# Global Dynamics of a Generalized SIRS Epidemic Model with Constant Immigration

Authors: Qianqian Cui; Qinghui Du; Li Wang
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

In this paper, we discuss the global dynamics of a general susceptible-infected-recovered-susceptible (SIRS) epidemic model. By using LaSalle’s invariance principle and Lyapunov direct method, the global stability of equilibria is completely established. If there is no input of infectious individuals, the dynamical behaviors completely depend on the basic reproduction number. If there exists input of infectious individuals, the unique equilibrium of model is endemic equilibrium and is globally asymptotically stable. Once one place has imported a disease case, then it may become outbreak after that. Numerical simulations are presented to expound and complement our theoretical conclusions.


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## Body

## 1. Introduction

In 1927, Kermack and McKendrick [1] proposed the compartment dynamic model, which uses differential equations research the dissemination rule of the infectious diseases. Since then, infectious disease dynamics has been developing vigorously and produced some important theoretical and practical valuable results. In classical Kermack-Mckendrick model, the whole population is divided into three subclasses: susceptible, infected, and recovered, which is also called SIR-type model. This type of model always assumed that the recovered individuals have permanent immunity. In fact, a plenty of diseases just have only a short period of acquired immunity, such as flu, tuberculosis, Hepatitis B, schistosomiasis, and the common cold. When acquired immunity disappears after some time, the recovered individuals become susceptible again [2]. To incorporate this phenomenon, SIRS epidemic models have emerged.According to the total population, the SIRS epidemic model can be divided into two kinds: one kind is with constant total population, and the other kind is with varying total population. Theoretically, the analysis on basic properties and global asymptotic stability of SIRS model with varying total population size is more difficult. A large portion of works focus on some special cases of this type of SIRS epidemic model (see [3–5] and the references therein). Mena-Lorca and Hethcote [2] discussed an SIRS epidemic model with simple mass action incidence and constant immigration. They proposed the following open question: when the basic reproduction number is greater than one, the unique endemic equilibrium is globally asymptotically stable. Ma et al. also proposed the same open question in [6].Nowadays, these open questions were gradually being solved. Li et al. [7] tried to study the global dynamical behaviors of SIRS model with incidence rate βNSI and constant immigration. They did obtain the global asymptotic stability of endemic equilibrium; however, this conclusion is valid only in the cases satisfying certain conditions but is not generally true. It is worth celebrating that the global asymptotic stabilities for classical SIRS model with general incidence rate are completely derived in [3, 8]. The method they used is LaSalle’s invariance principle and Lyapunov direct method. This is the most common method to study the local and global asymptotic stability of epidemic models. By using this method, Ma et al. [9] studied the local and global asymptotic stability of a SIQR model. Cui et al. [10] investigated the transmission dynamics of an epidemic model with vaccination, treatment, and isolation. Therefore, we wonder whether the global stability of SIRS model with general incidence rate and constant immigration can be obtained through this method or not.Motivated by these studies, in this paper, we considered an SIRS epidemic model with constant immigration, especially with immigration of infected individuals. Notice that transmission of infection occurs after adequate contacts between the susceptible and the infectious. To better simulate this phenomenon, we considered a general incidence rategIS. Like in [8], function gI is assumed as a real locally Lipschitz function on 0,+∞ and satisfied:(i)
g0=0 and gI>0 for I>0(ii)
limI⟶0gI/I=β>0, and gI/I is continuous and monotonously nonincreasing for I>0The same hypotheses on functiongI can be found in [11]. This type of incidence rate was first proposed by Capasso and Serio [12] to model the cholera epidemic spread in Bari in 1973. Since then, many authors studied the different special case of gI; for more details, one can see literature [11, 13–15].Due to the fact that the infectious individuals or the recovery individuals may move from one place to another, in this paper, we assume that the rate constant of input to the total population isA, of which fractions a,b, and ca+b+c=1anda,b,c≥0 attribute to susceptible, infected, and recovered compartments. That is, the infectious individuals immigrate to infectious class at time t at rate bA, and the recovery individuals immigrate to recovery class at time t at rate cA. The rate constant for nature death is d; thus, 1/d is the average lifetime. The disease-related death rate for infected individuals is α. After a period, the infected individuals might be declared cured of the disease and enter to the recovered class R at rate γ; thus, 1/γ is the average recovery time. Due to the fact that the recovered individuals can only have temporary immunity, thus, the recovered individuals might lose their immunity and move to the infected class I at rate δ. Therefore, the objective model in this paper can be given as(1)dSdt=aA−gIS+δR−dS,dIdt=bA+gIS−γ+d+αI,dRdt=cA+γI−δ+dR.In this paper, we mainly use LaSalle’s invariance principle and Lyapunov direct method to investigate the global dynamics of model (1). The whole paper is organized as follows. In section 2, the existence and local stability of each equilibrium will be stated. The globally asymptotically stable state of equilibria will be given in the next section. Numerical simulations and conclusion will be listed in the last two sections.


