On invariant measures and the asymptotic behavior of a stochastic delayed SIRS epidemic model

Abstract

In this paper, we consider a stochastic epidemic model with time delay and general incidence rate. We first prove the existence and uniqueness of the global positive solution. By using the Krylov-Bogoliubov method, we obtain the existence of invariant measures. Furthermore, we study a special case where the incidence rate is bilinear with distributed time delay. When the basic reproduction number R0 < 1, the analysis of the asymptotic behavior around the disease-free equilibrium E0 is provided while when R0 > 1, we prove that the invariant measure is unique and ergodic. The numerical simulations also validate our analytical results.

Sharp convex generalizations of stochastic Gronwall inequalities

“…The related results are mentioned in detail in Section 4. For the definition of the constants see (8). The sharpness of the constants α 1 , α 1 α 2 and β is discussed in detail in Section 3.3.…”

“…Due to convexity of η and η(c 0 ) = 0 there exists some K > 0 s.t. η(x) ≤ K(x − c 0 ) for x ∈ [c 0 , c] where c denotes the constant from the definition of G, see (8). This implies lim x→c 0 G(x) = −∞, in particular, domain(G −1 ) = range(G) = (−∞, lim x→∞ G(x)).…”

“…The constants α 1 , α 2 and β occuring in the following examples are defined in (8) and they only depend on p.…”

“…Since η p is non-decreasing and s → E F 0 [(X * s ) p ] is càdlàg, we may apply the deterministic Bihari-LaSalle inequality (see Lemma 1.1) to the previous inequality. Recall the following definition and its properties (see (8)):…”

“…The stochastic Gronwall inequality without supremum is applied to study various SDEs without memory, see e.g. [8], [11], [12], [15], [18], [23], [28], [29], [32], [35] and [37].…”