DDNS Discrete Dynamics in Nature and Society 1607-887X 1026-0226 Hindawi Publishing Corporation 10.1155/2016/4823815 4823815 Research Article Global Dynamics for a Novel Differential Infectivity Epidemic Model with Stage Structure http://orcid.org/0000-0002-6156-7989 Jin Yunguo 1,2 2 Zhang Zhengqiu 1 School of Statistics Chengdu University of Information Technology Chengdu Sichuan 610103 China cuit.edu.cn 2 Key Laboratory of Statistical Information Technology and Data Mining State Statistics Bureau Chengdu China 2016 19122016 2016 01 09 2016 17 11 2016 24 11 2016 2016 Copyright © 2016 Yunguo Jin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

Key Laboratory of Statistical Information Technology and Data Mining, State Statistics Bureau SDL201601
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 , 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.  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 .

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  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=12pikj=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=12pikj=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 Section 2, 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=limx0Gkxx>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,,InR+n+2:  S1S10,  S2S20,  S1+S2+k=1nIkbm_,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)Qi=12pikβijSi0δjmkn×n=i=12pi1βi1Si0δ1m1i=12pi1βinSi0δnm1i=12pinβi1Si0δ1mni=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 PAP 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 and B=[i=12pikβij] is irreducible.

If R01, then P0 is globally asymptotically stable in Γ.

If R0>1, then P0 is unstable and system (1) admits at least one endemic equilibrium in Γ.

Proof.

Let S=(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=1nωki=12pikj=1nβijSiGjIjmk-Ik=ω1,ω2,,ωnQS,IIT-ITω1,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 R01.

If R0>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  and a similar argument as in the proof of Proposition 3.3 of , 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  or Theorem D.3 of ). The proof is completed.

By Theorem 1, 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=12j=1nβijSiGjIj+d2S2,(8)aS1=j=1nβ2jS2GjIj+d2S2,(9)i=12pikj=1nβijSiGjIj=mkIk.Since φ is strictly decreasing on [0,+), we have(10)φS1-φS1p1k-p1kS1S10.

For convenience of notations, set(11)β¯kj=i=12pikβijSiGjIj,1k,jn,B¯=l1β¯1l-β¯21-β¯n1-β¯12l2β¯2l-β¯n2-β¯1n-β¯2nlnβ¯nl.Then, B¯ is also irreducible. It follows from Lemma 2.1 of  that the solution space of linear system,(12)B¯v=0,has dimension 1, with a basis(13)vv1,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=1nvkj=1ni=12pikβijSiGjIj=k,j=1ni=12pijβikSivjGkIk=k=1nj=1ni=12pijβikSiGkIkvjGkIkGkIk=k=1nj=1nβ¯jkvjGkIkGkIk=k=1nj=1nβ¯kjvkGkIkGkIk=k=1nvkj=1ni=12pikβijSiGjIjGkIkGkIk.

We further make the following assumption.

