Global Dynamics of a Multi-group SEIR Epidemic Model with Infection Age?

Vijay Pal BAJIYA Jai Prakash TRIPATHI Vipul KAKKAR Jinshan WANG Guiquan SUN

Abstract Consider the heterogeneity (e.g., heterogeneous social behaviour, heterogeneity due to different geography, contrasting contact patterns and different numbers of sexual partners etc.) of host population,in this paper,the authors propose an infection age multigroup SEIR epidemic model. The model system also incorporates the feedback variables,where the infectivity of infected individuals may depend on the infection age. In the direction of mathematical analysis of model, the basic reproduction number R0 has been computed. The global stability of disease-free equilibrium and endemic equilibrium have been established in the term of R0. More precisely, for R0 ≤1, the disease-free equilibrium is globally asymptotically stable and for R0 >1, they establish global stability of endemic equilibrium using some graph theoretic techniques to Lyapunov function method. By considering a numerical example, they investigate the effects of infection age and feedback on the prevalence of the disease. Their result shows that feedback parameters have different and even opposite effects on different groups. However, by choosing an appropriate value of feedback parameters, the disease could be eradicated or maintained at endemic level.Besides, the infection age of infected individuals may also change the behaviour of the disease, global stable to damped oscillations or damped oscillations to global stable.

Keywords Multi-group model, Infection age, Feedback, Graph-theoretic approach,Lyapunov function

1 Introduction

Mathematical Modelling in Epidemiology: Over the past several years, human health have continuously been threatened and thousands of people died of various infectious diseases every year (see [1—3]). By the World Health Report 1996 (WHO for short) (see [4]): “Nearly 50,000 men, women and children are dying every day from infectious diseases”. The report warns that on one side, several highly infectious diseases (e.g., incurable diseases, like, Ebola haemorrhagic fever, HIV/AIDS) are emerging to pose new threats; on the other hand, some of major diseases, for example, treatable and preventable diseases: Tuberculosis, Malaria and Cholera are making a deadly comeback in many parts of the world. Therefore, the study how a particular infectious disease progresses to ensure the likely outcome of an epidemic and information related to public health interventions become major concerns of public health. From last many decades, mathematical models are being used to understand the transmission dynamics, to predict the spreading patterns and future course of an outbreak and to evaluate suitable feedback strategies for various infectious diseases via relating the important factors of diseases to basic parameters of related models (see [5—7]). The Susceptible-Infected-Recovered(SIR for short) epidemic model proposed by Kermack and McKendrick [8] in the year 1927,is one of the very first compartmental models of infectious diseases. In which, they consider three compartments of homogeneous population (susceptible, infected and recovered compartments). The SIR epidemic model was very successful to capture too many observations of recorded epidemic data and also in predicting the behaviour of an outbreak. Further introducing one more compartment (exposed compartment) in SIR epidemic model, it was extend to Susceptible-Exposed-Infected-Recovered(SEIR for short) epidemic model (see [9]).

