Extreme values in SIR epidemic models with two strains and cross-immunity

J. Amador; D. Armesto; A. Gómez-Corral; J. Amador; D. Armesto; A. Gómez-Corral

doi:10.3934/mbe.2019098

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 4: 1992-2022. doi: 10.3934/mbe.2019098

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Research article

Extreme values in SIR epidemic models with two strains and cross-immunity

1.
School of Statistical Studies, Complutense University of Madrid, Madrid 28040, Spain
2.
School of Mathematical Sciences, Complutense University of Madrid, Madrid 28040, Spain
3.
Instituto de Ciencias Matem´aticas CSIC-UAM-UC3M-UCM, Calle Nicolás Cabrera 13-15, Campus de Cantoblanco, Madrid 28049, Spain

Received: 11 January 2018 Accepted: 16 February 2019 Published: 08 March 2019

The paper explores the dynamics of extreme values in an SIR (susceptible $\to$ infectious $\to$ removed) epidemic model with two strains of a disease. The strains are assumed to be perfectly distinguishable, instantly diagnosed and each strain of the disease confers immunity against the second strain, thus showing total cross-immunity. The aim is to derive the joint probability distribution of the maximum number of individuals simultaneously infected during an outbreak and the time to reach such a maximum number for the first time. Specifically, this distribution is analyzed by distinguishing between a global outbreak and the local outbreaks, which are linked to the extinction of the disease and the extinction of particular strains of the disease, respectively. Based on the mass function of the maximum number of individuals simultaneously infected during the outbreak, we also present an iterative procedure for computing the final size of the epidemic. For illustrative purposes, the two-strain SIR-model with cross-immunity is applied to the study of the spread of antibiotic-sensitive and antibiotic-resistant bacterial strains within a hospital ward.
- epidemics,
- extreme values,
- final size,
- multi-type SIR-model,
- QBD process
Citation: J. Amador, D. Armesto, A. Gómez-Corral. Extreme values in SIR epidemic models with two strains and cross-immunity[J]. Mathematical Biosciences and Engineering, 2019, 16(4): 1992-2022. doi: 10.3934/mbe.2019098

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Abstract

The paper explores the dynamics of extreme values in an SIR (susceptible $\to$ infectious $\to$ removed) epidemic model with two strains of a disease. The strains are assumed to be perfectly distinguishable, instantly diagnosed and each strain of the disease confers immunity against the second strain, thus showing total cross-immunity. The aim is to derive the joint probability distribution of the maximum number of individuals simultaneously infected during an outbreak and the time to reach such a maximum number for the first time. Specifically, this distribution is analyzed by distinguishing between a global outbreak and the local outbreaks, which are linked to the extinction of the disease and the extinction of particular strains of the disease, respectively. Based on the mass function of the maximum number of individuals simultaneously infected during the outbreak, we also present an iterative procedure for computing the final size of the epidemic. For illustrative purposes, the two-strain SIR-model with cross-immunity is applied to the study of the spread of antibiotic-sensitive and antibiotic-resistant bacterial strains within a hospital ward.

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