Dynamics of a time-delayed two-strain epidemic model with general incidence rates

https://doi.org/10.1016/j.chaos.2021.111527Get rights and content

Abstract

Two-strain time-delayed epidemic model with general incidence rates is suggested and studied in this paper. The model consists of four compartments that describe the interaction between the susceptible, the first strain infected individuals, the second strain infected ones and the recovered individuals. In order to interpret the infection incubation period for each strain, two time delays will be incorporated into the studied model. Our first mathematical study will concern the wellposedness of the suggested model in terms of the classical existence, positivity and boundedness results. In order to perform the global stability, four equilibria of the problem are given. The first one stands for the disease-free equilibrium, the second describes first strain endemic equilibrium, the third one represents the second strain equilibrium and the last one is called the both strains endemic equilibrium. It was established that the global stability of each equilibrium depends on the strain 1 basic reproduction number R01 and on the strain 2 basic reproduction number R02. Numerical simulations are performed with a various incidence functions, namely, bilinear, Beddington–DeAngelis, Crowley–Martin and non-monotonic incidence rates. The bifurcation analysis have been conducted depending on time delays. We will limit ourselves to the theoretical study of the Hopf bifurcation results. The numerical results are in good agreement with the theoretical results dealing with the equilibria stability. Moreover, it was revealed that the time-delays may play an essential role in changing the nature of the equilibria stability.

Introduction

Multi-strain infection models present nowadays an efficient tool to study and predict the dynamics of the infection under the presence of multiple acting strains. Indeed, many infections such as human immunodeficiency virus, dengue fever, tuberculosis and even the late coronavirus disease COVID-19 can be developed with the coexistence of two or more strains [1], [2], [3], [4], [5]. For instance, H1N1 flu virus infection is considered as a mutation of the seasonal influenza [6], [7] and COVID-19 is classified as a new strain of SARS-CoV-1 [8]. This process of mutation can generate other new strains, especially when there is still no discovered efficient drug. It will be worthy to notice that recent works [9], [10], [11], for instance, have point out the development of further new variants of the late COVID-19 pandemic. The study of COVID-19 mathematical model under the fractional order operators is also developed recently [12], [13], [14], [15]. Hence, modelling multi-strain diseases may permit to study the evolution of the infection and also to look for the different conditions allowing the coexistence of strains or to block their mutation processes.

To describe the infection latency during the long period of the disease, many works have considered the exposed class. Taking into account the infection incidence rates under either bilinear or nonlinear, the global dynamics of one-strain susceptible-exposed-infectious-recovered (SEIR) have been studied in Li and Muldowney [16], Li and Wang [17], Huang et al. [18]. The global analysis of two-strain SEIR model have been tackled in Bentaleb and Amine [19] by considering bilinear incidence function for the first strain and non-monotonic incidence rate for the second strain. In a recent work, the same issue was studied in Meskaf et al. [20] by taking into account two non-monotonic incidence rates. More recently, a multi-train SEIR epidemic model with generalized incidence rates was fullfilled in Khyar and Allali [5]. The authors perform the global stability of the equilibria and compare the numerical simulation with some clinical data.

Another mean to describe the disease long period of incubation is the infection time-delay. In [21] the local stability and the Hopf bifurcation analysis are investigated. The global dynamics of a delayed two-strain epidemic model have been studied in Xu et al. [22]. The authors have included two time delays describing the infection incubation periods for the two strains. Also the authors have considered that the two incidence rates are bilinear and only one strain subjected to vaccination process. The same issue was tackled in Kaymakamzade and Hincal [23], Chen and Xu [24] by considering two vaccinations instead of one. A delayed diffusive two-strain disease model was studied by Chen and Xu [25] by taking into account two bilinear incidence rate and by incorporating diffusion terms in all problem compartments. Knowing that the incidence rates can give more information about the disease infection, modelling with generalized incidences can be more appropriate to study the infection dynamics. In this context and motivated by the previous works, we will consider in our study a delayed two-strain epidemic model under the effect of generalized incidence rates. Hence, we will consider the following nonlinear system of differential equations in order to describe our problem:{dS(t)dt=Λg(S(t);I1(t))I1(t)h(S(t);I2(t))I2(t)δS(t),dI1(t)dt=g(S(tτ1);I1(tτ1))I1(tτ1)eδτ1(δ+d1+μ1)I1(t),dI2(t)dt=h(S(tτ2);I2(tτ2))I2(tτ2)eδτ2(δ+d2+μ2)I2(t),dR(t)dt=μ1I1(t)+μ2I2(t)δR(t),where S represents the susceptible individuals, I1 the first strain infected individuals, I2 the second strain infected individuals and R is the removed individuals. The parameters of problem are described as follows, Λ is the recruitment rate, δ is the death rate of the population, μ1(respectively,μ2) is the transfer rates from first strain (respectively, second strain) infected individuals to recovered, di(i=1,2) is in the infection-induce death rate of the i strain (i=1,2); g(S;I1) is the rate of transmission of susceptible individuals to strain 1 infected individuals, h(S;I2) is the rate of transmission of susceptible individuals to strain 2 infected individuals, τ1 is the time delay describing the strain 1 incubation period, τ2 is the time delay describing the strain 2 incubation period. The terms eδτ1 and eδτ2 stand for the each strain individual survival probability from time tτ1 to time t and from time tτ2 to time t, respectively.

