# Global Dynamics for a Novel Differential Infectivity Epidemic Model with Stage Structure

Authors: Yunguo Jin
Journal: Discrete Dynamics in Nature and Society
 (2016)

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## Abstract

A novel differential infectivity epidemic model with stage structure is formulated and studied. Under biological motivation, the stability of equilibria is investigated by the global Lyapunov functions. Some novel techniques are applied to the global dynamics analysis for the differential infectivity epidemic model. Uniform persistence and the sharp threshold dynamics are established; that is, the reproduction number determines the global dynamics of the system. Finally, numerical simulations are given to illustrate the main theoretical results.


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## Body

## 1. Introduction

Mathematical model that reflects the characteristics of an epidemic to some extent can help us to understand better how the disease spreads in the community and can investigate how changes in the various assumptions and parameter values affect the course of epidemic. In [1], Hyman et al. proposed a differential infectivity model that accounted for differences in infectiousness between individuals during the chronic stages and the correlation between viral loads and rates of developing AIDS. They assumed that the susceptible population was homogeneous and neglected variations in susceptibility, risk behavior, and many other factors associated with the dynamics of HIV spread. Ma et al. [2] presented several differential infectivity epidemic models under different assumptions.In the real world, some epidemics, such as malaria, dengue, fever, gonorrhea, and bacterial infections, may have a different ability to transmit the infections in different ages. For example, measles and varicella always occur in juveniles, while it is reasonable to consider the disease transmission in adult population such as typhus and diphtheria. In recent years, epidemic models with stage structure have been studied in many papers [3–10].In this paper, we formulate a differential infectivity epidemic model with stage structure. The proof of global stability of the endemic equilibrium utilizes a graph-theoretical approach [11–22] to the method of Lyapunov functions. Let S1 and S2 denote the immature susceptible and mature susceptible populations, respectively. The infectious population I was subdivided into n subgroups I1,I2,…,In. p1k and p2k denote the probabilities of an immature infectious individual and a mature infectious individual enter subgroup k, respectively, where ∑k=1np1k=∑k=1np2k=1. The disease incidence in the kth subgroup can be calculated as ∑i=12pik∑j=1nβijSiGj(Ij), where βij is the transmission coefficient between compartments Si and Ij. Gj(Ij) includes some special incidence functions in the literature. For instance, Gj(Ij)=Ij/(1+αjIj) (saturation effect). Since we do not assume that recovered individuals return into the susceptible class, the recovered class does not need to be explicitly modeled. Then, we obtain the following model:(1)S1˙=φS1-∑j=1nβ1jS1GjIj-aS1,S2˙=aS1-∑j=1nβ2jS2GjIj-d2S2,Ik˙=∑i=12pik∑j=1nβijSiGjIj-mkIk,k=1,2,…,n,where φ(S1)=b-d1S1 with b being the recruitment constant and d1 being the natural death rate. a is the conversion rate from an immature individual to a mature individual. d2 is the natural death rate of the mature susceptible class. mk=dkI+γk, where dkI is the death rate of I population in subgroup k and γk is the recovery rate in the kth subgroup. All parameter values are assumed to be nonnegative and b, a, mk, d1, d2>0.The organization of this paper is as follows. In Section2, we prove some preliminary results for system (1). In Section 3, the main theorem of this paper is stated and proved. In Section 4, numerical simulations which support our theoretical analysis are given.


