*3.1. Establishment of Recurrent SEAIR Models*

The recurrent SEAIR model divides the population into five categories: S—susceptible, E—latent, A—asymptomatic, I—symptomatic, and R—recovering. In the recurrent SEAIR model, the population always keeps a constant Q, and at time T, the number of susceptible people is S(t), the number of latent people is E(t), the number of asymptomatic people is A(t), the number of symptomatic people is I(t), and the number of recovered people is R(t). A class of SEAIR models with asymptomatic infection and secondary recurrence was established.

The law of infection is as follows: (1) After contacting the infected person, the susceptible person will be infected with a certain probability, from a healthy state to a latent state and then to an infected state. (2) The state of infected people includes two types: one is symptomatic infection and the other is asymptomatic infection. (3) After these two infected people recover their health, they will develop their own immunity. However, over time, it is possible that this immunity will disappear and the person will again be vulnerable to infection. The mechanism of infection of the recurrent SEAIR risk transmission model is shown in Figure 2.

**Figure 2.** Recurrent SEAIR model.

In the recurrent SEAIR model, it is assumed that (1) the natural mortality is *μ*, regardless of factors such as birth population and floating population; (2) the infection rate of symptomatic infected persons is *α*<sup>1</sup> and that of asymptomatic infected persons is *α*2; (3) the probability of transmission from latent to infected persons is *b*; (4) the mortality rate of asymptomatic infected persons is *c*<sup>1</sup> and that of symptomatic infected persons is *c*2;

and (5) the recovery rate of asymptomatic infected persons is *d*<sup>1</sup> and that of symptomatic infected persons is *d*2; (6) the probability of latent patients entering the asymptomatic period is *β*, and the probability of latent patients entering the symptomatic period is 1 − *β*. Those who recover will recover to latent patients with the probability of *β*; (7) *H* represents the recurrence rate of the disease.

The infectious disease model is as follows:

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$$\begin{aligned} \frac{dS(t)}{dt} &= Q - a\_1 S(t) A(t) - a\_2 S(t) I(t) - \mu S(t) \\ \frac{dE(t)}{dt} &= a\_1 S(t) A(t) + a\_2 S(t) I(t) - \mu E(t) - b \beta E(t) - b(1 - \beta) E(t) + HR(t) \\ \frac{dA(t)}{dt} &= b \beta E(t) - (c\_1 + \mu) A(t) - d\_1 A(t) \\ \frac{dI(t)}{dt} &= b(1 - \beta) E(t) - (c\_2 + \mu) I(t) - d\_2 I(t) \\ \frac{dR(t)}{dt} &= d\_1 A(t) + d\_2 I(t) - \mu R(t) - HR(t) \end{aligned} \tag{1}$$
