On epidemic spreading in metapopulation networks with time-varying contact patterns

Considering that people may change their face-to-face communication patterns with others depending on the season, we propose an epidemic model that incorporates a time-varying contact rate on a metapopulation network and its second-neighbor network. To describe the time-varying contact mode, we utilize a switched system and define two forms of the basic reproduction number corresponding to two different restrictions. We provide the theoretical proof for the stability of the disease-free equilibrium and confirm periodic stability conditions using simulations. The simulation results reveal that as the period of the switched system lengthens, the amplitude of the final infected density increases; however, the peak infected density within a specific period remains relatively unchanged. Interestingly, as the basic reproduction number grows, the amplitude of the final infected density within a period gradually rises to its maximum and then declines. Moreover, the contact rate that occupies a longer duration within a single period has a more significant influence on epidemic spreading. As the values of different contact rates progressively increase, the recovery rate, natural birth rate, and natural death rate all decrease, leading to a larger final infection density.