Dynamic analysis of a stochastically perturbed dysentery diarrhoea
epidemic model with controls
Abstract
To understand the transmission dynamics of diarrhea in random
environment, in this paper we propose a stochastically perturbed
dysentery diarrhoea epidemic model with controls. Using the theory of
stopping time, we first show the existence of global positive solution
of the model. Then, we study the stochastic dynamics of the model and
present a stochastic threshold $\mathcal{R}_0^S$
which determines the extinction and persistence of the disease. Based on
Khasminskii’s theory, we further prove that the model has a unique
ergodic stationary distribution under the condition of
$\mathcal{R}_0^S>1$. Numerical
simulations are carried out to verify the analytical results, showing
that the white noise, and the constant treatment and sanitation may have
certain inhibitory effects on disease transmission. Lastly, the model is
further extended to include colored noise and seasonal fluctuation to
study the long-term transmission dynamics of disease. It is found that
the method proposed in this paper is universal.