Abstract

In this paper we look at the two dimensional stochastic differential equation (SDE) susceptible-infected-susceptible (SIS) epidemic model with demographic stochasticity where births and deaths are regarded as stochastic processes with per capita disease contact rate depending on the population size. First we look at the SDE model for the total population size and show that there exists a unique non-negative solution. Then we look at the two dimensional SDE SIS model and show that there exists a unique non-negative solution which is bounded above given the total population size. Furthermore we show that the number of infecteds and the number of susceptibles become extinct in finite time almost surely. Lastly, we support our analytical results with numerical simulations using theoretical and realistic disease parameter values.

Highlights

  • One of the simplest possible models for how diseases spread amongst a population is the susceptible-infected-susceptible (SIS) epidemic model

  • The SIS epidemic model is one of the simplest models which is suitable in analysing diseases where individuals do not gain immunity after recovery, for example gonorrhea, pneumococcus, tuberculosis or the common cold

  • The deterministic stochastic differential equation (SDE) SIS model assumes that the total population size is constant susceptible or infected individuals who die are immediately replaced by new susceptible individuals

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Summary

Introduction

One of the simplest possible models for how diseases spread amongst a population is the susceptible-infected-susceptible (SIS) epidemic model. Most deterministic models for infectious diseases assume that the population size remains constant but there has been some work done on epidemic models with variable population size This is usually for a different reason, because either there is disease-related mortality so infected individuals die at an increased rate compared to susceptible ones or there is some sort of population density dependence in either the birth rate or the death rate, due for example to competition for scarce resources. The reader might argue that the SIS epidemic model given by (1.1) and (1.2) with transmission term βS(t)I(t), corresponding to per capita disease contact rate λ = βN, might not be realistic when analysing models where population size is allowed to change as the transmission rate β may not remain constant especially when N is large. Numerical simulations with theoretical parameter values and realistic parameter values for pneumococcus and the common cold are given in Sections 6 and 7 respectively

Demographic stochasticity for the two dimensional SDE SIS epidemic model
Extinction of the number of infecteds and the total number of individuals
Simulations to illustrate the analytical results
Simulations on the total number of individuals
Simulations on the total number of infecteds
Realistic simulations
Findings
Conclusion and discussion
Full Text
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