Abstract
In this paper, stochastic effect on the spread of the infectious disease with saturated incidence rate and the special transfer from infectious is discussed. The threshold dynamics is explored for the case of relatively small noise. Our results show that large noise will cause the elimination of the disease, which will help suppress the spread of the disease.
Highlights
Infectious diseases have always threatened the health of human beings and have brought enormous disasters to human beings
McKendrick proposed the classical epidemic model known as SIR model [8], in which the total population size is divided into three disjoint classes, namely the susceptible class (S), the infective class (I), and the recovery class (R)
SIR models have been investigated by many scholars, e.g., [9–13]
Summary
Infectious diseases have always threatened the health of human beings and have brought enormous disasters to human beings. Li et al [27] considered both transfer from the infectious to the susceptible and transfer from the recovery to the susceptible and proposed an SIRS epidemic model with nonlinear transmission rate as follows (see Fig. 1):. Our main purpose is to explore the threshold value associated with epidemic spread and try to establish the threshold dynamics of system (1.2). Lemma 2.1 For any given initial value (S(0), I(0), R(0)) ∈ R3+, system (1.2) has a unique positive solution (S(t), I(t), R(t)) ∈ R3+ on t ≥ 0, almost surely. Lemma 2.3 For any given initial value (S(0), I(0), R(0)) ∈ R3+, the solution (S(t), I(t), R(t)) of system (1.2) has the following properties:. }, the infectious disease of system (1.2) goes to extinction almost surely, namely large white noise stochastic disturbance is conducive to control infectious disease. R0 < 1, the infectious disease of system (1.2) goes to extinction almost surely, R0 is the threshold associated with the extinction of infectious disease
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