## 2. Basic Properties

The total population numberN=S+I+R and satisfies(2)dNdt=A−dN−αI≤A−dN,which implies that limsupt⟶∞Nt≤A/d. Therefore, the following domain(3)Γ=S,I,R∈ℝ+3∣S+I+R≤Ad.is the positively invariant set with respect to model (1).From the locally sign-preserving property of limit, we know thatlimI⟶0gI/I=β>0 implies gI/I=β in the small domain of zero. Then, the assumption gI/I is continuous and monotonously nonincreasing for I>0 implies that gI/I≤β for I>0, that is, gI≤βI for I≥0. As the assumption gI/I is continuous and monotonously nonincreasing and also implies that gI/I′≤0, a direct calculation derived that g′I≤gI/I. Then, it follows from g0=0 and limI⟶0gI/I=β that g′0+=limI⟶0+gI−g0/I−0=limI⟶0+gI/I=β. Therefore, from the assumptions (a) and (b) on gI, we have(4)gI≤βI,g′I≤gI/I,β=g′0+.To obtain the expressions of equilibria, we shall consider model (1) in two cases: one is b=0 and the other is b>0, which, respectively, imply model (1) without input of infectious individuals and with constant input of infectious individuals.Case 1.
b=0.Any equilibrium of model (1) must satisfy(5)aA−gIS+δR−dS=0,gIS−γ+d+αI=0,cA+γI−δ+dR=0.One can easily derive that model (1) always has a disease-free equilibrium E0=Aδ+ad/dδ+d,0,cA/δ+d. From van den Driessche and Watmough [16], we can derive that the basic reproduction number is(6)ℜ0=βAδ+addδ+dγ+d+α.Biologically, the basic reproduction numberℜ0 presents the average number of secondary infectious produced by single infective individuals, which are introduced into an entirely susceptible population.Except disease-free equilibrium (i.e., the solution of (5) with I=0), equations (5) may have one positive solution, which can be denoted by E∗=S∗,I∗,R∗. Then, a simple calculation derives the following:(7)S∗=γ+d+αI∗gI∗,R∗=cA+γI∗δ+d.AndI∗ is the nonzero solution of the function Φ(8)ΦI=dγ+d+αIgI+d+α+dγδ+dI−Aδ+adδ+d.It follows from assumption (ii) that(9)Φ′I=dγ+α+dIgI′+d+α+dγd+δ>0,and if,limI⟶0+ΦI=dγ+α+dβ−Aδ+adδ+d=dγ+α+dβ1−ℜ0<0.Moreover,(10)ΦAd=dγ+d+αA/dgA/d+d+α+dγδ+dAd−Aδ+adδ+d≥dγ+d+αA/dgA/d+α+dγδ+dAd>0.Therefore,ΦI has a unique positive root in 0,A/d if ℜ0>1.In sum, we have the following result.Theorem 1.
Whenb=0, we have the following:(i)
Model (1) always has a disease-free equilibrium E0(ii)
Ifℜ0>1, besides E0, model (1) also has a unique endemic equilibrium E∗S∗,I∗,R∗, where S∗andR∗ are given in (7) and the expression of I∗ is the unique positive root of ΦI given in (8)
By analyzing the Jacobian matrix, the local stability of equilibria can be given as follows.Theorem 2.
For model (1) with b=0, the disease-free equilibrium E0 is locally asymptotically stable when ℜ0<1, and it is unstable when ℜ0>1, while the unique endemic equilibrium E∗S∗,I∗,R∗ is locally asymptotically stable when ℜ0>1.Proof.
