Further dynamic analysis for a network sexually transmitted disease model with birth and death

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Highlights

  • The global stability of the endemic equilibrium is further investigated.

  • The final size relation for a network disease model is determined.

  • The asymptotic behavior of final size equations is analyzed in detail.

Abstract

In this paper, we further study a network sexually transmitted disease model with birth and death in detail. For 0 < p < 1, we prove that the endemic equilibrium of the model is globally asymptotically stable by using suitable Lyapunov functions. Specifically, for the permanent immunity case (δ=0), we establish the conclusion by applying a graph-theoretical result; for the waning immunity case (δ > 0), if the recovery rates of high-risk infected individuals and low-risk infected individuals are equal, we conclude the result from performing some mathematical techniques. Moreover, in the absence of birth and death, we use the model equations to determine the final size relation of a disease. In particular, we derive the final size relations and establish the asymptotic behavior of them. The results give a numerical algorithm to estimate the final size of a disease spreading in heterogeneous networks.

Introduction

As we know, network disease models assume that the population is classified as statistically equivalent groups based on the contact number of each individual (or the degree of each node), and recently they have attracted much attention, see, for example [1], [2], [3], [4], [5], [6]. On one hand, the exploration of the global dynamics of these models is continuing to be an important issue in network epidemiology. For the network SIS virus model first proposed by Pastor-Satorras and Vespignani [1], they obtained a propagation threshold and established the existence and uniqueness of equilibria, including virus-free equilibrium and virose equilibrium, but they did not determine the global behavior of the model from a mathematical point of view. Motivated by the approach to address a classical multi-group SIS model, d’Onofrio [7] first gave a mathematical proof of the global asymptotical stability of the virose equilibrium in the feasible region if R0 > 1. Later, Wang and Dai [8] provided another more skilled proof of the global asymptotical stability of the virose equilibrium by choosing suitable monotone sequences. However, the application of this monotone iteration method is closely related to the mathematical structure of a system, and it is not easy to extend it to other more sophisticated network epidemic models, especially for those with differential infectivity. By combining Lyapunov functions with an original graph-theoretical result [9], Cao et al. [5], [6] have discussed the global asymptotic stability of the endemic equilibrium for coupled network disease models. Zhang et al. [10] studied an SIS model in adaptive networks and determined the exact parameter space for degenerate Hopf bifurcation, homoclinic bifurcation and Bogdanov-Takens bifurcation.

In [11], Wang et al. discussed the global asymptotic stability of the equilibria of the following network sexually transmitted disease model with demography{dsk(t)dt=b(1ski1ki2krk)kskΘ(t)μsk+δrk,di1k(t)dt=(1p)kskΘ(t)γ1i1kμi1k,di2k(t)dt=pkskΘ(t)γ2i2kμi2k,drk(t)dt=γ1i1k+γ2i2kμrkδrk,k=1,,n,whereΘ(t)=1kj=1njP(j)(β1i1j+β2i2j),and for k=1,,n, the initial values are specified bysk(0)=sk0R+,i1k(0)=i1k0R+,i2k(0)=i2k0R+,rk(0)=rk0R+.Moreover, the model variables and parameters are explained as follows. For k=1,,n, sk, i1k, i2k and rk label the relative density of susceptible, low-risk infected, high-risk infected and recovered individuals of degree k, respectively. Parameters b, μ and δ signify the birth rate, death rate and decay rate of immunity, respectively; γ1 and γ2 describe the recovery rates of low-risk and high-risk infected individuals, respectively; β1 (β2) captures the infection rate between a low-risk (high-risk) infected individual and a susceptible individual (β1 < β2). The rate for taking no measures is denoted by p, such as the non-usage of condom or higher-risk sexual activities.

Summing the equations of system (1) according to degree k and noting thatlim supt+(sk(t)+i1k(t)+i2k(t)+rk(t))=b/(b+μ),we obtain the limit system of (1) as follows{dsk(t)dt=bμb+μβ1kskΘ1(t)β2kskΘ2(t)μsk+δrk,di1k(t)dt=(1p)(β1kskΘ1(t)+β2kskΘ2(t))(γ1+μ)i1k,di2k(t)dt=p(β1kskΘ1(t)+β2kskΘ2(t))(γ2+μ)i2k,drk(t)dt=γ1i1k+γ2i2k(μ+δ)rk,k=1,,n,where Θ1(t)=j=1njP(j)i1j(t)/k and Θ2(t)=j=1njP(j)i2j(t)/k. It is easy to verify that the compact regionΓ{(s1,i1k,i2k,rk,,sn,i1n,i2n,rn)R+4n:sk+i1k+i2k+rkbb+μ},k=1,2,,n, is a positive invariant set for system (1).

