Cross immunity protection and antibody-dependent enhancement in a distributed delay dynamic model

Vanessa Steindorf; Sergio Oliva; Jianhong Wu; Vanessa Steindorf; Sergio Oliva; Jianhong Wu

doi:10.3934/mbe.2022136

Mathematical Biosciences and Engineering

2022, Volume 19, Issue 3: 2950-2984. doi: 10.3934/mbe.2022136

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Cross immunity protection and antibody-dependent enhancement in a distributed delay dynamic model

1.
Mathematical and Theoretical Biology Group, Basque Center for Applied Mathematics, BCAM, Bilbao, Spain
2.
Applied Mathematics Department, Institute of Mathematics and Statistics, University of São Paulo, São Paulo, SP, Brazil
3.
Laboratory for Industrial and Applied Mathematics, Department of Mathematics and Statistics, Faculty of Science and Engineering, York University, Toronto, Canada

Received: 11 November 2021 Revised: 13 December 2021 Accepted: 11 January 2022 Published: 17 January 2022

Dengue fever is endemic in tropical and subtropical countries, and certain important features of the spread of dengue fever continue to pose challenges for mathematical modelling. Here we propose a system of integro-differential equations (IDE) to study the disease transmission dynamics that involve multi-serotypes and cross immunity. Our main objective is to incorporate and analyze the effect of a general time delay term describing acquired cross immunity protection and the effect of antibody-dependent enhancement (ADE), both characteristics of Dengue fever. We perform qualitative analysis of the model and obtain results to show the stability of the epidemiologically important steady solutions that are completely determined by the basic reproduction number and the invasion reproduction number. We establish the global dynamics by constructing a suitable Lyapunov functional. We also conduct some numerical experiments to illustrate bifurcation structures, indicating the occurrence of periodic oscillations for a specific range of values of a key parameter representing ADE.
- integro-differential equations,
- multi-strain model,
- temporary immunity,
- antibody-dependent enhancement,
- distributed delay
Citation: Vanessa Steindorf, Sergio Oliva, Jianhong Wu. Cross immunity protection and antibody-dependent enhancement in a distributed delay dynamic model[J]. Mathematical Biosciences and Engineering, 2022, 19(3): 2950-2984. doi: 10.3934/mbe.2022136

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Abstract

Dengue fever is endemic in tropical and subtropical countries, and certain important features of the spread of dengue fever continue to pose challenges for mathematical modelling. Here we propose a system of integro-differential equations (IDE) to study the disease transmission dynamics that involve multi-serotypes and cross immunity. Our main objective is to incorporate and analyze the effect of a general time delay term describing acquired cross immunity protection and the effect of antibody-dependent enhancement (ADE), both characteristics of Dengue fever. We perform qualitative analysis of the model and obtain results to show the stability of the epidemiologically important steady solutions that are completely determined by the basic reproduction number and the invasion reproduction number. We establish the global dynamics by constructing a suitable Lyapunov functional. We also conduct some numerical experiments to illustrate bifurcation structures, indicating the occurrence of periodic oscillations for a specific range of values of a key parameter representing ADE.

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