Analysis of a fractional SIR model with general incidence function
This paper presents a stability analysis for a fractional hepatitis B virus epidemic model. With \(S\), \(I\), \(R\) representing the susceptible, infectious, and recovered groups, the model featuring a general incidence function \(g(S,I)\) is given by \begin{align*} D^{\alpha}S(t)&=\lambda^{\alpha}-g(S,I)-(\mu_0^{\alpha}+\nu^{\alpha})S,\\ D^{\alpha}I(t)&=g(S,I)-(\mu_0^{\alpha}+\mu_1^{\alpha}+\beta^{\alpha})I,\text{ and }\\ D^{\alpha}R(t)&=\beta^{\alpha}I+\nu^{\alpha}S-\mu_0^{\alpha}R, \end{align*} where \(D^{\alpha}\) is the Caputo fractional derivative and the parameters are described as follows. The birth rate is \(\lambda\), the transmission rate of disease is \(\alpha\), the disease induced death rate and the natural death rate are \(\mu_1\) and \(\mu_0\) respectively, and the recovery rate is \(\beta\). Moreover, \(\nu\) is the vaccination rate. Under some conditions, the equilibrium points and the basic reproduction number \(R_0\) are derived for this model. It is shown that if \(R_0<1\) the disease-free equilibrium is locally asymptotically stable, and if \(R_0>1\) then this point is unstable. When \(R_0\le1\), the disease-free equilibrium of the system is shown to be globally asymptotically stable.
scientific article; zbMATH DE number 7258062
scientific article