In [11], Wang et al. have claimed the following conclusion.

Theorem A

If R0 < 1, the disease-free equilibrium E0 is the only equilibrium of system (2) and it is globally asymptotically stable in Γ. If R0 > 1, E0 is unstable and system (2) has a unique endemic equilibrium E* in the interior of Γ. Furthermore, for R0 > 1, if p=0 or p=1, E* is globally asymptotically stable in the interior of Γ.

In this article, we discuss the global stability of endemic equilibrium for system (2) with 0 < p < 1, which is unsolved in [11]. Specifically, for the permanent immunity case (δ=0), the endemic equilibrium is shown to be globally asymptotically stable by combining a Lyapunov function with a graph-theoretical result; for the waning immunity case (δ > 0), if the recovery rates of high-risk infected individuals and low-risk infected individuals are equal (γ1=γ2=γ), the endemic equilibrium is shown to be globally asymptotically stable by combining a Lyapunov function with mathematical techniques.

On the other hand, in the absence of population demography, i.e., birth and death processes, we use system (2) to determine the final size relations. For the networks of regular or homogeneous type, that is, the degree distribution follows Delta or Poisson distribution, the final size relation of an epidemic may be understood similar to classical SIR model (see [12], [13], [14], [15], [16] and references cited therein). However, results on final size relation of an epidemic on heterogeneous networks are scarce [6], [17], [18], [19], [20]. Moreno et al. [17] first derived the final size relation of SIR epidemic on annealing networks and related the epidemic prevalence to network model parameters. Newman [18] used the percolation method to describe the outbreak size of SIR epidemic as the “giant component” of a quenched network, but this method could not track the time evolution of an epidemic. Miller [19] formulated low-dimensional (only two ordinary differential equations) edge-based SIR equations on quenched networks and showed that the final size relation was equivalent to that of Newman [18]. Zhang and Jin [20] calculated the final size relation of an epidemic on annealing networks with community structure from model equations, but they did not analyze the qualitative behavior in detail. Similarly, Cao et al. [6] established the final size relation of an epidemic with a carrier state on annealing networks based on model equations. To the best of our knowledge, there are no rigorous results on the asymptotic behavior for the final size of an epidemic on annealing networks, especially for a disease with differential infectivity.

This paper is organized as follows. In Section 2, we recall the basic reproduction number R0 and some known results on system (2). Section 3 is devoted to the global stability of the endemic equilibrium of system (2) for 0 < p < 1 and the final size equations of disease in the absence of birth and death processes. Specifically, in the first two sections of Section 3, we discuss the global stability of the endemic equilibrium using Lyapunov functions, while in the third part of Section 3, we use the model to estimate the final size and analyze its qualitative behavior. Some concluding remarks are given in Section 4.

Section snippets

Preliminaries

In this section, we give a brief introduction to mathematical results of model (2) that are established by Wang et al. [11].

Main results

In this section, under the assumption R0 > 1 and 0 < p < 1, we further discuss the global stability of E* for system (2) and provide two theorems utilizing the graph-theoretical result and mathematical techniques.

Concluding remarks

In this paper, by applying the Lyapunov functions to a network sexually transmitted disease model with birth and death, we have established further results on the global asymptotic stability of endemic equilibrium of system (2) with 0 < p < 1 (see Theorems 3.1 and 3.2), which is unsolved in [11]. Specifically, for the permanent immunity case (δ=0), a Lyapunov function and a graph-theoretical result are combined to prove the global asymptotic stability of endemic equilibrium; for the waning

Acknowledgments

This project was supported by (i) the National Natural Science Foundation of China under Grants 11801532, 61833005 and 11571326, (ii) the China Postdoctoral Science Foundation under Grant 2018M630490 and 2019T120372, (iii) the Hubei Provincial Natural Science Foundation of China under Grant 2018CFB260, (iv) the Fundamental Research Funds for the Central Universities, China University of Geosciences (Wuhan) (No. CUG170622).

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