The incidence rates functions g(S;I1) and h(S;I2) are assumed to be continuously differentiable in the interior of R+2 and satisfies the following hypotheses as in Khyar and Allali [5], Hattaf et al. [26]:g(0;I1)=0,h(0;I2)=0,Ii0,i=1,2.gS(S;I1)>0,hS(S;I2)>0,S>0,Ii0,i=1,2,gI1(S;I1)0,hI2(S;I2)0,S0,Ii0,i=1,2,The three above properties (H1), (H2) and (H3), for the both functions g and h, are straightforwardly verified by many classical biological incidence rates such as the bilinear incidence function αS [27], [28], [29], the saturated incidence function αS1+σ1S or αS1+σ2I [30], [31], Beddington–DeAngelis incidence function αS1+σ1S+σ2I [32], [33], [34], Crowley–Martin incidence function αS1+σ1S+σ2I+σ1σ2SI [35], [36], [37] and non-monotone incidence function αS1+σI2 [38], [39], [40], [41], [42]. The flowchart of our two-strain time-delayed epidemiological model is illustrated in Fig. 1.

The paper is organized as follows. The next section will be devoted to the wellposedness of our problem in terms of proving the positivity and the boundedness results. The steady states will be given in the same section depending on the strain basic reproduction numbers. The global stability of the equilibria is fulfilled in Section 3 by choosing adequate Lyapunov functionals. The Hopf bifurcation analysis is devoted in Section 4. Section 5 will be dedicated to some numerical simulations in order to confirm our theoretical findings. The last section will conclude the present work.

Section snippets

Wellposedness and steady states

In this section, we will establish the wellposedness of our problem in terms of demonstrating the existence, positivity and boundedness results. Moreover, we will show that there exists a disease-free equilibrium point and three endemic equilibrium points namely, the first strain endemic equilibrium, the second strain equilibrium and the both strains endemic equilibrium.

Global stability of the problem equilibria

In this section, we will prove the global stability of the equilibrium points. To this end, some suitable Lyapunov functionals will be used to prove the global stability.

Hopf bifurcation analysis

In this section, we will analyze the bifurcation at the both strains endemic equilibrium. The characteristic equation of the model (1.1) at the both strains endemic equilibrium Et is as follows:λ4+A3λ3+A2λ2+A1λ+A0+(B3λ3+B2λ2+B1λ+B0)eλτ1+(C3λ3+C2λ2+C1λ+C0)eλτ2+(D2λ2+D1λ+D0)eλ(τ1+τ2)=0, where

  • A3=M+N+a+b+2δ,

  • A2=δ2+(2δ+M+N)(a+b)+ab,

  • A1=(M+N+δ)[δ(a+b)+ab]+abδ,

  • A0=abδ(δ+M+N),

  • B3=(PQ)eδτ1,

  • B2=(PQ)(N+b+2δ)eδτ1,

  • B1=(PQ)(δ2+Nb+2bδ+Nδ)eδτ1,

  • B0=(PQ)(bδ2+Nbδ)eδτ1,

  • C3=(RT)eδτ2,

  • C2=(RT)(M+a+2δ)eδτ2,

Numerical simulations

Numerical simulations will be performed in this section in order to check numerically the stability of all the equilibria and to confirm our theoretical results. To this end, different incidence functions will be taken into consideration to have a wide view on the infection dynamics. More precisely, numerical simulations will be fulfilled for four cases, the first one is to consider the problem (1.1) under the simplest two bilinear incidence rates g(S,I1)=α1S and h(S,I2)=α2S, the second case

Conclusion

In this work, we have studied a time-delayed two-strain infection model with the presence of two generalized incidence rates. First, the well-posedness of our problem is established in terms of proving the existence, the positivity and boundedness of solution. The problem have four steady states namely the disease-free equilibrium, the first strain endemic equilibrium, the second strain endemic equilibrium and the both strain endemic equilibrium. Next, the global stability of the problem have

Declaration of Competing Interest

Authors declare that they have no conflict of interest.

References (49)

  • J.J. Wang et al.

    Analysis of an SIR model with bilinear incidence rate

    Nonlinear Anal Real World Appl

    (2010)
  • X. Liu et al.

    Stability analysis of an SEIQV epidemic model with saturated incidence rate

    Nonlinear Anal Real World Appl

    (2012)
  • Y. Zhao et al.

    The threshold of a stochastic SIRS epidemic model with saturated incidence

    Appl Math Lett

    (2014)
  • R.S. Cantrell et al.

    On the dynamics of predator-prey models with the Beddington–DeAngelis functional response

    J Math Anal Appl

    (2001)
  • V. Capasso et al.

    A generalization of the Kermack–Mckendrick deterministic epidemic model

    Math Biosci

    (1978)
  • S. Ruan et al.

    Dynamical behavior of an epidemic model with a nonlinear incidence rate

    J Differ Equ

    (2003)
  • P. Van den Driessche et al.

    Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission

    Math Biosci

    (2002)
  • X. Li et al.

    On the zeros of a fourth degree exponential polynomial with applications to a neural network model with delays

    Chaos Solitons Fractals

    (2005)
  • K. Hattaf

    Global stability and Hopf bifurcation of a generalized viral infection model with multi-delays and humoral immunity

    Phys A

    (2020)
  • J.E. Golub et al.

    Delayed tuberculosis diagnosis and tuberculosis transmission

    Int J Tuber

    (2006)
  • J.M. Brenchley et al.

    Microbial translocation is a cause of systemic immune activation in chronic HIV infection

    Nat Med

    (2006)
  • D.J. Gubler

    Dengue and dengue hemorrhagic fever

    Clin Microbiol Rev

    (1998)
  • A.E. Gobalenya et al.

    The species severe acute respiratory syndrome-related coronavirus: classifying 2019- nCoV and naming it SARS-CoV-2

    Nat Microbiol

    (2020)
  • O. Khyar et al.

    Global dynamics of a multi-strain SEIR epidemic model with general incidence rates: application to COVID-19 pandemic

    Nonlinear Dyn

    (2020)
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