Mathematical Analysis of 2-Strain Endemic Equilibrium using a SEIR Model with Behavioral Effect

Abdulvahid H. Hasmani; Viralkumar D. Patel

doi:10.22342/jims.v32i2.2106

From Equations to Action: Mathematical Models Driving Epidemics Control Strategies

Abdulvahid H. Hasmani ⁽¹⁾ , Viralkumar D. Patel ⁽²⁾

(1) Department of Mathematics, Sardar Patel University, India,

(2) Department of Mathematics, Shri Alpesh N. Patel Post Graduate Institute of Science & Research, India

https://doi.org/10.22342/jims.v32i2.2106

Issue
Vol. 32 No. 2 (2026): JUNE

Submitted
June 23, 2025

Accepted
February 15, 2026

Published
May 11, 2026

Keywords:

multi-strain epidemic dynamics, behavioural heterogeneity, Lyapunov stability, strain coexistence, principle of competitive exclusion

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Abstract

Multi-strain epidemic dynamics have been analyzed using two-strain SEIR models. We have taken into consideration a two-strain SEIR model that incorporates behavioural alterations for strain 2 among the susceptibles, with non-monotone incidence for strain 2 and bilinear incidence for strain 1. We have focused on the co-dynamics of both strains and analytically derived a sufficient condition, $1+\mathcal{R}_k^*<\mathcal{R}_{01}<\mathcal{R}_{02}$, for the existence of the two-strain endemic equilibrium, where $\mathcal{R}_{01}$ and $\mathcal{R}_{02}$ are basic reproduction numbers of respective strains and, $\mathcal{R}_k^*>0$ is a value derived from endemic state of the model. Under this sufficient condition, we establish the global stability of the two-strain endemic equilibrium using the Lyapunov function method in terms of reproduction numbers and behavioural parameter $k$. Additionally, the condition, $\mathcal{R}_{02}=\mathcal{R}_{01}>1$, result in instability of the two-strain endemic equilibrium. Lastly, to support these analytical results regarding the existence and stability of the two-strain endemic equilibrium, we have presented numerical simulations. Interestingly, the study reveals that the principle of competitive exclusion relaxing in this model, as behavioural heterogeneity among susceptibles effectively partitions the population. This partitioning reduces inter-strain competition and enables strain coexistence even when $ \mathcal{R}_{02} > \mathcal{R}_{01} > 1 $.

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Authors

Abdulvahid H. Hasmani

Department of Mathematics, Sardar Patel University

Viralkumar D. Patel

Department of Mathematics, Shri Alpesh N. Patel Post Graduate Institute of Science & Research

atviralmaths@gmail.com (Primary Contact)

Hasmani, A. H., & Patel, V. D. (2026). Mathematical Analysis of 2-Strain Endemic Equilibrium using a SEIR Model with Behavioral Effect. Journal of the Indonesian Mathematical Society, 32(2), 2106. https://doi.org/10.22342/jims.v32i2.2106

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1. INTRODUCTION

Mathematical modelling of infectious diseases has evolved over more than two centuries, beginning with early formal attempts by Daniel Bernoulli, who analysed smallpox dynamics in 1766 [1,2], and followed by William Farr’s statistical studies of cholera outbreaks in 1852 [3,2]. Later, Ronald Ross developed a threshold-based compartmental framework for malaria transmission between 1908 and 1911 [4,5], providing a conceptual foundation for modern infectious-disease modelling. This foundation was significantly strengthened in 1927 by Kermack and McKendrick, who introduced the classical SIR model and the epidemic threshold theorem [6,7,8]. Subsequent extensions, including SEIR-type formulations incorporating latent periods [9,10], and the formalization of the modern use of the basic reproduction number $(R_0)$ by Dietz, Anderson, and May[11], shaped contemporary epidemiological theory.

Despite major advances in the 21st century, infectious diseases continue to impose substantial global burdens. The recent COVID-19 pandemic illustrated the dificulty of managing multi-strain diseases driven by emergent variants[12]. Similar mutational mechanisms are observed in diseases such as tuberculosis, dengue, and HIV [13,14,15]. Modelling these systems requires accounting for multi-strain structure and incidence mechanisms that incorporate infectiousness, behavioural responses, public awareness, and healthcare availability.