Multi-group Modelling: Heterogeneous Population: Although, while modelling simple SIRtype models, we assume the homogeneity (each individual is supposed to be similarly having random contacts) of population (see [10—13]). However, in general, during the modelling of an epidemic system, one may easily encounter an important theme i.e., population heterogeneity(in some sense, the individuals in the respective population are not similar to one another). In various aspects of disease transmission processes, one may realize heterogeneity, for instance,in case of sexually transmitted diseases, heterogeneous social behaviour, heterogeneity due to different geography, contrasting contact patterns and different number of sexual partners,different age groups with non-homogeneous susceptibility, heterogeneity in spatial distribution,heterogeneity due to multi-hosts pathogens in many diseases like West Nile Virus, subtypes of Influenza A,Plague(see[14—16]). Thus heterogeneity in host can come due to many population factors and its impact on respective epidemic dynamics has been studied in references [17—18]. To understand the transmission dynamics of infectious diseases (e.g., Mumps, HIV/AIDS,Measles, Gonorrhea etc.) in an heterogeneous host population, the population may be divided into various homogeneous groups depending on different types of heterogeneity, e.g., contact patterns of individuals such as those among children and adults for Measles, or having distinct number of sexual partners for HIV/ADIS and other sexually transmitted diseases (STDs for short),age of host individuals,professions of hosts,modes of disease transmission or geographic distribution of host population such as communities,cities and countries(as in the transmission of cholera). In general,multi-group modelling in epidemiology is used to recognize the role of the heterogeneity in population. This kind of multi-group modelling also helps to model the intergroup interaction and interactions within the groups. To include irregularity of infectiousness of the disease agent, we divide the host population into various groups according to their epidemiological characteristics. In the existing literature of epidemiological modelling, the multi-group epidemic models have been used to investigate the transmission pattern and to describe the transmission dynamics of many infectious diseases in heterogenous population such as Cholera(see [19]), Measles, HIV/ADIS (see [20]) and Gonorrhea(see [18]). In particular, to study the impact of variations in infectiousness of HIV, Hyman et al. [20] proposed two simple models. In the first model,during the chronic phase of infection,they considered different levels of virus between individuals. The second model was based on standard hypothesis depending upon an individual current disease stage and his/her infectiousness, the infected individuals progress via a series of infection phases. The Gonorroea multi-group model is one of the earliest work in the area of multi-group epidemic modelling. This particular model was used to describe the prevalence of Gonorroea and proposed by Lajmanovich and Yorke [18]. In this particularn-group Susceptible-Infected-Susceptible (SIS for short) model, the authors discussed global dynamics by establishing global stability of unique endemic equilibrium,using a quadratic global Lyapunov function. Following the work of Lajmanovich and Yorke [18], Beretta and Capasso[21] studied a multi-group SIR epidemic model assuming constant population in each group.In references [18, 21], the authors discussed the global dynamics of the model by establishing sufficient conditions for global stability of endemic equilibrium. A multi-group infectious disease model with temporary immunity of recovered population was studied by Hethcote [22]. They obtained threshold criterion to determine the immunization rates relating the eradication of the disease. Afterwards, various forms of multi-group epidemic models have been discussed in[23—26]. From the above critical review of epidemic modelling of heterogeneous population, it may easily be observed that establishing the global stability of endemic equilibrium is one of the main challenges while dealing the theoretical aspect of the model.

Age of Infection in an Epidemic Model: Most of the time, we assume the homogeneity of infected individuals (i.e., all infected individuals in that class have the same epidemiological parameters) after dividing a particular host population in different classes. However, it may be a unrealistic assumption. In reality, as time progresses, the disease develops within the host individuals with the different infectivity or many times one may easily observe different infectiousness of an infected individual at different stages of infection (e.g., Cholera, Typhoid).In particular, there are diseases, whose transmissibility increases with age of infection, for example, Ebola. On the other hand, there are diseases in which the infectiousness of infected individuals increases upto a threshold level with age of infection and then starts decreasing,like Influenza (see [27]). This suggests that infectivity of host might continuously change with time and infection age may be one of the informative factor to model some of the infectious diseases. Therefore, we may consider the infectivity as the function of infection age of infected individuals. Some of the epidemic models incorporating an individual infection for some particular diseases are: Tubercolosis (see [28]), HIV/AIDS (see [29]), Chagas disease (see [30]), or pandemic Influenza (see [31]). Feng et al. [28] studied the qualitative behaviour of system of ordinary equations and system of integral-differential equations, where both models describe the dynamics of Tuberculosis (TB for short) disease. The authors found that the dynamical behaviour of age structured TB model(model with variable latent period)is very similar to that of TB model with ordinary differential equations. Thieme et al. [29] explored the dynamics of HIV/AIDS model incorporating long and variable periods of infectiousness,variable infectivity.They established the conditions under which the endemic equilibrium is locally stable. They found that undamped oscillations may also occur if the variable infectivity is at a higher level at certain incubation period. In particular, Rost et al. [32] studied an SEIR epidemic model with varying infectivity by considering infection age of infected individuals. Recently,McCluskey[33]proposed an SEIR epidemic model including infected individuals with infection-age structure to allow the varying infectivity. Because in the varying infectivity, the incidence term has the formβS(t)hence, Li et al. [34] extended the results of Rost et al. [32] and McCluskey [33] to a multi-group SEIR epidemic model with distributed delays. The authors discussed the global dynamics of extended model by using a graph-theoretical approach to the method of Lyapunov functionals.