Based ong0=0 and g′0+=β, the Jacobian matrix at equilibrium E0 of model (1) can be given as(11)JE0=−d−βAδ+addδ+dδ0βAδ+addδ+d−γ+d+α00γ−δ+d.
One can easily obtain that the eigenvalues of the matrixJE0 are all less than zero when ℜ0<1. This implies that E0 is locally asymptotically stable. If ℜ0>1, the matrix JE0 has a positive eigenvalue, which implies that E0 is unstable.
For the endemic equilibriumE∗, the Jacobian matrix is given by(12)JE∗=−gI∗−d−S∗g′I∗δgI∗a2200γ−δ+d,where a22=S∗g′I∗−γ+d+α. The characteristic equation of JE∗ is(13)λ3+a1λ2+a2λ+a3=0,where(14)a1=gI∗+δ+2d−a22,a2=γ+α+dgI∗+γ+dgI∗+d−δ+2da22,a3=δ+dγ+α+dgI∗−δγgI∗−δ+dda22.
It follows from the expression ofS∗ and the conditions (4) of gI that a22≤0 if ℜ0>1. Hence, ai>0,i=1,2,3 and(15)a1a2−a3=gI∗−a22a2+δ+2da2−a3>gI∗−a22a2>0.
By Routh–Hurwitz criterion, we obtain that the endemic equilibriumE∗ is locally asymptotically stable if ℜ0>1.Case 2.
b>0.In this case, the existence and local stability of equilibrium for model (1) are stated as follows.Theorem 3.
Whenb>0, model (1) has only one equilibrium, which is the endemic equilibrium E^=S^,I^,R^, where(16)S^=N^−I^−R^,I^=A−dN^α,R^=cα+γA−γdN^αδ+dandN^is the unique solution of,gI^I^δ+dα+d+γdN−δ+cα+γ+dA=αδ+dγ+d+α−bαAA−dN.in the domain 0,A/d.Proof.
Due toN=S+I+R, any equilibrium of model (1) can be given as(17)bA+gIN−I−R−γ+d+αI=0,cA+γI−δ+dR=0,A−dN−αI=0.
One can easily verify that all the solutions of equation (17) can not contain I=0, which means that model (1) does not have disease-free equilibrium. Denote one solution of (17) by I,R,N=I^,R^,N^. Then, one can derive that(18)I^=A−dN^α,R^=cα+γA−γdN^αd+δ,
Substituting (18) into the first equation of (17), we have that N^ is the solution of(19)gI^I^δ+dα+d+γdN−δ+cα+γ+dA=αδ+dγ+d+α−bαAA−dN,
Next, we prove that equation (19) has only one solution in domain 0,A/d. Denote(20)ϕN=gI^I^δ+dα+d+γdN−δ+d+cα+γA,ψN=αδ+dγ+d+α−bαAA−dN.
Notice thatgI/I is continuous and monotone nonincreasing for I>0, that is, d/dIgI/I≤0. It follows from the expression of I^ in (18) that(21)ddNgII=ddIgIIdIdN=−dαddIgII≥0.
Therefore,ϕ′N≥0, which means that ϕN is a nondecreasing function of N in 0,A/d. Furthermore, it follows from the expression of I and limI⟶0+gI/I=β>0 that(22)limN⟶A−/dϕN=limI⟶0+gIIαA/d1−cd+δ=αβA/d1−cd+δ>0,limN⟶0+ϕN=−gA/dA/dδ+d+cα+γA<0.
Meanwhile, one can verify thatψ′N<0 in 0,A/d, limN⟶0+ψN>0, and limN⟶A/dψN=−∞. Therefore, there exists only one solution in domain 0,A/d for equation (19). This completes the proof.
Similar to the proof of Theorem2, the eigenvalues of Jacobian matrix JE^ are all negative, which implies the following theorem.Theorem 4.
Whenb>0, the unique endemic equilibrium E^ of model (1) is locally asymptotically stable.


