Abstract
In this paper, we analyze the quasi-stationary distribution of the stochastic SVIR (Susceptible, Vaccinated, Infected, Recovered) model for the measles. The quasi-stationary distributions, as discussed by Danoch and Seneta, have been used in biology to describe the steady state behaviour of population models which exhibit discernible stationarity before to become extinct. The stochastic SVIR model is a stochastic SIR (Susceptible, Infected, Recovered) model with vaccination and recruitment where the disease-free equilibrium is reached, regardless of the magnitude of the basic reproduction number. But the mean time until the absorption (the disease-free) can be very long. If we assume the effective reproduction number Rp , the quasi-stationary distribution can be closely approximated by geometric distribution. β and δ stands respectively, for the disease transmission coefficient and the natural rate.
Highlights
Measles is a highly contagious viral infection that manifests as a rash associated with signs of respiratory infections
Our stochastic model is a stochastic SVIR (Susceptible, Vaccinated, Infected, Recovered) model for the measles [8], where the process ( ) Xt = St, It t≥0 is a continuous-time Markov chain resulting from a set of transient states E0 which evolves until it escapes to a set of absorbing states corresponding to disease-free equilibrium
We study the long time behavior of the process conditioned on non extinction, which leads us to consider the quasi-stationary distribution introduced by Danoch and Seneta in biology
Summary
Measles is a highly contagious viral infection that manifests as a rash associated with signs of respiratory infections. Our stochastic model is a stochastic SVIR (Susceptible, Vaccinated, Infected, Recovered) model for the measles [8], where the process ( ) Xt = St , It t≥0 is a continuous-time Markov chain resulting from a set of transient states E0 which evolves until it escapes to a set of absorbing states corresponding to disease-free equilibrium. We study the long time behavior of the process conditioned on non extinction, which leads us to consider the quasi-stationary distribution introduced by Danoch and Seneta in biology It allows to describe the steady state behaviour of population models which exhibit discernible stationarity before to become extinct [9]. For the continuous time SVIR model under some conditions on the effective reproduction number [12], the quasi-stationary distribution of the number of infected exists and can be closely approximated by geometric distribution. In the last section, we discuss our stochastic approach and scientific conclusions
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