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First passage time and peak size probability distributions for a complex epidemic model

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Abstract

The potential of zoonotic diseases to cross-species physical boundaries makes them a persistent threat to global public health. Of all the zoonoses, the bubonic plague serves as a historical and modern model. In this work, we investigate a bubonic plague stochastic model using a continuous-time Markov chain (CTMC) model to study the disease dynamics in rats and humans. Using a Galton–Watson multi-type (GWbp) branching process, we have derived an analytical expression for disease extinction probability at the beginning of the epidemic. There is a variation in disease extinction probability calculated via the branching process and numerical simulations, which is the consequence of the discrete assumption of an infected flea instead of being considered in terms of an infected rat in the numerical simulation. An analytical expression for the distribution of first passage time (FPT) to spillover is also obtained in this work using the probability generating function (PGF) technique, and it agrees well with the FPT distribution obtained numerically. This distribution is defective since infected fleas can die before infecting human. Furthermore, the effect of the initial rat population on the FPT of spillover is also shown, which suggest that as the initial rat population increases, the likelihood of spillover decreases, and it increases as the rat population decreases. Additionally, we have derived an expression for the mean and variance of the first passage time to spillover, and we incorporate the impact of various parameters on the mean first passage time of spillover. Lastly, we have performed numerical simulations to estimate the peak size of each infected class and the related time to attain peak infection. The peak of the infected flea, which is a disease carrier, is much higher as compared to that of rat and human. Also, there is a delay in attaining peak size for the human population, which is the consequence of the human population being a secondary host for fleas to survive.

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Data Availability Statement

Data sharing is not applicable to this article as no new data were created or analysed in this study.

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Acknowledgements

Suman Kumari’s research is supported by research fellowship from MHRD, Government of India, Partha Sarathi Mandal’s research is supported by CSIR Project [File No: 25(0303)/19/EMR-II].

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  1. Department of Mathematics, NIT Patna, Patna, Bihar, 800005, India

    Suman Kumari, Partha Sarathi Mandal & Moitri Sen

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Correspondence to Partha Sarathi Mandal.

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Kumari, S., Mandal, P.S. & Sen, M. First passage time and peak size probability distributions for a complex epidemic model. Eur. Phys. J. Plus 139, 324 (2024). https://doi.org/10.1140/epjp/s13360-024-05108-z

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