## 3. Global Dynamics of Model (1)

For the global dynamical behaviors of model (1), we study them in two cases: b=0 and b>0, as discussed in section 2. To obtain the globally asymptotic stability of equilibria, we can rewrite model (1) as the following equivalent model:(23)dIdt=bA+gIN−I−R−γ+d+αI,dRdt=cA+γI−δ+dR,dNdt=A−dN−αI.The dynamical behaviors of model (1) and model (23) are consistent.We first discuss the global dynamics of model (1) in the case of b=0.Case 1.
b=0.In this case, by considering Lyapunov functionV=I and using LaSalle’s invariance principle [17], the global stability of the disease-free equilibrium E0 can be derived. We can conclude this with following theorem.Theorem 5.
Whenb=0 and ℜ0<1, then the disease-free equilibrium E0 of model (1) is globally asymptotic stability in Γ.
For the global stability of endemic equilibriumE∗, we have the following theorem.Theorem 6.
Whenb=0 and ℜ0>1, the endemic equilibrium E∗ for model (1) is globally asymptotically stable in Γ.Proof.
Notice thatI∗,R∗,N∗ is the unique solution of (17) when ℜ0>1. When b=0, it follows from the first equation of (17) that I∗,R∗,N∗ satisfies(24)N∗−I∗−R∗−γ+d+αI∗gI∗=0.Then, we can rewrite dI/dt in (23) as(25)dIdt=gIN−I−R−γ+d+αIgI=gIN−I−R−γ+d+αIgI−N∗−I∗−R∗−γ+d+αI∗gI∗.
Hence, the following model(26)dIdt=gIN−N∗−I−I∗−R−R∗−gIγ+d+αIgI−I∗gI∗,dRdt=γI−I∗−δ+dR−R∗,dNdt=−dN−N∗−αI−I∗,is the equivalent model of (23).
Define a Lyapunov function(27)V=12αN−N∗2+∫I∗Iu−I∗gudu+12γR−R∗2.
One can testify thatVI,R,N>0 except at I=I∗,R=R∗,N=S∗+I∗+R∗. The derivative of V along solutions of (26) is(28)dVdt|3.2=12αN−N∗dNdt+I−I∗gIdIdt+12γR−R∗dRdt−dαN−N∗2−I−I∗N−N∗−I−I∗2+I−I∗N−N∗−I−I∗R−R∗−γ+d+αI−I∗IgI−I∗gI∗+I−I∗R−R∗−δ+dγR−R∗2−dαN−N∗2−I−I∗2−δ+dγR−R∗2−γ+d+αI−I∗IgI−I∗gI∗.The assumption that gI/I is a continuous and monotonously nonincreasing function for I≥0, which means that(29)I−I∗IgI−I∗gI∗≥0.
Therefore,dV/dt is a negative definite in Γ. Using the Lyapunov stability theorem in [18], we know that I,R,N=I∗,R∗,N∗ of (23) is globally asymptotically stable. This completes the proof.Remark 1.
Ifc=0, then a=1, and model (1) becomes(30)dSdt=A−gIS+δR−dS,dIdt=gIS−γ+d+αI,dRdt=γI−δ+dR,
From Theorems5 and 6, one can directly obtain that disease-free equilibrium and endemic equilibrium for this model are both globally asymptotically stable. In fact, model (30) is also a special case of literature [8].Case 2. b>0ForgI, except assumptions (i) and (ii), we add one assumption (iii). That is, for this case, function gI satisfies.(i)
g0=0 and gI>0 for I>0(ii)
limI⟶0gI/I=β>0, where gI/I is continuous and monotonously nonincreasing for I>0(iii)
gI as a nondecreasing function of I>0
There are many incidence rates that satisfy assumptions (i)-(iii), such as bilinear incidencegIS=βIS stated in [1], saturated incidence rate gIS=βIS/1+αI discussed in [12].
Similar to the proof of the caseb=0, the following theorem holds.Theorem 7.
Whenb>0, the endemic equilibrium E^ of model (1) is globally asymptotically stable in Γ.Proof.
Similarly, we can rewrite model (23) as the following model:(31)dIdt=gIN−N^−I−I^−R−R^−gIγ+d+αIgI−I^gI^+bA1gI−1gI^,dRdt=γI−I^−δ+dR−R^,dNdt=−dN−N^−αI−I^.
Define the Lyapunov function by(32)W=12αN−N^2+∫I^Iu−I^gudu+12γR−R^2.which is the same as given in the case of b=0.
Taking the derivative along (31) derives that(33)dWdt|3.5=−dαN−N^2−I−I^2−δ+dγR−R^2−γ+d+αI−I^IgI−I^gI^+bAI−I^1gI−1gI^.The assumption gI as a nondecreasing function of I>0 implies that I−I^1/gI−1/gI^≤0. Therefore, dW/dt|3.5 is a negative definite. By the Lyapunov stability theorem [18], the globally asymptotic stability of endemic equilibrium is proved.