Feedback in Epidemic Model: In case of many epidemics, when complete eradication of disease is not possible, then whether we may change the endemicity level of disease, becomes an important question. This means we can maintain the endemic level below a threshold value so that outbreak of disease can be avoided. In some cases, we may also want to change the endemicity of the existing equilibrium maintaining its stability. On the other hand, in real world, ecological systems are continuously disrupted by unpredictable disturbances persisting for finite period of times. The presence of such unpredictable forces may also result in alteration of various parameters like survival rate. In the language of control theory, these unpredictable forces are called the feedback variables. By introducing feedback variables, we make improvement in the associated epidemic model system which may provide us population stabilizing at lower value of equilibrium. Sometimes, the disease can also be made endemic or extinct by choosing suitable values of associated feedback variables. The understanding of the feedback technique might be implemented by means of some biological and reasonable feedback or by some harvesting mechanism (see [35]). In ecology, one important and practical issue related to feedback variables is the “ecology balance”: Whether the considered ecosystem would be able to persist in the presence of such unpredictable disturbances. Indeed, in the time period of the last few decades, the dynamical behaviours of the population models with feedback have been deliberated significantly (see [36—41]). Fan et al. [42] proposed a logistic model incorporating feedback variable. They established the global stability of positive equilibrium using a new method combined with lower and upper solution technique. This method is much simple and convenient than Lyapunov function method of global stability. Yang et al. [43] studied an autonomous cooperative system with single feedback. They showed that the stable species could become die out or change its position of stable state maintaining its stability by choosing appropriate values of control parameters. An Susceptible-infected(SI for short)epidemic model with feedbacks in a patchy environment was investigated in reference[44]. They obtained global stability criteria of the disease-free and endemic states. The authors determined the global stability of endemic state using some results of graph theory. The global stability of epidemic models with feedback and the effects of feedback on the transmission dynamics of disease have been investigated in [45—47]. Thus, we rarely find few studies on the effect of feedback in multi-group epidemic models incorporating the effect of age of infection. Moreover,the impact of the infection age on the susceptible individuals has not been addressed in the presence of feedback variables. However, the study of the multi-group epidemic model with infection age and feedback can significantly contribute to the control of infection in more realistic situations.Motivated from above cited works,we propose a multi-group SEIR epidemic model system with infectivity as a function of infection age and feedback with the following objectives:

1. To establish the global dynamics of the model system using graph theoretic results.

2. To improve the understanding how feedback and age of infection influence the transmission dynamics of infectious diseases.

Graph-Theoretic Approach to Multi-group Epidemic Model: To understand the dynamics of model, basic reproduction number (R0) plays an important role. One tries to the study the caseR0≤1 andR0>1. For the caseR0≤1, there exists only one (disease free)equilibria and for the caseR0>1, there exist at least two equilibrium, one is called disease free equilibrium and the other one is known as endemic equilibrium. To study the global stability of an equilibrium, one tries to construct a suitable Lyapunov function. In general,the construction of a Lyapunov function is a difficult problem. In references [48—49], the authors elaborated a graph theoretic technique to study the global dynamics of endemic equilibrium by constructing a Lyapunov function. The method involves the complete description of the complicated pattern to construct a Lyapunov function. There are a few multi-group epidemic models (see [50—54]), in which this graph theoretic approach is used to determine the global stability of endemic equilibria of models. This method has been used to determine the global stability of unique endemic equilibrium of a multi-group SIR epidemic model which is described by ordinary differential equations (see [48]).

