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Mathematical Analysis of Epidemic Dynamics Under Periodic Transmission Patterns

Author(s) Neeraj Kumar
Country India
Abstract Periodic changes in disease transmission are critical to understand the temporal behaviour of infectious diseases. The dynamics of an epidemic can be affected by seasons, environmental conditions, population behavior, contact patterns, and periodic social activities, and hence, the transmission rate of an epidemic can vary systematically over time. This paper presents a mathematical analysis of epidemic dynamics under periodic transmission patterns for a non-autonomous compartmental modelling framework. The transmission coefficient is represented as a positive periodic function to allow for recurrent changes in the transmission of the disease and to still keep the basic complexity of classical epidemic models. We consider the existence, positivity and boundedness of the epidemic solutions and study the behaviour of disease-free and endemic states under time-dependent transmission. In particular, we pay special attention to epidemic thresholds, local stability, disease persistence and extinction in periodically forced epidemic systems. We discuss the impact of transmission amplitude, periodicity, recovery rate and demographic parameters on the path of the epidemic. The mathematical framework suggests that one should distinguish between constant and periodically varying transmission since systems that have similar average transmission rates may have very different time-dependence. We introduce the concept of Floquet theory to understand the stability of periodic epidemic states and relate them to mathematical theories of recurrent outbreaks. We then discuss the importance of periodic transmission for vaccination, treatment and other time-dependent intervention strategies and solutions to the above problems. We find that the temporal periodicity of epidemic models provides a more flexible theoretical framework to interpret recurrent outbreaks, disease persistence, and long-term behaviour of infectious diseases.
Keywords Periodic transmission; Mathematical epidemiology; Epidemic dynamics; Non-autonomous epidemic models; SIR model; SEIR model; Seasonal forcing; Disease persistence; Disease extinction; Epidemic threshold; Periodic solutions; Floquet theory; Infectious diseases; Transmission dynamics; Mathematical modelling
Published In Volume 8, Issue 1, January-March 2017
Published On 2017-02-17
DOI https://doi.org/10.71097/IJSAT.v8.i1.11596
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Crossref DOI prefix of IJSAT is 10.71097/IJSAT

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