This paper addresses, motivated by mathematical work on infectious disease models, the impacts of environmental noise and media coverage on the dynamics of recovery-relapse infectious diseases. A susceptibleinfectious-recovered-infectious model is formulated with both vertical transmission and horizontal transmission. The existence and uniqueness of the positive global solution is studied by constructing suitable Lyapunov-type function. Then, the existence of positive periodic solutions is verified by applying Khasminskii's theory. The existence of positive periodic solutions indicates the continued survival of the diseases. Besides, sufficient conditions for the extinction of the diseases are obtained. Numerical simulations then demonstrate the dynamics of the solutions. The paper extends the results of the corresponding deterministic system.

Analysis of a stochastic recovery-relapse epidemic model with periodic parameters and media coverage

In this paper, we prove the existence of a critical traveling wave solution for a delayed diffusive SIR epidemic model with saturated incidence. Moreover, we establish the nonexistence of traveling wave solutions with nonpositive wave speed for this model. Our results solve some open problems left in the recent paper (Z. Xu in Nonlinear Anal. 111:66–81, 2014).

/pdf/existence-of-traveling-wave-solutions-with-critical-speed-in-2incyatn3h.pdf

Existence of traveling wave solutions with critical speed in a delayed diffusive epidemic model

In this paper, we study the traveling wave solutions for a nonlocal dispersal SIR epidemic model with standard incidence rate and nonlocal delayed transmission. The existence and nonexistence of traveling wave solutions are determined by the basic reproduction number of the corresponding reaction system and the minimal wave speed. To prove these results, we apply the Schauder’s fixed point theorem and two-sided Laplace transform. The main difficulties are that the complexity of the incidence rate in the epidemic model and the lack of regularity for nonlocal dispersal operator.

https://www.mdpi.com/2227-7390/7/7/641/pdf

Traveling Waves in a Nonlocal Dispersal SIR Model with Standard Incidence Rate and Nonlocal Delayed Transmission

Positive traveling waves in a diffusive epidemic system with distributed delay and constant external supplies

The focus of this paper is to investigate the exact solutions of a diffusive susceptible‐infectious‐recovered (SIR) epidemic model, characterized by a nonlinear incidence. A four‐dimensional Lie point symmetry algebra is obtained for this model. We utilize the Lie symmetries to deduce the optimal system of one‐dimensional subalgebras. The reductions and group‐invariant solutions are obtained with the aid of these subalgebras. We also derive new group‐invariant solutions and reductions for the underlying model via subalgebras that are related to the optimal system by adjoint maps. We developed the diffusive susceptible‐infectious‐quarantined (SIQ) model with quarantine‐adjusted incidence function to understand the transmission dynamics of COVID‐19. 

The Exact Solutions of a Diffusive SIR Model via Symmetry Groups

In this paper, we study the traveling wave solutions of a delayed diffusive SIR epidemic model with nonlinear incidence rate and constant external supplies. We find that the existence of traveling wave solutions is determined by the basic reproduction number of the corresponding spatial-homogenous delay differential system and the minimal wave speed. The existence is proved by applying Schauder's fixed point theorem and Lyapunov functional method. The non-existence of traveling waves is obtained by two-sided Laplace transform. Copyright © 2016 John Wiley & Sons, Ltd.

Traveling wave solutions for a delayed diffusive SIR epidemic model with nonlinear incidence rate and external supplies

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Traveling wave solutions for a delayed diffusive SIR epidemic model with nonlinear incidence rate and external supplies