For basic notions of the graph theory, we refer the interested readers to [55—56]. Now, we mention some results to be used in the present paper.

Definition 1.1(see [55])Let B= (βkj)n×nbe a real matrix. If βkjare nonnegative for all k and j, then B is called non-negative matrix(i.e., B≥0). If B and a F= (fkj)n×nare both non-negative, then B?F≥0if and only if βkj≥fkjfor all k and j.

Definition 1.2(see [55])Let B= (βkj)n×nbe a non-negative matrix. If B satisfies one of the following properties, then B is called reducible

1.n=1and B=0,

2.n≥2, there exits a permutation matrix P, such that

where B1and B2are square matrices and PTis the transpose of matrix P. Otherwise, B is called irreducible.

We consider the linear system

where

is the Laplacian matrix of the directed graphG(B) associated to the matrixB.

Lemma 1.1(see [57])If B is non-negative and irreducible, then

(1)the spectral radius ρ(B)of B is a simple eigenvalue of B,and B has a positive eigenvector c=(c1,c2,···,cn)corresponding to ρ(B).

(2)If B≤F, then ρ(B)≤ρ(F)and furthermore, if B

(3)If F is a diagonal and positive(i.e., all entries of F are positive)matrix, then BF is irreducible.

Lemma 1.2(see[48])If the matrix B is irreducible and n≥2,then the following properties hold.

(1)The solution of linear system(1.1)is the space of dimension1, with a basis(v1,v2,···,vn) = (K11,K22,···,Knn), where Kiiis the cofactor of the i-th diagonal entry of matrix B,1 ≤i≤n.

(2)For all1 ≤i≤n,

whereTiis the set of all spanning subtrees of vertices of G(B)that are rooted at vertex i and E(T)is the set of all arcs of directed tree T.

The rest of the paper is organized as follows. In Section 2, we mathematically formulate our problem considering some basic assumptions. In Section 3, we prove the well-posedness(positivity and boundedness) of the proposed model system. In Section 4, we prove our main results about the global asymptotic stability of the disease free equilibrium and the endemic equilibrium of model system. In Section 5,numerical evaluations have been presented to support our theoretical results by taking an example of 2-group populations. Finally, in Section 6, we discuss our results including some ideas about future scope.

2 Mathematical Model Formulation

The heterogeneous host population is divided intonhomogeneous groups of population according to gender, age, profession, education levels and geographical distribution for their heterogeneity to disease transmission. Further, we divide anyk-th group into four compartments(Sk,Ek,Ik,Rk) to study our problem as compartmental model of epidemic, where 1 ≤k≤n.LetSkbe the susceptible individuals who are at risk of infection of disease,Ekbe the individuals of exposed class who are infected by disease but do not liable to spread disease, the infectious individualsIkwho are infected by disease and they will influence susceptible individuals, the recovered individualsRkwho are the recovered and have permanent immunity against the disease. We also assume that the susceptible individualsSk, the exposed individualsEkand the recovered individuals are homogeneous at any timetin thek-th group and the infectious individualsIkis structured by the infection ageθandik(t,θ)is the density of infectious individuals with infection ageθat the timetink-th group. We assume thatik(t,θ) = 0 for allθ > θ?(finite), whereθ?is the maximum infection age,for which the infectious individual can survive,that means an individual can stay in infectious class forθ?unit time. ThenIk(t)=is the total number of infectious individuals at timet. We make the following assumptions for our model system:

(A1) Ink-th group, new recruits in the total population take place at a rate Γk>0 at any moment of time. In which(1?pk)proportion of new recruits enter in the susceptible population and remainingpkproportion enter in the recovered population. That means some proportion of new recruiting population could not be susceptible to infection due to having permanent immunity.

(A2) Ink-th group, by using the vaccine against the virus/bacteria and provided immunity against the various diseases, the susceptible individuals (Sk) enter to recovered class (Rk) at a constant rateδk>0.

