Stochastic regime switching SIR model driven by Lévy noise

Persistence and extinction of a stochastic SIS epidemic model with regime switching and Lévy jumps

“…The corresponding deterministic systems of (27) and 28have a unique globally asymptotically stable endemic equilibrium (S * , I * ) = ( 6 5 , 2 5 ) and (S * , I * ) = ( 16 9 , 8 45 ), respectively (for more details, see for example [30]). From Figure 1, we see that in regime 1, the solution of (2) oscillates around the deterministic endemic equilibrium (S * , I * ) = ( 6 5 , 2 5 ) after some initial transients; in regime 2, the curve of susceptible population to (2) oscillates around the deterministic endemic equilibrium S * = 16 9 , and the disease I(t) of (2) with jumps is extinct eventually. Under regime-switching, however, the solution of (2) with parameters (29) is persistent and the curves of S(t) and I(t) of (2) oscillate in some appropriate region (see Figure 2).…”

“…For the bilinear incidence g(S, I) = SI, Zhang and Wang [34] studied the asymptotical behavior of (35) without regime-switching; Guo [16] obtained a unique global positive solution and investigated the asymptotical behavior of the stochastic SIR model (35). Comparing with [16,34], we weaken the restrictions in the assumptions and improve their results to some degree. By choosing appropriate Lyapunov functions, Zhou and Zhang [38] investigated the extinction and persistence of the disease of Zhou and Zhang [38] concluded that if R 0 < 1 then the disease of system (36) ultimately vanishes from the population, and that if R 0 > 1 then the disease of system (36) persists in the population, where R 0 is given in (6).…”

“…There are a number of results about persistence and extinction of the disease in the SIR model (35). For the bilinear incidence g(S, I) = SI, Zhang and Wang [34] studied the asymptotical behavior of (35) without regime-switching; Guo [16] obtained a unique global positive solution and investigated the asymptotical behavior of the stochastic SIR model (35). Comparing with [16,34], we weaken the restrictions in the assumptions and improve their results to some degree.…”

Stabilisation in distribution by delay feedback control for stochastic differential equations with Markovian switching and Lévy noise

“…This class of systems is often used to model many realistic systems, such as mathematical finance, biology and so on, see refs. [1][2][3][4][5]. As an important aspect of the study of SDEs-MS-LN, asymptotic stability analysis has been broadly studied in refs.…”

Analysis of a stochastic predator–prey system with fear effect and Lévy noise

“…Furthermore, the population system may suffer sudden environmental shocks, e.g., earthquakes, hurricanes, epidemics, etc. However, the stochastic Lotka-Volterra model (1.2) cannot explain such phenomena [24][25][26]. Just as Scheffer et al [27] pointed out that studies on lakes, coral reefs, oceans, forests, and arid lands had shown that smooth change could be interrupted by sudden drastic switches to a contrasting state.…”