An SEIRS epidemic model with stochastic transmission

Peter J Witbooi

doi:10.1186/s13662-017-1166-6

Abstract

For an SEIRS epidemic model with stochastic perturbations on transmission from the susceptible class to the latent and infectious classes, we prove the existence of global positive solutions. For sufficiently small values of the perturbation parameter, we prove the almost surely exponential stability of the disease-free equilibrium whenever a certain invariant mathcal{R}_{sigma} is below unity. Here mathcal{R}_{sigma}< mathcal{R}, the latter being the basic reproduction number of the underlying deterministic model. Biologically, the main result has the following significance for a disease model that has an incubation phase of the pathogen: A small stochastic perturbation on the transmission rate from susceptible to infectious via the latent phase will enhance the stability of the disease-free state if both components of the perturbation are non-trivial; otherwise the stability will not be disturbed. Simulations illustrate the main stability theorem.

Highlights

In recent years, a number of articles have been published on stochastic differential equation models of population dynamics of infectious diseases
In many cases it has been proved that the introduction of stochastic perturbations into an ode epidemic model system can possibly render an unstable disease-free equilibrium of the ode system to become stable in the stochastic differential equation system
6 Conclusion In this paper we constructed an SEIRS model, with stochastic perturbations which can be viewed as linked to the transmission rate out of the class of susceptibles

Summary

Introduction

A number of articles have been published on stochastic differential equation models of population dynamics of infectious diseases. Implies that if a disease-free equilibrium is locally asymptotically stable with respect to the underlying ode-system, it is almost surely exponentially stable with respect to the stochastic model, in particular, the perturbations do not disrupt the stability of the disease-free equilibrium. We present the main result of this paper, which proves that the stochastic perturbation improves the stability of the disease-free equilibrium for small values of the perturbation parameter. Since at least one of f and i must be non-zero, it follows that < The following theorem asserts that, for sufficiently large values of the perturbation parameter σ , the disease will eventually vanish from the population. (other than condition ( ) just being utilized in the proof ) In this case the deterministic model has a non-trivial equilibrium value I∗ for I. This time the parameters selection fulfills all the conditions of Theorem . , and what we see appears to be in line with the assertion of the theorem

Conclusion

Methods