(A3)Fork-th group,Ukis feedback variable which satisfies certain differential equation and it influences susceptible individuals by a ratebkUk, wherebkis the feedback parameter.

(A4)Ink-th group,after the latent period,the individuals of exposed class turn into infectious individual class with a constant rateεk>0.

(A5)The infected individuals inIkclass can recover(through treatment or automatically)and get permanent immunity against disease, i.e.,γkis the recovery rate ofIk.

(A6) Here, we consider the cross-infection from all groups of infectious individuals to a group of susceptible individuals. Lethk(θ) be a bounded and non-negative continuous function ofθ,which represents the infectivity of infected individuals ofθinfection age ofk-th group.

(A7) Ink-th group, the coefficient of infection transmission for susceptible individualsSkturning into exposed individualsEkisβkj≥0, in which a susceptible individual makes contact with infectious individualsIkof thej-th group. For any two distinct groups (k-th andj-th),individuals ofIjcan infect individuals ofSkin direct or indirect mode, i.e.,βkjis irreducible matrix.

Under the above assumptions and discussions,the proposed multi-group SEIR model system is

wherek=1,2,···,nandUkis thek-th feedback variable.

The initial and boundary conditions of model system (2.1) are given by:

whereL+(0,∞) is the space of the non-negative functions.

In an age-structured infection age model,the variableik(t,θ)has two interpretations. Firstly, it can be interpreted as the density of infected individuals of infection ageθat timet(this means the actual number of infected individuals at timetbetween two infection agesθ1andθ2will be the integral of the density functionik(t,θ) overθ∈(θ1,θ2).Secondly, the density functionik(t,θ) evaluated at (t,a) has the interpretation of being the rate at timetat which individuals pass through agea. As a consequence,ih(t,0) is the overall birth rate for infected individuals (rate at which the exposed individuals become infected).

The total number of infected individuals is found by integrating the densityik(t,θ) over all agesθ∈[0,∞) as follows:

Note that we assume that the disease confers permanent immunity in the above model.This assumption makes the first four equations of model system (2.1) independent fromRk.Therefore, the dynamics of our model is governed by the following reduced system:

Now, letThenψ(θ)is the probability of an infected individual in thej-th group surviving to infection ageθ. Integrating the third equation of model system (2.1)and incorporating the initial conditions, we obtain

Letfj(θ) =hj(θ)εje?(dIj+γj)θbe the general kernel function. Then by (2.3) and (2.4), we obtain the following model system:

Here the kernel functionfk(θ)(≥0) is a continuous function of the infection ageθwithOur assumption on the kernel functionfk(θ) iswhereλkis a positive number,k=1,2,···,n. System(2.5)can be interpreted as a multigroup(sayn-group) epidemic model for an infectious disease whose latent periodθin the host is obtained from the general age of infection model (2.1). Here, we also realize that the model system (2.5) is an epidemic model with distributed time delay.

The dynamical behaviour of model system (2.3) is equivalent to that of system (2.5). Once we determine the solution of system (2.5), we can calculateik(t,θ) from(2.4). So, the stability of equilibria of model system (2.1) is the same as that of system (2.5). From here onward, we shall focus on the model system (2.5).

3 The Well-Posedness of Model

Due to infinite delay,it is necessary to determine the suitable phase space of state varibales.Therefore, we define the following Banach space of fading memory type (see [58]) for anyλk∈(0,dIk+γk),

Then, by standard theory of functional differential equations (see [59]), we haveEkt∈Ck.Therefore, we consider model system (2.5) in the phase space

Considering the biological meaning, we are only interested in the solutions which are nonnegative and bounded. Now, we show that all the solutions of model system (2.5) with initial conditions (3.1) are non-negative, i.e.,Sk(t) ≥0,Ek(t) ≥0,Uk(t) ≥0 for allt≥0 andk∈{1,2,···,n}.LetSk(t)>0 with initial valueSk(0)>0 for allk∈{1,2,···,n}. We suppose that it is not correct,then there exist a positive numbert1and somek1∈{1,2,···,n}such that

