Measuring Infection Transmission in a Stochastic SIV Model with Infection Reintroduction and Imperfect Vaccine
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Measuring Infection Transmission in a Stochastic SIV Model with Infection Reintroduction and Imperfect Vaccine M. Gamboa1 · M. J. Lopez‑Herrero1 Received: 14 February 2019 / Accepted: 28 December 2019 © Springer Nature B.V. 2020
Abstract An additional compartment of vaccinated individuals is considered in a SIS stochastic epidemic model with infection reintroduction. The quantification of the spread of the disease is modeled by a continuous time Markov chain. A well-known measure of the initial transmission potential is the basic reproduction number R0 , which determines the herd immunity threshold or the critical proportion of immune individuals required to stop the spread of a disease when a vaccine offers a complete protection. Due to repeated contacts between the typical infective and previously infected individuals, R0 overestimates the average number of secondary infections and leads to, perhaps unnecessary, high immunization coverage. Assuming that the vaccine is imperfect, alternative measures to R0 are defined in order to study the influence of the initial coverage and vaccine efficacy on the transmission of the epidemic. Keywords Stochastic Markovian epidemic · Imperfect vaccine · Basic reproduction number Mathematics Subject Classification 92D30 · 60J28 · 60J22
The authors would like to thank the referees and the editor for their useful comments and suggestions, which helped us to improve the manuscript. Financial support for this work was provided by the Government of Spain (Department of Science, Technology and Innovation) and the European Commission through project MTM 2014-58091-P. The first author is grateful for the economical support of Banco Santander and Universidad Complutense of Madrid (Pre-doctoral researcher contract CT42/18-CT43/18) * M. J. Lopez‑Herrero [email protected] M. Gamboa [email protected] 1
Faculty of Statistical Studies, Complutense University of Madrid, Madrid, Spain
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M. Gamboa, M. J. Lopez‑Herrero
1 Introduction Mathematical modeling is an essential tool to represent the progress of an epidemic through a population. It is commonly accepted that the work of Kermack and McKendrick (1927) is the prototype of almost all epidemiological models based on a classification of individuals by their epidemic status. Since then, many other compartmental models have been developed to investigate a diverse range of communicable diseases to obtain a better knowledge of their transmission mechanisms (Anderson and Britton 2000; Kretzschmar et al. 2001; Aguiar et al. 2010; Artalejo and Lopez-Herrero 2014; Liu et al. 2018). A common assumption is that the communicable disease spreads in a community of constant size. The population can be closed in the sense that infectious individuals can infect only other individuals within the population under study during the epidemic’s time span. However, assuming reintroduction of the disease through contact with infected individuals from other areas could represent a more realistic scenario (Marchette and Wierman 2004; S
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