A two-strain model of infectious disease spread with asymmetric temporary immunity periods and partial cross-immunity

Matthew D. Johnston; Bruce Pell; David A. Rubel; Matthew D. Johnston; Bruce Pell; David A. Rubel

doi:10.3934/mbe.2023718

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 9: 16083-16113. doi: 10.3934/mbe.2023718

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A two-strain model of infectious disease spread with asymmetric temporary immunity periods and partial cross-immunity

Department of Mathematics + Computer Science, Lawrence Technological University, 21000 W 10 Mile Rd, Southfield, MI 48075, USA

Academic Editor: Raluca Eftimie

Received: 26 June 2023 Revised: 26 July 2023 Accepted: 01 August 2023 Published: 08 August 2023

We introduce a two-strain model with asymmetric temporary immunity periods and partial cross-immunity. We derive explicit conditions for competitive exclusion and coexistence of the strains depending on the strain-specific basic reproduction numbers, temporary immunity periods, and degree of cross-immunity. The results of our bifurcation analysis suggest that, even when two strains share similar basic reproduction numbers and other epidemiological parameters, a disparity in temporary immunity periods and partial or complete cross-immunity can provide a significant competitive advantage. To analyze the dynamics, we introduce a quasi-steady state reduced model which assumes the original strain remains at its endemic steady state. We completely analyze the resulting reduced planar hybrid switching system using linear stability analysis, planar phase-plane analysis, and the Bendixson-Dulac criterion. We validate both the full and reduced models with COVID-19 incidence data, focusing on the Delta (B.1.617.2), Omicron (B.1.1.529), and Kraken (XBB.1.5) variants. These numerical studies suggest that, while early novel strains of COVID-19 had a tendency toward dramatic takeovers and extinction of ancestral strains, more recent strains have the capacity for co-existence.
- epidemiology,
- mathematical model,
- multi-strain,
- cross-immunity,
- reproduction number
Citation: Matthew D. Johnston, Bruce Pell, David A. Rubel. A two-strain model of infectious disease spread with asymmetric temporary immunity periods and partial cross-immunity[J]. Mathematical Biosciences and Engineering, 2023, 20(9): 16083-16113. doi: 10.3934/mbe.2023718

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Abstract

We introduce a two-strain model with asymmetric temporary immunity periods and partial cross-immunity. We derive explicit conditions for competitive exclusion and coexistence of the strains depending on the strain-specific basic reproduction numbers, temporary immunity periods, and degree of cross-immunity. The results of our bifurcation analysis suggest that, even when two strains share similar basic reproduction numbers and other epidemiological parameters, a disparity in temporary immunity periods and partial or complete cross-immunity can provide a significant competitive advantage. To analyze the dynamics, we introduce a quasi-steady state reduced model which assumes the original strain remains at its endemic steady state. We completely analyze the resulting reduced planar hybrid switching system using linear stability analysis, planar phase-plane analysis, and the Bendixson-Dulac criterion. We validate both the full and reduced models with COVID-19 incidence data, focusing on the Delta (B.1.617.2), Omicron (B.1.1.529), and Kraken (XBB.1.5) variants. These numerical studies suggest that, while early novel strains of COVID-19 had a tendency toward dramatic takeovers and extinction of ancestral strains, more recent strains have the capacity for co-existence.

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