## 2. Preliminaries

We assume the following.(A1)Gk is continuous and Lipschitz on [0,+∞),  Gk(x)/x is nonincreasing on (0,+∞), and(2)δk=limx→0⁡Gkxx>0exists.From our assumptions, it is clear that system (1) has a unique solution for any given initial data (S1(0),S2(0),I1(0),…,In(0)) with S1(0)>0,   S2(0)>0, and  Ik(0)>0 for k=1,2,…,n and the solution remains nonnegative. We see that system (1) exits in a disease-free equilibrium P0=(S10,S20,0,…,0), where S20=aS10/d2. Let m_=min⁡{d1,d2,m1,…,mn}. Then, we derive from (1) that the region, (3)Γ=S1,S2,I1,…,In∈R+n+2:S1≤S10,S2≤S20,S1+S2+∑k=1nIk≤bm_,is a forward invariant compact absorbing set with respect to (1). Also let Γ∘ denote the interior of Γ. The next generation matrix for system (1) is(4)Q≔∑i=12pikβijSi0δjmkn×n=∑i=12pi1βi1Si0δ1m1⋯∑i=12pi1βinSi0δnm1⋮⋱⋮∑i=12pinβi1Si0δ1mn⋯∑i=12pinβinSi0δnmn.Then, we define the basic reproduction number as the spectral radius of Q, R0=ρ(Q). A square matrix is said to be reducible, if there is a permutation matrix P, such that P′AP is a block upper triangular matrix; otherwise it is irreducible.


