Investigating seasonal disease emergence and extinction in stochastic epidemic models

Seasonal disease outbreaks are common in many infectious diseases such as seasonal influenza, Zika, dengue fever, Lyme disease, malaria, and cholera. Seasonal outbreaks are often due to weather patterns affecting pathogens or disease-carrying vectors or by social behavior. We investigate disease emergence and extinction in seasonal stochastic epidemic models. Specifically, we study disease emergence through seasonally varying parameters for transmission, recovery, and vector births and deaths in time-nonhomogeneous Markov chains for SIR, SEIR, and vector-host systems. A branching process approximation of the Markov chain is used to estimate the seasonal probabilities of disease extinction. Several disease outcome measures are used to compare the dynamics in seasonal and constant environments. Numerical investigations illustrate and confirm previous results derived from stochastic epidemic models. Seasonal environments often result in lower probabilities of disease emergence and smaller values of the basic reproduction number than inconstant environments, and the time of peak emergence generally precedes the peak time of the seasonal driver. We identify some new results when both transmission and recovery vary seasonally. If the relative amplitude of the recovery exceeds that of transmission or if the periodicity is not synchronized in time, lower average probabilities of disease emergence occur in a constant environment than in a seasonal environment. We also investigate the timing of vector control. This investigation provides new methods and outcome measures to study seasonal infectious disease dynamics and offers new insights into the timing of prevention and control.