The stochastic extinction and stability in the mean of a family of SEIRS
malaria models with a general nonlinear incidence rate is presented. The
dynamics is driven by independent white noise processes from the disease
transmission and natural death rates. The basic reproduction number
R0∗​, the expected survival probability of the plasmodium
E(e−(μv​T1​+μT2​)), and other threshold values are calculated.
A sample Lyapunov exponential analysis for the system is utilized to obtain
extinction results. Moreover, the rate of extinction of malaria is estimated,
and innovative local Martingale and Lyapunov functional techniques are applied
to establish the strong persistence, and asymptotic stability in the mean of
the malaria-free steady population. %The extinction of malaria depends on
R0∗​, and E(e−(μv​T1​+μT2​)). Moreover, for either
R0∗​<1, or E(e−(μv​T1​+μT2​))<R0∗​1​,
whenever R0∗​≥1, respectively, extinction of malaria occurs.
Furthermore, the robustness of these threshold conditions to the intensity of
noise from the disease transmission rate is exhibited. Numerical simulation
results are presented.Comment: arXiv admin note: substantial text overlap with arXiv:1808.09842,
arXiv:1809.03866, arXiv:1809.0389