An SIRS Epidemic Model Supervised by a Control System for Vaccination and Treatment Actions Which Involve First-Order Dynamics and Vaccination of Newborns

This paper analyses an SIRS epidemic model with the vaccination of susceptible individuals and treatment of infectious ones. Both actions are governed by a designed control system whose inputs are the subpopulations of the epidemic model. In addition, the vaccination of a proportion of newborns is considered. The control reproduction number Rc of the controlled epidemic model is calculated, and its influence in the existence and stability of equilibrium points is studied. If such a number is smaller than a threshold value Rc, then the model has a unique equilibrium point: the so-called disease-free equilibrium point at which there are not infectious individuals. Furthermore, such an equilibrium point is locally and globally asymptotically stable. On the contrary, if Rc>Rc, then the model has two equilibrium points: the referred disease-free one, which is unstable, and an endemic one at which there are infectious individuals. The proposed control strategy provides several free-design parameters that influence both values Rc and Rc. Then, such parameters can be appropriately adjusted for guaranteeing the non-existence of the endemic equilibrium point and, in this way, eradicating the persistence of the infectious disease.

Threshold dynamics for a class of stochastic SIRS epidemic models with nonlinear incidence and Markovian switching

In this paper, we consider a stochastic SIRS epidemic model with nonlinear incidence and Markovian switching. By using the stochastic calculus background, we establish that the stochastic threshold R_{ swt} can be used to determine the compartment dynamics of the stochastic system. Some examples and numerical simulations are presented to confirm the theoretical results established in this paper.

On the Sirs Epidemic Model with Free Boundaries

We investigate an epidemic non-linear reaction-diffusion system with two free boundaries. A free boundary is introduced to describe the expanding front of the infectious environment. A priori estimates of the required functions are established, which are necessary for the correctness and global solvability of the problem. We get sufficient conditions for the spread or disappearance of the disease. It has been proven that with a base reproductive number the disease disappears in the long term if the initial values and the initial area are sufficiently small.

Modeling the dynamics of epidemics of infectious diseases under conditions of diffusion perturbations

The paper proposes a modification of the SIRS epidemic model to take into account the influence of diffusion perturbations on the dynamics of the spread of an infectious disease. A singularly perturbed model problem with delay is reduced to a sequence of problems without delay. The sought functions are represented in asymptotic series as perturbations of solutions of the corresponding degenerate problems. The results of numerical experiments illustrating the influence of spatially distributed diffusion redistributions on the spread of an infectious disease are presented.

Dynamics of a stochastic SIRS epidemic model with standard incidence under regime switching

The goal of this paper is to introduce and initiate a study of a stochastic SIRS epidemic model with standard incidence which is perturbed by both white and telegraph noises. We first show persistence in the mean and then establish the sufficient conditions for extinction of the disease. Moreover, in the case of persistence, we obtain sufficient conditions for the existence of positive recurrence of the solutions by means of structuring suitable stochastic Lyapunov function with regime switching. Meanwhile, the threshold between persistence in the mean and extinction of the stochastic system is also obtained. Finally, we test our theory conclusion by simulations.

Threshold dynamics and optimal control on an age-structured SIRS epidemic model with vaccination

<abstract><p>We consider a vaccination control into a age-structured susceptible-infective-recovered-susceptible (SIRS) model and study the global stability of the endemic equilibrium by the iterative method. The basic reproduction number $ R_0 $ is obtained. It is shown that if $ R_0 &lt; 1 $, then the disease-free equilibrium is globally asymptotically stable, if $ R_0 &gt; 1 $, then the disease-free and endemic equilibrium coexist simultaneously, and the global asymptotic stability of endemic equilibrium is also shown. Additionally, the Hamilton-Jacobi-Bellman (HJB) equation is given by employing the Bellman's principle of optimality. Through proving the existence of viscosity solution for HJB equation, we obtain the optimal vaccination control strategy. Finally, numerical simulations are performed to illustrate the corresponding analytical results.</p></abstract>