(A2) Gk is strictly increasing on [0,+), and(16)GkxkIkGkIkxk+GkIkGkxk-Ikxk1,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=1nvki=12pikSi-Si-SilnSiSi+IkIkGkx-GkIkGkxdx.Differentiating V along the solution of system (1), we obtain (18)V˙=k=1nvkp1kφS1-aS1+p2kaS1-d2S2-mkIk-p1kS1S1φS1-j=1nβ1jS1GjIj-aS1-p2kS2S2aS1-j=1nβ2jS2GjIj-d2S2-GkIkGkIki=12pikj=1nβijSiGjIj-mkIk=k=1nvkφS1p1k-p1kS1S1+d2S2p2k-p2kS2S2+i=12pikj=1nβijSiGjIj+mkGkIkIkGkIk-mkIk+aS1p1k-p1kS1S1+p2kS1S1-p2kS1S2S1S2-GkIkGkIki=12pikj=1nβijSiGjIj.From (7), we know that(19)φS1p1k-p1kS1S1=i=12j=1nβijSiGjIj+d2S2p1k-p1kS1S1.It follows from (8), (9), and (19) that (20)V˙=k=1nvkφS1-φS1p1k-p1kS1S1+d2S2p2k-p2kS2S2+i=12pikβijSijGjIjGkIkIkGkIkIk+GjIjGjIj-IkIk+j=1nβ1jS1GjIjp1k-p1kS1S1+d2S2+j=1nβ2jS2GjIj2p1k-p1kS1S1-p1kS1S1+p2kS1S1-p2kS1S2S1S2-GkIkGkIki=12pikj=1nβijSiGjIj.From (10), (15), and (16), we obtain(21)V˙k=1nvkd2S2p2k-p2kS2S2+i=12pikβijSijGjIj+j=1nβ1jS1GjIjp1k-p1kS1S1+d2S2+j=1nβ2jS2GjIj2p1k-p1kS1S1-p1kS1S1+p2kS1S1-p2kS1S2S1S2-GkIkGkIki=12pikj=1nβijSiGjIj=k=1nvkd2S2p2k+2p1k-p1kS1S1-p1kS1S1+p2kS1S1-p2kS1S2S1S2-p2kS2S2+j=1nβ2jS2GjIj2p1k+p2k-p1kS1S1-p1kS1S1+p2kS1S1-p2kS1S2S1S2+j=1nβ1jS1GjIj2p1k-p1kS1S1-GkIkGkIki=12pikj=1nβijSiGjIj=-k=1nvkp1k-p2kd2S2+j=1nβ2jS2GjIjS1S1+S1S1+k=1nvkd2S2p2k+2p1k-p2kS1S1+S1S2S1S2+S2S2+j=1nβ2jS2GjIj2p1k+p2k-p2kS1S1+S1S2S1S2+S2GkIkGjIjS2GkIkGjIj+j=1nβ1jS1GjIj2p1k-p1kS1S1+S1GkIkGjIjS1GkIkGjIjB1.By k=1nvkp1k-p2k0 and the arithmetic-geometric mean, we easily see that(22)B1-k=1nvk2p1k-2p2kd2S2+j=1nβ2jS2GjIj+k=1nvkd2S2p2k+2p1k-3p2k+j=1nβ2jS2GjIj2p1k+p2k-3p2kGkIkGjIjGkIkGjIj1/3+j=1nβ1jS1GjIj2p1k-2p1kGkIkGjIjGkIkGjIj1/2=k=1nvkj=1nβ2jS2GjIj3p2k-3p2kGkIkGjIjGkIkGjIj1/3+j=1nβ1jS1GjIj2p1k-2p1kGkIkGjIjGkIkGjIj1/2B2.We can rewrite B2 as (23)B2=k=1nvk3j=1np2kβ2jS2GjIj1-GkIkGjIjGkIkGjIj1/3+lnGkIkGjIjGkIkGjIj1/3+2j=1np1kβ1jS1GjIj1-GkIkGjIjGkIkGjIj1/2+lnGkIkGjIjGkIkGjIj1/2-i2j=1npikβijSiGjIjlnGkIkGjIjGkIkGjIj.Using the fact that 1-x+lnx0, where equality holds if only if x=1, we obtain(24)B2-k=1nvki2j=1npikβijSiGjIjlnGkIkGjIjGkIkGjIj=k,j=1nvkβ¯kjlnGkIkGjIjGkIkGjIj.In the following, we will show that(25)H¯nk,j=1nvkβ¯kjlnGkIkGjIjGkIkGjIj0.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/GkIkGjIj). Formula (13) gives v1=β¯21 and v2=β¯12 in this case. Expanding H¯2 yields (26)H¯2=β¯21β¯11lnG1I1G1I1G1I1G1I1+β¯12β¯22lnG2I2G2I2G2I2G2I2+β¯21β¯12lnG1I1G2I2G1I1G2I2+β¯12β¯21lnG2I2G1I1G2I2G1I1=β¯12β¯21lnG1I1G2I2G1I1G2I2+lnG2I2G1I1G2I2G1I1=0.For more general n, by a similar argument as in the proof of k,j=1nvkβ¯kjlnEkEj/EkEj0 in , we obtain k,j=1nvkβ¯kjlnGk(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=1nβ1jS1GjIj-aS1,0=aS1-j=1nβ2jS2GjIj-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-d2S2mk=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˙=p11j=12β1jS1GjIj+p21j=12β2jS2GjIj-m1I1,I2˙=p12j=12β1jS1GjIj+p22j=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.