It follows from the first equation of model system (2.5)

Using the comparison theorem (see [60]) and taking the limit ast→t1, we obtain

which contradicts our supposition thatSk1(t1) = 0.Therefore, we conclude that ifSk(0)>0 thenSk(t)>0 for allt≥0 andk∈{1,2,···,n}. From the continuity of solution of system(2.5) around the initial condition, we have that ifSk(0) ≥0, thenSk(t) ≥0 for allt≥0 andk∈{1,2,···,n}.

Similarly, letEk(t)>0 with initial valueEk(0)>0 for allk∈{1,2,···,n}. We assume that it is not correct and there exist a positive numbert2andk2∈{1,2,···,n} such that,here,00.Then from the second equation of model system (2.5), we obtain0 andEk2(t)>0 for allt > t2. This is a contradiction to our assumption. Therefore,from continuity of solution of system(2.5)around the initial value,we conclude thatEk(0)≥0.ThusEk(t)≥0 for allt≥0 andk∈{1,2,···,n}, we have

Now, from the last equation of system (2.5):

By the comparison theorem (see [60]), we obtain

Thus, we conclude that ifUk(0)≥0, thenUk(t)≥0 for allt≥0 andk∈{1,2,···,n}.

Now, we show that solutions of system (2.5) with initial and boundary conditions (3.1) are bounded. For this, we haveSk(t)>0 fort >0.Thus, from the first equation of system (2.5),we obtainS′k(t)≤(1 ?pk)Γk?(dsk+δk)Sk(t).Hence,

By adding all the equations of system (2.5) for eachk, we obtain

whered?k=min{(dsk+δk?ek),(dEk+εk),fk} and (dsk+δk?ek)>0.

Thus, we obtain

Therefore, the following region is positively invariant for system (2.5)

and

is interior set ofξ.Thereforeξis also positively invariant for system (2.5).

4 Equilibria and Their Global Stability

System (2.5) always has a disease free equilibriumP0= (S01,0,U01,S02,0,U02,···,S0n,0,U0n)∈R3n+,where

The endemic equilibrium of system (2.5) is given by

and it is calculated by solving the following system of equations

whereaj=In epidemiology, the basic reproduction numberR0is defined as the total expected number of secondary cases produced by an infected individual during its total infectious period, in an entirely susceptible population (see [61]). The basic reproduction numberR0is determined by the spectral radius of a matrixQ0defined asQ0=qkj, where

Therefore

whereρ(Q0) denotes the spectral radius of matrixQ0. In epidemiology,R0plays a major role to determine the dynamical behavior of system and it acts as threshold. We shall determine the dynamical behavior of system (2.5) completely in terms ofR0.

4.1 Global stability of disease free equilibrium (DFE for short)

LetS=(S1,S2,···,Sn) andS0=(S01,S02,···,S0n).ThenQ0=Q(S0). Since 0 ≤Sk≤S0kfor allk, hence, we have 0 ≤Q(S)≤Q(S0)=M0. IfS/=S0, thenQ(S)

Theorem 4.1Assume that B= (βkj)n×nis irreducible. If R0≤1, thenDFEof system(2.5)is globally asymptotically stable. Moreover, if R0>1, thenDFEis unstable.

ProofWe know that matrixQ=Q0=is irreducible. Therefore, by Lemma 1.1, the matrixQhas a positive left eigenvector (w1,w2,···,wn) corresponding to the spectral radius of matrix (ρ(Q)>0). In particular,ρ(Q) =ρ(Q0) =R0≤1.Letck=andThus,ak(0)=

Now, we consider the following Lyapunov function:

Now, by using the disease free stationary state and integration by parts, we obtain

whereE(t)=(E1(t),E2(t),···,En(t))T.

