Wave propagation in a diffusive SEIR epidemic model with nonlocal transmission and a general nonlinear incidence rate

We introduce a diffusive SEIR model with nonlocal delayed transmission between the infected subpopulation and the susceptible subpopulation with a general nonlinear incidence. We show that our results on existence and nonexistence of traveling wave solutions are determined by the basic reproduction number $R_{0}=\partial _{I}F(S_{0},0)/\gamma $ R 0 = ∂ I F ( S 0 , 0 ) / γ of the corresponding ordinary differential equations and the minimal wave speed $c^{*}$ c ∗ . The main difficulties lie in the fact that the semiflow generated here does not admit the order-preserving property. In the present paper, we overcome these difficulties to obtain the threshold dynamics. In view of the numerical simulations, we also obtain that the minimal wave speed is explicitly determined by the time delay and nonlocality in disease transmission and by the spatial movement pattern of the exposed and infected individuals.

The Dynamics of a Stochastic SIR Epidemic Model with Nonlinear Incidence and Vertical Transmission

In this study, we build a stochastic SIR epidemic model with vertical infection and nonlinear incidence. The influence of the fluctuation of disease transmission parameters and state variables on the dynamic behaviors of the system is the focus of our study. Through the theoretical analysis, we obtain that there exists a unique global positive solution for any positive initial value. A threshold R 0 s is given. When R 0 s < 1 , the diseases can be extincted with probability one. When R 0 s > 1 , we construct a stochastic Lyapunov function to prove that the system exists an ergodic stationary distribution, which means that the disease will persist. Then, we obtain the conditions that the solution of the stochastic model fluctuates widely near the equilibria of the corresponding deterministic model. Finally, the correctness of the results is verified by numerical simulation. It is further found that the fluctuation of disease transmission parameters and infected individuals with the environment can reduce the threshold of disease outbreak, while the fluctuation of susceptible and recovered individuals has a little effect on the dynamic behavior of the system. Therefore, we can make the disease extinct by adjusting the appropriate random disturbance.