Modeling the Effects of Meteorological Factors and Unreported Cases on Seasonal Influenza Outbreaks in Gansu Province, China

Abstract

Influenza usually breaks out seasonally in temperate regions, especially in winter, infection rates and mortality rates of influenza increase significantly, which means that dry air and cold temperatures accelerate the spread of influenza viruses. However, the meteorological factors that lead to seasonal influenza outbreaks and how these meteorological factors play a decisive role in influenza transmission remain unclear. During the epidemic of infectious diseases, the neglect of unreported cases leads to an underestimation of infection rates and basic reproduction number. In this paper, we propose a new non-autonomous periodic differential equation model with meteorological factors including unreported cases. First, the basic reproduction number is obtained and the global asymptotic stability of the disease-free periodic solution is proved. Furthermore, the existence of periodic solutions and the uniformly persistence of the model are demonstrated. Second, the best-fit parameter values in our model are identified by the MCMC algorithm on the basis of the influenza data in Gansu province, China. We also estimate that the basic reproduction number is 1.2288 (95% CI:(1.2287, 1.2289)). Then, to determine the key parameters of the model, uncertainty and sensitivity analysis are explored. Finally, our results show that influenza is more likely to spread in low temperature, low humidity and low precipitation environments. Temperature is a more important factor than relative humidity and precipitation during the influenza epidemic. In addition, our results also show that there are far more unreported cases than reported cases.

Appendix A

See Figs. 14 and 15.

Fig. 14

Numerical simulation of the global stability of the disease-free periodic solution $P_{0}$, where the values of the parameters are $\beta (t)=6.4\times 10^{-8}+8\times 10^{-10}\sin (\frac{\pi }{6}t+2)$, $\theta =0.3$, $\delta =0.2$, $\rho (t)=6.4\times 10^{-9}+9\times 10^{-10}\sin (\frac{\pi }{6}t+2)$, $\alpha =590$, $\mu (t)=32+20\sin (\frac{\pi }{6}t+2)$, $\kappa =0.09$, ${\varLambda }=25813$, $d=1/(73\times 12)$, $q=0.9$, $\sigma =30/4$, $\gamma _{1}=30/7$ and $\gamma _{2}=30/10$. The initial value of model (1) is $(S(0), E(0), I_{C}(0), I_{N}(0), R(0), V(0)) = (22,000,000, 500, 500, 1000, 2000, 20,00,000)$. According to this set of parameters, we get $R_{0}=0.99127<1$ (For an explanation of the reference to color in this illustration, the reader is referred to the Web version of this article)

Fig. 15

The existence of the positive periodic solution, where the values of the parameters are $\beta (t)=7\times 10^{-7}+8\times 10^{-8}\sin (\frac{\pi }{6}t+2)$, $\theta =0.3$, $\delta =0.2$, $\rho (t)=1\times 10^{-7}+9\times 10^{-8}\sin (\frac{\pi }{6}t+2)$, $\alpha =590$, $\mu (t)=32+20\sin (\frac{\pi }{6}t+2)$, $\kappa =0.09$, ${\varLambda }=25813$, $d=1/(73\times 12)$, $q=0.9$, $\sigma =30/4$, $\gamma _{1}=30/7$ and $\gamma _{2}=30/10$. The initial value of model (1) is $(S(0), E(0), I_{C}(0), I_{N}(0), R(0), V(0)) = (10,000,000, 500, 500, 1000, 2000, 20,00,000)$. According to this set of parameters, we get $R_{0}=15.4955>1$ (For an explanation of the reference to color in this illustration, the reader is referred to the Web version of this article) (Colour figure online)