LetY= {(S1,E1(·),U1,···,Sn,En(·),U1) ∈ξ V′= 0} andZbe the largest compact invariant subset ofY.Now, we prove thatZ= {P0}.From inequality (4.3) and assumptionck>0, ifL′= 0 then= 0 and (Uk?U0k)2= 0.Therefore,Sk(t) =andUk=U0k/=0. Hence, from the first equation of system (2.5), we have

and thus

Note thatB=(βkj)n×nis irreducible. So, for each 1 ≤j≤nandk/=j,we haveβkj/=0.Therefore, we obtain

which givesEjt(r)=0,r∈(?∞,0],j=1,2,···,n.Therefore,Z={P0}.

Now, using the LaSalle-Lyapunov invariance principle (see [62]),P0is globally stable inξ,ifR0≤1.

4.2 Global stability of endemic equilibrium

In this section,we consider thatR0>1. In this case,it follows from Theorem 4.1 that DFEP0is unstable. Form the uniform persistence results of [63] and similar argument as in the proof of[64,Proposition 3.3],we conclude that instability ofP0implies the uniform persistence of system (2.5) in positively invariant setξ. This means that, there exists a constantc(>0)such that

provided that (S1(0),E10(θ),U1(0),···,Sn(0),En0(θ),Un(0))∈ξ.

The uniform persistence of model system (2.5) along with boundedness of the solutions inξ, implies existence of an endemic equilibriumP?inξ, which is summarized in the following proposition.

Proposition 4.1If R0>1, then system(2.5)is uniformly persistent and there exists at least one endemic equilibrium P?in ξ.

Now, we prove our one main result using Proposition 4.1.

Theorem 4.2Assume that B= (βkj)is irreducible. If R0>1, then system(2.5)has a globally asymptotically stable endemic equilibrium P?inΞ.

ProofLetP?= (S?1,E?1,U?1,S?2,E?2,U?2,···,S?n,E?n,U?n),whereS?k,E?k,U?k>0 denote the endemic equilibrium of system (2.5). Let=βkjajS?kE?jandbe given by (1.1). Note thatis the Laplacian matrix of ().SinceBis irreducible,is also irreducible. Let{v1,v2,···,vn}(vk>0,1 ≤k≤n) be a basis for linear system (1.1) as discussed in Lemma 1.2. Now, we consider the Lyapunov function to establish the following global stability of endemic equilibrium:

where

and

Sinceψ(x) =x?1 ?lnx≥0 for allx >0.It is clear thatL(t) is always bounded for allt >0. L(t) ≥0 and the equality holds if and only ifSk=S?k, E?k=E(t?θ) =E?k, Uk=U?k.Now, differentiatingL1along the solution of system (2.5), we obtain

Using the equilibrium state equation of model system (2.5), we obtain

Now, differentiatingL2along the solution of model system (2.5) and using the integration by parts, we obtain

Now, we calculateL′(t)=L′1+L′2,

Here, we use the following notation

Now, we will show thatL′≤0 and for this we need to show thatHn=0. Here we consider the casesn=1 andn=2 separately and finallyn≥3.

Case-1: If we taken=1, then obviouslyH1=0. ThenL′(t)≤0 with equality satisfying if and only ifS1(t)=S?1,E1(t)=E1(t?θ)=E?1,U1(t)=U?1for allt≥0,θ∈[0,θ+].

Case-2:If we taken=2, thenH2=H2(E1,E2)=

Now, from Lemma 1.2, we obtainBy expandingH2, we have

Hence,H2= 0 andL′(t) ≤0.Equality satisfies if and only ifSk(t) =S?k,Ek(t) =Ek(t?θ)=E?k,Uk(t)=U?k,for allt≥0,θ∈[0,θ+], wherek=1,2.

Case-3:Letn≥3. The functionHnbecomes complicated. It is difficult to solve it manually. So we will use the graph theoretic technique. LetHn=H1n+H2n, whereH1n=In this case, we first prove thatH1n==0.