## 4. Numerical Example

Theorems5 and 6 show that when b=0, the global dynamical behavior of SIRS models completely relies on the basic reproduction number ℜ0. Theorems 4 and 7 show that when b>0, the SIRS model has only one endemic model, which is globally asymptotically stable. In this section, some numerical simulations are run to demonstrate our theoretical results. To do this, we choose gI=βI and fix the parameters at A=1,α=0.032,γ=0.1,δ=0.009,andd=0.026.Case 1.
a=1,b=c=0. If β=0.0037, we have ℜ0=0.9<1 and E0=38.46,0,0. Then, it follows from Figure 1(a) that the numerical solutions of model (1) converge to E0, which is consistent with Theorem 1. Furthermore, we choose β=0.005. It follows that ℜ0=1.22>1 and the unique endemic equilibrium E∗=31.6,1.356,3.85. Figure 1(b) shows that the numerical solutions of model (1) converge to E∗, which is also consistent with Theorem 5.Case 2.
a=0.8,b=0,andc=0.2. When taking β=0.0046, then ℜ0=0.95<1 and E0=34.35,0,5.71. It follows from Figure 2(a) that all of the numerical solutions of model (1) converge to E0, which is consistent with Theorem 6. By choosing β=0.006, then ℜ0=1.24>1 and the unique endemic equilibrium E∗=26.33,1.26,9.32. It follows from Figure 2(b) that all the numerical solutions converge to E∗, which is also consistent with Theorem 6.Case 3.
a=0.6,b=0.1,andc=0.3. In this case, model (1) only owns the endemic equilibrium. If we choose β=0.001, Figure 3 shows that all solutions of model (1) converge to the unique equilibrium E∗=26.03,0.76,10.74, while if we choose β=0.005, all the solutions of model (1) converge to E∗=20.62,1.82,13.78, which is consistent with Theorem 7.Case 4.
The constant immigration isa=0.8,b=0,c=0.2 for the first 100 days, and then it becomes a=0.8,b=0,c=0.2. For this case, we choose β=0.0046, and Figure 4(a) is one scenario of Case 2 when ℜ0=0.95. In this case, the disease dies out before 100 days. However, if there is an imported disease case at 100 days, this disease will become outbreak again and become an endemic disease if no control measures are taken. As the results showed in our former work [19, 20], the 2019 novel coronavirus (COVID-19) will quickly become outbreak in all China in some sense due to the fact that the exposed or infected individuals move from Wuhan (the source of COVID-19) to their hometown during the Spring Festival.


## 5. Conclusion

In this paper, we considered the global properties of SIRS epidemic models with general incidence rate and constant immigration. The results show that whenb=0, the stability of the model entirely depends on the reproduction number ℜ0. We should point out that our conclusion is a general result, and it holds for different SIRS models once its incidence rate satisfies our assumptions. In fact, many incidence rates satisfy these assumptions, such as nonmonotone incidence rate with psychological effect gIS=βIS/1+αI2 and gIS=βIS/1+kI+αI2, respectively, investigated in [21, 22], which satisfy assumptions (i)-(ii), and the bilinear incidence gIS=βIS stated in [1], the Hattaf-Yousfi functional response βIS/α0+α1S+α2I+α3IS [23] with α0=1,α1=α3=0 discussed in [12], which satisfy assumptions (i)–(iii).Figure 1
Time plots of populations for the casea=1,b=0,andc=0 with different initial conditions. (a) β=0.0037. (b) β=0.005.
(a)(b)Figure 2
Time plots of populations for the casea=0.8,b=0,c=0.2 with different initial conditions. (a) β=0.0046. (b) β=0.006.
(a)(b)Figure 3
Time plots of populations for the casea=0.6,b=0.1,c=0.3 with different initial conditions. (a) β=0.001. (b) β=0.005.
(a)(b)Figure 4
Time plots of populations for the caseβ=0.0046 for different constant immigration. (a) a=0.8,b=0,c=0.2. (b) a=0.8,b=0.1,c=0.1.
(a)(b)Biologically, the global asymptotic stability of disease-free equilibrium means that the disease will die out in the whole population, while the globally asymptotic stability of endemic equilibrium means that the disease will persist in the whole population. Therefore, Theorem6 implies that the disease will extinct if ℜ0<1 and Theorem 6 reveals that the disease will persist if ℜ0>1. When b>0, the infection cannot be eliminated due to the constant input of new infections. In this case, model (1) always has only one equilibrium E^. Theorem 7 shows that the unique endemic equilibrium E^ is globally asymptotically stable, which means that the disease always exists if the input of infectious individuals size is not zero. Therefore, to control the transmit of infectious disease, we should first shrink the input of infected individuals to zero and then keep the basic reproduction number less than one. For example, COVID-19 has become a widespread outbreak in China, which in part is due to the massive population shift in China during the Spring Festival. Fortunately, the daily new confirmed COVID-19 cases are declining, and we are confident that the disease will soon become extinct under the Chinese government and people’s efforts.



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