## 3. Main Results

In the section, we will study the global asymptotical stability of equilibria of system (1).Theorem 1.
Assume that (A1) holds andB=[∑i=12pikβij] is irreducible. (1)
IfR0≤1, then P0 is globally asymptotically stable in Γ.(2)
IfR0>1, then P0 is unstable and system (1) admits at least one endemic equilibrium in Γ∘.Proof.
LetS=(S1,S2),  S0=(S10,S20),  I=(I1,I2,…,In), and Q(S,I)=∑i=12(pikβijSiGjIj)/mkIjn×n. Notice that B is irreducible, and then Q is also irreducible. Hence, there exists ωk>0,   k=1,2,…,n, such that(5)ω1,ω2,…,ωnρQ=ω1,ω2,…,ωnQ.Define L=∑k=1n(ωkIk/mk). Then (6)L˙=∑k=1nωk∑i=12pik∑j=1nβijSiGjIjmk-Ik=ω1,ω2,…,ωnQS,IIT-IT≤ω1,ω2,…,ωnQIT-IT=ρQ-1ω1,ω2,…,ωnIT.We see that the only compact invariant subset of the set where L˙=0 is the singleton {P0}. By LaSalle’s Invariance Principle, P0 is globally asymptotically stable in Γ if R0≤1.
IfR0>1, by continuity, we obtain that L˙=(ω1,ω2,…,ωn)Q(S,I)IT-IT>0 in a neighborhood of P0 in Γ∘. This implies that P0 is unstable. From a uniform persistence result of [23] and a similar argument as in the proof of Proposition 3.3 of [24], we can deduce that the instability of P0 implies the uniform persistence of system (1) in Γ∘. This together with the uniform boundedness of solutions of system (1) in Γ∘ implies that system (1) has an endemic equilibrium in Γ∘ (see Theorem 2.8.6 of [25] or Theorem D.3 of [26]). The proof is completed.By Theorem1, we have the idea that if B=[∑i=12pikβij] is irreducible, (A1) holds and R0>1, and then system (1) exists in endemic equilibrium P∗ in Γ∘. Let P∗=(S1∗,S2∗,I1∗,…,In∗), and then the components of P∗ satisfy(7)φS1∗=∑i=12∑j=1nβijSi∗GjIj∗+d2S2∗,(8)aS1∗=∑j=1nβ2jS2∗GjIj∗+d2S2∗,(9)∑i=12pik∑j=1nβijSi∗GjIj∗=mkIk∗.Since φ is strictly decreasing on [0,+∞), we have(10)φS1-φS1∗p1k-p1kS1∗S1≤0.For convenience of notations, set(11)β¯kj=∑i=12pikβijSi∗GjIj∗,1≤k,j≤n,B¯=∑l≠1β¯1l-β¯21⋯-β¯n1-β¯12∑l≠2β¯2l⋯-β¯n2⋮⋮⋱⋮-β¯1n-β¯2n⋯∑l≠nβ¯nl.Then, B¯ is also irreducible. It follows from Lemma 2.1 of [11] that the solution space of linear system,(12)B¯v=0,has dimension 1, with a basis(13)v≔v1,v2,…,vnT=c1,c2,…,cnT,where ck denotes the cofactor of the kth diagonal entry of B¯. Note that from (12) we have(14)∑j=1nB¯kjvk=∑j=1nB¯jkvj,k=1,2,…,n.From (14), we have(15)∑k=1nvk∑j=1n∑i=12pikβijSi∗GjIj=∑k,j=1n∑i=12pijβikSi∗vjGkIk=∑k=1n∑j=1n∑i=12pijβikSi∗GkIk∗vjGkIkGkIk∗=∑k=1n∑j=1nβ¯jkvjGkIkGkIk∗=∑k=1n∑j=1nβ¯kjvkGkIkGkIk∗=∑k=1nvk∑j=1n∑i=12pikβijSi∗GjIj∗GkIkGkIk∗.We further make the following assumption.(A2)Gk is strictly increasing on [0,+∞), and(16)GkxkIkGkIkxk+GkIkGkxk-Ikxk≤1,k=1,2,…,n,where xk>0 is chosen in an arbitrary way and equality holds if Ik=xk.Theorem 2.
Assume that (A1) and (A2) hold,∑k=1nvk(p1k-p2k)≥0, and B=[∑i=12pikβij] is irreducible. If R0>1, then P∗ is globally asymptotically stable in Γ∘ and thus is the unique endemic equilibrium.Proof.
