SEIRQ model
In this study, according to the characteristics of the COVID-19 transmission, the whole population at time t is divided into seven compartments which include the susceptible individuals S(t), exposed individuals E(t), infectious individuals I(t), removed individuals R(t), quarantined susceptible individuals S q(t), quarantined exposed individuals E q(t) and quarantined infectious individuals I q(t). The COVID-19 disease is transmitted from I(t) to S(t) with the incidence rate of β, and from E(t) to S(t) with the incidence rate of σβ, respectively. The susceptible individuals S(t) is partly quarantined with the rate of q 1(t). We assume that exposed individuals E(t) and quarantined exposed individuals E q(t) are transmitted to infectious individuals I(t) and quarantined infectious individuals I q(t) with the same transition rate of ν. The quarantined rates of exposed individuals E(t) and infectious individuals I(t) are q 1(t) and q 3. The death rate induced by the COVID-19 disease is α in both infectious individuals I(t) and quarantined infectious individuals I q(t) which removed to the removed individuals R(t) . γ(t) is the recovery rate of quarantined infected individuals I q(t) which is the mainly part of removed individuals R(t).
Moreover, based on the population migration, we assume that the input population and output population have constant numbers. Susceptible individuals S(t), exposed individuals E(t) and infectious individuals I(t) have their respective input individuals of p 1(t)A(t), p 2(t)A(t) and p 3(t)A(t), and the parameters p i(t), i  = 1, 2, 3 are the rates of susceptible individuals, exposed individuals, infectious individuals in the total input number of A(t) from other provinces. The output population are B 1, B 2 and B 3 for the susceptible individuals S(t), exposed individuals E(t), infectious individuals I(t). The COVID-19 disease transmission and population migration are demonstrated by Fig. 1 in details.
Figure 1 Flowchart of COVID-19 SEIRQ epidemic model.
The SEIRQ epidemic model can be described by the following system of ordinary differential equations(1) S′=p1(t)A(t)−βSI−σβSE−q1(t)S−B1,E′=p2(t)A(t)+βSI+σβSE−νE−q2(t)E−B2,I′=p3(t)A(t)+νE−q3I−αI−B3,R′=γ(t)Iq+αI+αIq,Sq′=q1(t)SEq′=q2(t)E−νEqIq′=q3I+νEq−γ(t)Iq−αIqwhere the prime (′) denotes the differentiation with respect to time t. Here, parameters 0 <  β, ν, γ(t), α  < 1 and the quarantined rates 0 ≤  q 1(t), q 2(t), q 3  ≤ 1. All the initial values of different individual groups: S(0), E(0), I(0), R(0), S q(0), E q(0), I q(0) are non-negative.