If β11=β21=0.001,β12=β22=0.0001, then we obtain R00.55. By Theorem 1, the disease dies out in both subgroups. Numerical simulation illustrates this fact (see Figure 1).

Dynamical behavior of system (31) with parameter values in (32) and Case 1. R00.55. 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.

If β11=β21=0.01,β12=β22=0.001, then we have R05.53, P(171.0530,222.9491,32.0764,28.1767), v1=β¯21=i=12pi2βi1SiG1(I1)154.06, v2=β¯12=i=12pi1βi2SiG2(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).

Dynamical behavior of system (31) with parameter values in (32) and Case 1. R05.53. 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.

Competing Interests

The author declares that there are no competing interests.

Acknowledgments

This work was supported by Key Laboratory of Statistical Information Technology and Data Mining, State Statistics Bureau (SDL201601).

Hyman J. M. Li J. Stanley E. A. The differential infectivity and staged progression models for the transmission of HIV Mathematical Biosciences 1999 155 2 77 109 10.1016/s0025-5564(98)10057-3 2-s2.0-0033073528 Ma Z. Liu J. Li J. Stability analysis for differential infectivity epidemic models Nonlinear Analysis. Real World Applications 2003 4 5 841 856 10.1016/s1468-1218(03)00019-1 MR1978564 Alexanderian A. Gobbert M. K. Fister K. R. Gaff H. Lenhart S. Schaefer E. An age-structured model for the spread of epidemic cholera: analysis and simulation Nonlinear Analysis: Real World Applications 2011 12 6 3483 3498 10.1016/j.nonrwa.2011.06.009 MR2832986 2-s2.0-80052023465 Liu Y. Gao S. Luo Y. Impulsive epidemic model with differential susceptibility and stage structure Applied Mathematical Modelling 2012 36 1 370 378 10.1016/j.apm.2011.07.035 MR2835018 2-s2.0-80051807926 Zhang X.-B. Huo H.-F. Sun X.-K. Fu Q. The differential susceptibility SIR epidemic model with stage structure and pulse vaccination Nonlinear Analysis. Real World Applications 2010 11 4 2634 2646 10.1016/j.nonrwa.2009.09.011 MR2661931 2-s2.0-77955775111 Shi X. Cui J. Zhou X. Stability and Hopf bifurcation analysis of an eco-epidemic model with a stage structure Nonlinear Analysis. Theory, Methods & Applications 2011 74 4 1088 1106 10.1016/j.na.2010.09.038 MR2746791 2-s2.0-78650512399 Wu C. Weng P. Stability analysis of a SIS model with stage structured and distributed maturation delay Nonlinear Analysis: Theory, Methods & Applications 2009 71 12 e892 e901 10.1016/j.na.2008.12.058 2-s2.0-72149083300 Inaba H. Endemic threshold results in an age-duration-structured population model for HIV infection Mathematical Biosciences 2006 201 1-2 15 47 10.1016/j.mbs.2005.12.017 MR2252076 2-s2.0-33646824718 Feng Z. Huang W. Castillo-Chavez C. Global behavior of a multi-group SIS epidemic model with age structure Journal of Differential Equations 2005 218 2 292 324 10.1016/j.jde.2004.10.009 MR2177462 2-s2.0-27744440076 Tian B. Jin Y. Zhong S. Chen N. Global stability of an epidemic model with stage structure and nonlinear incidence rates in a heterogeneous host population Advances in Difference Equations 2015 2015, article 260 10.1186/s13662-015-0594-4 MR3386210 2-s2.0-84940213274 Guo H. Li M. Y. Shuai Z. Global stability of the endemic equilibrium of multigroup SIR epidemic models Canadian Applied Mathematics Quarterly 2006 