylactic state than the recovered state. The heat maps of behavioral change assume payoffs {u S , u P , u I , u R } of (
The effects of progressively reducing the recovered payoff are more evident for Disease 2.
Reducing the recovered payoff means that lower levels of prevalence will be sufficient for agents 236 to change their behavior. In the case of equal value for recovered and susceptible payoffs, agents 237 consider changing behavior only in narrow parameter range of protection efficacy and planning 238 horizon values (Fig. 4E) . Progressively reducing the recovered payoff, i.e. moving from case 1 239 The copyright holder for this preprint (which was not peer-reviewed) is the author/funder. It . https://doi.org/10.1101/069013 doi: bioRxiv preprint consider changing behavior) and the disease prevalence necessary for such change to occur 242 decreases (i.e. gradual change of the color towards blue).
In addition to this numerical analysis, we have also obtained analytical results for case 2 244 (payoff ordering u S > u R > u P > u I ) to identify the general conditions necessary for the 245 existence of one or more switch points. Mathematically, switch points occur where the utility 246 functions for S and P are equal (Eqs. (9) and (10)). Replacing the expected time notation 247 E T Y|DX in the utility U S and U P by the more concise T Y|X , where X ∈ {S, P} and 248 Y ∈ {S, P, I, R}, we have
Let K 1 = uS−uR uI−uR , which weights the benefits of S and I, and K 2 = uP−uR uS−uR , which weights the 251 benefit of S and P. Then,
Because the payoff ordering u S > u R > u P > u I and noting that T I|S ≥ T I|P and K −1 1 = uI−uR uS−uR , 253 we have that K −1 1 T I|S − T I|P ≤ 0 and 0 < K 2 < 1. From these we analyze some general 254 cases. 255 First, we analyze the case in which K 2 T P|P − T S|S ≥ 0, then K 2 T P|P ≥ T S|S . Because
This case always produces a switch point and occurs when
We turn now to understand how the above conditions for behavioral change may influence The copyright holder for this preprint (which was not peer-reviewed) is the author/funder. It . https://doi.org/10.1101/069013 doi: bioRxiv preprint of the disease prevalence; for κ > 1, the perceived disease prevalence is inflated and κ reflects 323 an increase in the risk perception of being infected; for κ < 1, the perceived disease prevalence 324 is reduced below its true value.
Distorting the perception of a disease prevalence can lead to changes in the decision making 326 process, and consequently on epidemic dynamics, as illustrated in Fig. 7 (see Fig. S2 for Disease 327
2). Figure 7A shows the proportion of infectious agents above which the prophylactic behavior 328 is more advantageous than non-prophylactic behavior assuming agents know the real disease 329 prevalence (κ = 1). By distorting the perceived disease prevalence to increase the risk 330 perception of being infected (κ = 1.5), the real proportion of infectious agents necessary for 331 agents to engage in prophylactic behavior is reduced as shown in Fig. 7B . Hence, the distortion 332 on disease prevalence makes agents engage in prophylactic behavior even when the chance of 333 being infected is low. This affects the epidemic dynamics by reducing the peak size but Proportion of infectious agents above which the prophylactic behavior is more advantageous than the non-prophylactic behavior considering the percentage of protection obtained for adopting the prophylactic behavior (1 − ρ) × 100 and the planning horizon H. (A) No perception distortion, thus κ = 1; while (B) Distortion factor κ of 1.5, which reduces the proportion of infectious agents above which the prophylactic behavior is more advantageous. (C) Epidemic dynamics for different distortion factors that shows how increasing κ reduces the peak size and prolongs the epidemic. 335
. CC-BY-NC 4.0 International license is made available under a The copyright holder for this preprint (which was not peer-reviewed) is the author/funder. It . https://doi.org/10.1101/069013 doi: bioRxiv preprint Individuals acting in their own self-interest make behavioral decisions to reduce their likelihood 337 of getting infected in response to an epidemic. We explore a decision making process that 338 integrates the prophylaxis efficacy and the current disease prevalence with individuals' payoffs 339 and planning horizon to understand the conditions in which individuals adopt prophylactic 340 behavior.
Our results show that the adoption of prophylactic behavior is sensitive to a planning horizon. 342
Individuals with a short planning horizon (i.e. "live for the moment") do not engage in 343 prophylactic behavior because of its adoption costs. Individuals with a long planning horizon 344 also fail to adopt prophylactic behavior, but for different reasons. They prefer to "get it over Moreover, the agents take into consideration only the payoffs of being susceptible and recovered 356 when optimizing the contact rates. In the SPIR model, however, agents maintain a constant 357 contact rate, yet adopt prophylactic behavior that reduces the chance of getting infected. When 358 agents are deciding to engage in prophylactic behavior, they take into account the payoff of all 359 possible epidemiological states. The fact that we reach the same conclusion using different 360 models further supports the claim that the planning horizon is a relevant decision making factor 361 in understanding epidemic dynamics.
Although associated with the prevalence of disease, the adoption of prophylactic behavior is 363 not always monotonically associated with it. Its adoption depends on the behavioral decision The copyright holder for this preprint (which was not peer-reviewed) is the author/funder. It . https://doi.org/10.1101/069013 doi: bioRxiv preprint severe diseases with shorter recovery time, e.g. Disease 2 (Fig. 4E-H) . Therefore, understanding 367 the payoffs related to each disease is critical to proposing effective public policies, especially 368 because there is not a "one-size-fits-all" solution.
Another aspect to highlight is that the beneficial adoption of prophylactic behavior can be 370 achieved through two different public policies: change the risk perception or introduce 371 incentives that reduce the difference between the susceptible and prophylactic payoffs. The 372 problem with increasing the risk perception is that if it is overdone, it leads to the opposite result 373 to the one that is desired. Because individuals perceive their risk of getting the disease as highly 374 probable, they prefer to "get it over with" and enjoy the benefits of being recovered. In contrast, 375 the more the prophylaxis is incentivized the better the results, e.g. reduction of epidemic peak 376 size.
Similar to our SPIR model, Perra et al. [15] and Del Valle et al. [4] also proposed an 378 extension to the SIR model and included a new compartment that reduces the transmission rate 379 between the susceptible and infectious states. A clear distinction between these models and the 380 SPIR model is that their agents do not take into account the costs associated with moving 381 between the susceptible compartment and this new compartment. While in Perra et al. [15] 382 agents make the decision to move between compartments based on the disease prevalence, in 383 Del Valle et al. [4] new constant transfer rates are defined to handle the transition.
In addition to these differences, an advantage of the SPIR model with respect to all other 385 models that implement some behavioral change is the distinction between the disease dynamics 386 and behavioral models. This distinction renders the model flexible by making it easier to, e.g. The copyright holder for this preprint (which was not peer-reviewed) is the author/funder. It . https://doi.org/10.1101/069013 doi: bioRxiv preprint by the National Institutes of Health grant number P30GM103324 that provided us computer resources to perform this study. This research made use of the resources of the High Performance Computing Center at Idaho National Laboratory, which is supported by the Office of Nuclear Energy of the U.S. Department of Energy under Contract No. DE-AC07-05ID14517.
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