Question

A SEIRS model with stochastic transmission :project proposal

Answer 1

SEIRS is a 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 Rσ is below unity. Here Rσ< 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.

The Model :

Melesse and Gumel present a model for a disease of SEIRS type that may cause different stages of infectiousness in a patient. In a special case of the mentioned model, in this paper we study the effect of stochastic perturbations on the stability of the disease-free equilibrium. The population, which at any time t consists of N(t) individuals, is regarded as being divided into four compartments or classes. These are called the susceptible, ex-posed, infectious and removed classes. Their sizes, at any time t,are denoted by S(t), E(t),I(t) and R(t), respectively. The equations of motion of the system are assumed to be given by the system of stochastic differential equations. If σ=  then the system reduces to a system ofode, which can be called the underlying deterministic model or the underlying system of ode. For the system the underlying system of ode coincides with a special case of the model. Inflow into the population is assumed to be all into the class of susceptibles, and it is at a rate μK. Additionally there is flow from the recovered class into the class of susceptibles due to loss of infection-acquired immunity. Themortality rates in the different classes are denoted by μi(i=,,,)and this allows for higher mortality rates in classes which have been affected by the disease,of Berettaet al. Hence the condition The symbol βdenotes the effective contact rate. The parameters αand determine the rates at which individuals in the population pass from classes E to Iand (respectively) from Ito R We further assume that Wis a standard Brownian motion. The aim is to have stochastic perturbations on the transmission rate. We do this by introducing two complementary pairs of stochastic perturbation terms. The non-negative constants σ,p and q are such that σ determines the intensity of the perturbation, while weights attached to the split parts of the perturbation.This motivates the presence of the factor E in the first pair of complementary perturbation terms. This form of the first pair of terms is particularly significant since we are specifically concerned with what happens near disease-free equilibrium. The second pair of complementary perturbation terms can be understood in view of the infection ultimately driving the susceptibles (via the E class) into the I class. The shorter the average latent period, the more relevant does the latter perturbation become. All the parameters are non-negative or positive constants.