14 3 259 284 MR2327745 Sun R. Shi J. Global stability of multigroup epidemic model with group mixing and nonlinear incidence rates Applied Mathematics and Computation 2011 218 2 280 286 10.1016/j.amc.2011.05.056 MR2820490 2-s2.0-79960834712 Yuan Z. Wang L. Global stability of epidemiological models with group mixing and nonlinear incidence rates Nonlinear Analysis. Real World Applications 2010 11 2 995 1004 10.1016/j.nonrwa.2009.01.040 MR2571270 2-s2.0-70449709108 Kuniya T. Global stability analysis with a discretization approach for an age-structured multigroup SIR epidemic model Nonlinear Analysis: Real World Applications 2011 12 5 2640 2655 10.1016/j.nonrwa.2011.03.011 MR2813210 2-s2.0-79957901175 Yuan Z. Zou X. Global threshold property in an epidemic model for disease with latency spreading in a heterogeneous host population Nonlinear Analysis. Real World Applications 2010 11 5 3479 3490 10.1016/j.nonrwa.2009.12.008 MR2683806 2-s2.0-77955513015 Sun R. Global stability of the endemic equilibrium of multigroup SIR models with nonlinear incidence Computers & Mathematics with Applications 2010 60 8 2286 2291 10.1016/j.camwa.2010.08.020 MR2725319 2-s2.0-77957355690 Li M. Y. Shuai Z. Wang C. Global stability of multi-group epidemic models with distributed delays Journal of Mathematical Analysis and Applications 2010 361 1 38 47 10.1016/j.jmaa.2009.09.017 MR2567280 2-s2.0-70349384940 Shu H. Fan D. Wei J. Global stability of multi-group SEIR epidemic models with distributed delays and nonlinear transmission Nonlinear Analysis: Real World Applications 2012 13 4 1581 1592 10.1016/j.nonrwa.2011.11.016 MR2890994 2-s2.0-84856551488 Ding D. Ding X. Global stability of multi-group vaccination epidemic models with delays Nonlinear Analysis. Real World Applications 2011 12 4 1991 1997 10.1016/j.nonrwa.2010.12.015 MR2800994 2-s2.0-79955522107 Chen H. Sun J. Global stability of delay multigroup epidemic models with group mixing and nonlinear incidence rates Applied Mathematics and Computation 2011 218 8 4391 4400 10.1016/j.amc.2011.10.015 MR2862109 2-s2.0-81855167021 Zhang C. Li W. Wang K. Graph theory-based approach for stability analysis of stochastic coupled systems with Lévy noise on networks IEEE Transactions on Neural Networks and Learning Systems 2015 26 8 1698 1709 10.1109/tnnls.2014.2352217 MR3454966 2-s2.0-84937431923 Zhang C. Li W. Wang K. Graph-theoretic approach to stability of multi-group models with dispersal Discrete and Continuous Dynamical Systems Series B 2015 20 1 259 280 10.3934/dcdsb.2015.20.259 MR3331655 2-s2.0-84908615593 Freedman H. I. Tang M. X. Ruan S. G. Uniform persistence and flows near a closed positively invariant set Journal of Dynamics and Differential Equations 1994 6 4 583 600 10.1007/bf02218848 Li M. Y. Graef J. R. Wang L. Karsai J. Global dynamics of a SEIR model with varying total population size Mathematical Biosciences 1999 160 2 191 213 10.1016/s0025-5564(99)00030-9 MR1706068 2-s2.0-0032812296 Bhatia N. P. Szegö G. P. Dynamical Systems: Stability Theory and Applications 1967 New York, NY, USA Springer Lecture Notes in Mathematics, No. 35 MR0219843 Smith H. L. Waltman P. The Theory of the Chemostat: Dynamics of Microbial Competition 1995 Cambridge, UK Cambridge University Press 10.1017/cbo9780511530043 MR1315301