Diferent strains often vary in both their transmissibility and their perceived severity, leading to behavioural heterogeneity among susceptible individuals. As a result, distinct incidence functions may more realistically capture strain-specific transmission. The literature documents several incidence functions, including bilinear[16], saturated [17,18], Beddington–DeAngelis [19,20], Crowley–Martin [21,22], and non-monotone structures [23,24,25,26].

Researchers have therefore developed multi-strain SEIR models. Some studies [27,28,29] assume bilinear incidence rate for all strains, while others [30,31] incorporate non-monotone incidence functions. A recent study[32]considers general incidence forms for all strains and derives necessary conditions for equilibria and global stability under certain structural assumptions. However, the epidemiological interpretation of these assumptions is not clearly articulated, and suficient conditions for the existence of equilibria are not explicitly established. Moreover, all of these studies efectively impose homogeneous behavioural responses across strains, which limits their ability to capture realistic behaviour-driven diferences in transmission. Despite these advances, there remains no general framework that provides behaviourally interpretable, mathematically rigorous conditions for the existence and stability of two-strain endemic equilibria under heterogeneous incidence structures.

Behavioural heterogeneity plays an important role in multi-strain outbreaks. Populations often react more cautiously to newly emerged or mutated strains because of uncertainty and perceived risk, whereas older, better-understood strains tend to elicit weaker precautionary responses. This motivates modelling the original strain using bilinear incidence and the mutated strain using non-monotone incidence. Such a formulation captures realistic, behaviour-driven diferences in contact patterns. A model of this type is analysed in[33]. For such multi-strain diseases, understanding strain coexistence requires analysing the co-dynamics before either strain is eradicated. Thus, the study of the two-strain endemic equilibrium is a foundational step.

The authors in[33]show that a two-strain (co-existence) endemic equilibrium exists and is globally stable if $\mathcal { R } _ { 0 1 } = \mathcal { R } _ { 0 2 } > 1$. However, because the strains difer in transmission mechanisms and behavioural responses, and because the condition does not involve any behavioural parameter, this symmetry requirement is rarely satisfied in realistic settings. Motivated by this, we re-examine the co-dynamics of a two-strain SEIR model incorporating behavioural heterogeneity and derive general, epidemiologically interpretable conditions for the existence and stability of a two-strain endemic equilibrium.

In this article, we study a two-strain SEIR model in which one strain follows bilinear incidence and the other a non-monotone incidence rate reflecting behavioural efect. The key contributions of this manuscript are: (i) the development of mathematically rigorous, necessary and suficient conditions for the existence of the two-strain endemic equilibrium; (ii) the formulation of global stability criteria expressed in terms of the basic reproduction numbers and the behavioural parameter; and (iii) the epidemiological interpretation of these analytical conditions. Finally, all analytical results are supported by numerical simulations to verify consistency between theory and model behaviour.

2. Model Description

We consider a two-strain SEIR model, as illustrated inFigure 1, with a bilinear incidence rate for strain 1 and a non-monotone incidence rate for strain 2 to reflect behavioural changes among susceptible individuals. The model consists of six compartments: susceptible $(S)$, exposed to strain-1 and strain-2 $\left( E _ { 1 } , E _ { 2 } \right)$， infectious with strain-1 and strain-2 $( I _ { 1 } , I _ { 2 } )$, and recovered $(R)$, described inTable 1; following the structure in[33]with a key modification.

To capture behavioural responses to strain–2, we adopt the non-monotone incidence function $\frac {\alpha_ {2} S I _ {2}}{1 + k I _ {2} ^ {2}},$

with $k > 0$. Since behavioural responses to infection typically intensify in a nonlinear manner as disease signals become more prominent, the quadratic term represents the minimal nonlinear form capable of reflecting this efect. It provides a stronger-than-linear reduction in transmission while avoiding the unnecessary complexity, instability, or steepness introduced by cubic or higher-order exponents. In[33], the parameter

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Mathematical Analysis of 2-Strain Endemic Equilibrium using a SEIR Model with Behavioral Effect

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