Consider a Lyapunov functional(17)V=∑k=1nvk∑i=12pikSi-Si∗-Si∗ln⁡SiSi∗+∫Ik∗IkGkx-GkIk∗Gkxdx.Differentiating V along the solution of system (1), we obtain (18)V˙=∑k=1nvkp1kφS1-aS1+p2kaS1-d2S2-mkIk-p1kS1∗S1φS1-∑j=1nβ1jS1GjIj-aS1-p2kS2∗S2aS1-∑j=1nβ2jS2GjIj-d2S2-GkIk∗GkIk∑i=12pik∑j=1nβijSiGjIj-mkIk=∑k=1nvkφS1p1k-p1kS1∗S1+d2S2∗p2k-p2kS2S2∗+∑i=12pik∑j=1nβijSi∗GjIj+mkGkIk∗IkGkIk-mkIk+aS1∗p1k-p1kS1S1∗+p2kS1S1∗-p2kS1S2∗S1∗S2-GkIk∗GkIk∑i=12pik∑j=1nβijSiGjIj.From (7), we know that(19)φS1∗p1k-p1kS1∗S1=∑i=12∑j=1nβijSi∗GjIj∗+d2S2∗p1k-p1kS1∗S1.It follows from (8), (9), and (19) that (20)V˙=∑k=1nvkφS1-φS1∗p1k-p1kS1∗S1+d2S2∗p2k-p2kS2S2∗+∑i=12pikβijSij∗GjIj∗GkIk∗IkGkIkIk∗+GjIjGjIj∗-IkIk∗+∑j=1nβ1jS1∗GjIj∗p1k-p1kS1∗S1+d2S2∗+∑j=1nβ2jS2∗GjIj∗2p1k-p1kS1∗S1-p1kS1S1∗+p2kS1S1∗-p2kS1S2∗S1∗S2-GkIk∗GkIk∑i=12pik∑j=1nβijSiGjIj.From (10), (15), and (16), we obtain(21)V˙≤∑k=1nvkd2S2∗p2k-p2kS2S2∗+∑i=12pikβijSij∗GjIj∗+∑j=1nβ1jS1∗GjIj∗p1k-p1kS1∗S1+d2S2∗+∑j=1nβ2jS2∗GjIj∗2p1k-p1kS1∗S1-p1kS1S1∗+p2kS1S1∗-p2kS1S2∗S1∗S2-GkIk∗GkIk∑i=12pik∑j=1nβijSiGjIj=∑k=1nvkd2S2∗p2k+2p1k-p1kS1∗S1-p1kS1S1∗+p2kS1S1∗-p2kS1S2∗S1∗S2-p2kS2S2∗+∑j=1nβ2jS2∗GjIj∗2p1k+p2k-p1kS1∗S1-p1kS1S1∗+p2kS1S1∗-p2kS1S2∗S1∗S2+∑j=1nβ1jS1∗GjIj∗2p1k-p1kS1∗S1-GkIk∗GkIk∑i=12pik∑j=1nβijSiGjIj=-∑k=1nvkp1k-p2kd2S2∗+∑j=1nβ2jS2∗GjIj∗S1∗S1+S1S1∗+∑k=1nvkd2S2∗p2k+2p1k-p2kS1∗S1+S1S2∗S1∗S2+S2S2∗+∑j=1nβ2jS2∗GjIj∗2p1k+p2k-p2kS1∗S1+S1S2∗S1∗S2+S2GkIk∗GjIjS2∗GkIkGjIj∗+∑j=1nβ1jS1∗GjIj∗2p1k-p1kS1∗S1+S1GkIk∗GjIjS1∗GkIkGjIj∗≕B1.By ∑k=1nvkp1k-p2k≥0 and the arithmetic-geometric mean, we easily see that(22)B1≤-∑k=1nvk2p1k-2p2kd2S2∗+∑j=1nβ2jS2∗GjIj∗+∑k=1nvkd2S2∗p2k+2p1k-3p2k+∑j=1nβ2jS2∗GjIj∗2p1k+p2k-3p2kGkIk∗GjIjGkIkGjIj∗1/3+∑j=1nβ1jS1∗GjIj∗2p1k-2p1kGkIk∗GjIjGkIkGjIj∗1/2=∑k=1nvk∑j=1nβ2jS2∗GjIj∗3p2k-3p2kGkIk∗GjIjGkIkGjIj∗1/3+∑j=1nβ1jS1∗GjIj∗2p1k-2p1kGkIk∗GjIjGkIkGjIj∗1/2≕B2.We can rewrite B2 as (23)B2=∑k=1nvk3∑j=1np2kβ2jS2∗GjIj∗1-GkIk∗GjIjGkIkGjIj∗1/3+ln⁡GkIk∗GjIjGkIkGjIj∗1/3+2∑j=1np1kβ1jS1∗GjIj∗1-GkIk∗GjIjGkIkGjIj∗1/2+ln⁡GkIk∗GjIjGkIkGjIj∗1/2-∑i2∑j=1npikβijSi∗GjIj∗ln⁡GkIk∗GjIjGkIkGjIj∗.Using the fact that 1-x+ln⁡x≤0, where equality holds if only if x=1, we obtain(24)B2≤-∑k=1nvk∑i2∑j=1npikβijSi∗GjIj∗ln⁡GkIk∗GjIjGkIkGjIj∗=∑k,j=1nvkβ¯kjln⁡GkIkGjIj∗GkIk∗GjIj.In the following, we will show that(25)H¯n≔∑k,j=1nvkβ¯kjln⁡GkIkGjIj∗GkIk∗GjIj≡0.We first give the proof of (25) for n=2, which would give a reader the basic yet clear ideas without being hidden by the complexity of terms caused by larger values of n. When n=2, we have H¯2=∑k,j=12vkβ¯kjln⁡(GkIkGjIj∗/GkIk∗GjIj). Formula (13) gives v1=β¯21 and v2=β¯12 in this case. Expanding H¯2 yields (26)H¯2=β¯21β¯11ln⁡G1I1G1I1∗G1I1∗G1I1+β¯12β¯22ln⁡G2I2G2I2∗G2I2∗G2I2+β¯21β¯12ln⁡G1I1G2I2∗G1I1∗G2I2+β¯12β¯21ln⁡G2I2G1I1∗G2I2∗G1I1=β¯12β¯21ln⁡G1I1G2I2∗G1I1∗G2I2+ln⁡G2I2G1I1∗G2I2∗G1I1=0.For more general n, by a similar argument as in the proof of ∑k,j=1nvkβ¯kjln⁡Ek∗Ej/EkEj∗≡0 in [16], we obtain ∑k,j=1nvkβ¯kjln⁡Gk(Ik)Gj(Ij∗)/Gk(Ik∗)Gj(Ij)≡0.
From (21), (22), and (24), we see that if V˙=0, then(27)Si=Si∗,i=1,2.If (27) holds, it follows from (1) that(28)0=φS1∗-∑j=1nβ1jS1∗GjIj-aS1∗,0=aS1∗-∑j=1nβ2jS2∗GjIj-d2S2∗.Then, we obtain that (29)Ik˙=p1kφS1∗-aS1∗+p2kaS1∗-d2S2∗-mkIk,k=1,2,…,n.This implies that(30)limt→+∞⁡Ik=p1kφS1∗-aS1∗+p2kaS1∗-d2S2∗mk=Ik∗.By the characteristics of V, we obtain the idea that the largest invariant subset of the set where V˙=0 is the singleton {P∗}. By LaSalle’s Invariance Principle, P∗ is globally asymptotically stable for R0>1. This completes the proof.


