Dynamic modeling and analysis for the periodic spread of influenza: A case study in Shanghai

In this paper, an $SIRS$ epidemic model with periodic transmission and vaccination is proposed to study the periodic transmission phenomenon of influenza. The threshold dynamics of the model are completely determined by the basic reproduction number ${\mathcal{R}}_0$, i.e., the disease-free periodic steady state is globally asymptotically stable if ${\mathcal{R}}_0<1$, while the disease is uniformly persistent if ${\mathcal{R}}_0>1$. With the help of control theory, the resonance frequency ${\omega }_r$ of our model is obtained, which is numerically validated in that there exists a resonance phenomenon of the disease spread (i.e., there is a maximum amplitude of the oscillatory spread of diseases) when the frequency $\omega ={\omega }_r$. By fitting the real data of influenza from Shanghai, we estimate the parameter values of the model. Results show that there is no resonance phenomenon in the spread of influenza in Shanghai. Based on the expression of resonance frequency ${\omega }_r$, the effects of the mean transmission rate $\beta$ and the transition rate $\delta$ on the resonance frequency ${\omega }_r$ are displayed numerically.