Abstract
We derive and analyze the dynamic of a stochastic SEI epidemic model for disease spread. Fluctuations in the transmission rate of the disease bring about stochasticity in model. We discuss the asymptotic stability of the infection-free equilibrium by first deriving the closed form deterministic (R0) and stochastic (R0) basic reproductive number. Contrary to some author’s remark that different diffusion rates have no effect on the stability of the disease-free equilibrium, we showed that even if no epidemic invasion occurs with respect to the deterministic version of the SEI model (i.e., R0<1), epidemic can still grow initially (if R0>1) because of the presence of noise in the stochastic version of the model. That is, diffusion rates can have effect on the stability by causing a transient epidemic advance. A threshold criterion for epidemic invasion was derived in the presence of external noise.
Highlights
Many mathematical models have been developed in order to understand disease transmissions and behavior of epidemics
Contrary to some author’s remark that different diffusion rates have no effect on the stability of the disease-free equilibrium, we showed that even if no epidemic invasion occurs with respect to the deterministic version of the SEI model (i.e., R0 < 1), epidemic can still grow initially because of the presence of noise in the stochastic version of the model
Among these model is the SEI susceptible-exposed-infectious model. This model is used by some author in studying disease transmission of the Severe Acute Respiratory Syndrome (SARS) disease
Summary
Many mathematical models have been developed in order to understand disease transmissions and behavior of epidemics Among these model is the SEI susceptible-exposed-infectious model. We are interested in studying the effect of stochastic fluctuations in the disease transmission rates in the susceptible-exposed-infected epidemic model. Due to this reason, we develop a Stratonovich stochastic dynamic SEI model by introducing noise in the transmission rates. By linearizing the Ito version of the stochastic SEI model around the infection-free equilibrium, we give a closed form expectation of the susceptible, exposed and infected. This is used to discuss and analyze the stability of the infection-free equilibrium.
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