## 4. Numerical Examples

In the section, numerical simulations are presented to support and complement the theoretical findings. We consider the following model:(31)S1˙=φS1-∑j=12β1jS1GjIj-aS1,S2˙=aS1-∑j=12β2jS2GjIj-d2S2,I1˙=p11∑j=12β1jS1GjIj+p21∑j=12β2jS2GjIj-m1I1,I2˙=p12∑j=12β1jS1GjIj+p22∑j=12β2jS2GjIj-m2I2,where Gj(Ij)=Ij/(1+αjIj). Clearly, (A1) and (A2) hold. We fix the parameters as follows:(32)b=100,d1=0.001,d2=0.3,a=0.5,α1=α2=0.1,m1=0.5,m2=0.6,p11=p22=0.6,p12=p21=0.4.Then, we have P0≈(199.6008,332.6680,0,0).Case 1.
Ifβ11=β21=0.001,β12=β22=0.0001, then we obtain R0≈0.55. By Theorem 1, the disease dies out in both subgroups. Numerical simulation illustrates this fact (see Figure 1).Figure 1
Dynamical behavior of system (31) with parameter values in (32) and Case 1. R0≈0.55. The initial conditions are S1(0)=100, S2(0)=80, I1(0)=1, and I2(0)=2. We see that the disease dies out in both subgroups.Case 2.
Ifβ11=β21=0.01,β12=β22=0.001, then we have R0≈5.53, P∗≈(171.0530,222.9491,32.0764,28.1767), v1=β¯21=∑i=12pi2βi1Si∗G1(I1∗)≈154.06, v2=β¯12=∑i=12pi1βi2Si∗G2(I2∗)≈73.81, and ∑k=12vk(p1k-p2k)≈16.05. By Theorem 2, the disease persists in both subgroups. Numerical simulation illustrates this fact (see Figure 2).Figure 2
Dynamical behavior of system (31) with parameter values in (32) and Case 1. R0≈5.53. The initial conditions are S1(0)=100, S2(0)=80, I1(0)=1, and I2(0)=2. We see that the disease persists in both subgroups.


## 5. Conclusions

A differential infectivity epidemic model with stage structure has been used to describe the spreading of such a disease. We have focused on the theoretical analysis of the equilibriums. Using a graph-theoretic approach to the method of Lyapunov functions, we have proved the global stability of the endemic equilibrium. We have established uniform persistence and the sharp threshold. The work has potential extensions and improvements, which remains to be discussed in the future.



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