From (1.1), we have

By puttingwe obtain

Using (4.7), we obtain

Therefore,

Now, we show thatH2n=0,i.e.,=0 holds forE1,E2,···,En>0.

By Lemma 1.2,vk=Kkkis a sum of all rooted directed spanning subtreesTofGof root at vertexk.If we add a directed arc (k,j) from root vertexkto another vertexj,then we get a unicyclic subgraphXofGand each termvkβkjis the weightw(X) of unicyclic subgraphX.Further, we observe that the arc (k,j) is an arc of the unique cycleCXofX.Moreover, we can form the same unicylicXby adding each arc ofCXto corresponding treeT. Thus, the meaning of double sum (overkandj) inH2ncan be considered as a sum over all the arcs in the cycle of all the unicyclic subgraphHcontaining vertices {1,2,···,n} of graphG,that is

whereE(CX) is the set of all arcs of unique cycleCX. Then, we have

ThusH2n,X= 0 for each unicylic subgraphX.For example, letn= 2. The unique cycleCXhas two vertices {1,2} and makes a cycle 1 →2 →1.HereE(CX)={(1,2),(2,1)} and

Then we have= 0 holds forE1,E2,···,En>0,which gives thatHn= 0 forE1,E2,···,En>0.

Therefore,L′≤0 for all (S1,E1,U1,···,Sn,En,Un) ∈ξand equality holds if and only ifSk(t)=S?k, Ek(t)=Ek(t?θ)=E?k, Uk=U?kandHn=0. Hence, we conclude that the only invariant set of system (2.5) inis the endemic equilibriumP?.Thus, ifR0>1,thenP?is globally asymptotically stable and unique in theξ-region(LaSalle’s Invariance Principle, see [62]).

5 Numerical Simulations

In this section,we show the feasibility of our main theoretical results. We discuss the effects of feedback variables and infection age on the transmission of the infectious disease. For the simplicity of our model,we consider an example withn=2 for numerical simulation to support our main results.

A numerical example of associated ODE model system withn=2 would be given

In Figure 1, we show the disease transmission diagram of model system (3).

Figure 1 The transfer diagram for model system (2.1), where Pj =∫hj(θ)ij(t,θ)dθ and Pk =hk(θ)ik(t,θ)dθ. The red line is corresponding to the transmission process.

Figure 2 The graph shows changes in infectivity of infected individuals(for both groups)with respect to infection age.

In Figure 2,we describe the infectivity of infected individuals for varying infection age. Here we observe that the infectivity increases and becomes saturated after a threshold of infection age. This situation can arise to a disease which becomes more and more transmissible with increasing infection age. This type of infectivity function could be applicable to Ebola disease(see [27]).

Table 1 Model parameters and their definitions.

Table 2 Numerical values of parameters.

Figure 3 shows the long-time dynamics of our considered model (5.1) which supports our main mathematical results. Here we observe that the solution curves converge to the diseasefree equilibrium forR0≤1 (see Figure 3a). IfR0>1,then the solution curves converge to endemic state of disease (see Figure 3b). Thus, Figure 3 also ensures the global stability of the both equilibria of our model system (5.1).

Figure 3 The graphs show the long time dynamics of our model system (5.1). Here,blue dashed curves stand for group-1 and green solid curves for group-2. Numerical values of all parameters are given in Table 1.

In Figure 4a, we observe that if we take the infection age from Region-2 (2< θ <20)of Figure 2, then the infected population of that infection age converges to endemic state of disease (after reaching peak point). For Region-1 (θ <2) and Region-3 (θ >20) of Figure 2,the infected population converges with damped oscillations to the endemic state of disease. In Figure 4b,we easily observe that in the infection age model,damped oscillations become visible but the associated ODE model system (5.2) does not show any such oscillations.

Figure 4 The graphs show the effects of age of infection on long time dynamics of infected populations, where numerical values of all parameters are given in Table 2.

Moreover, the effects of four feedback parameters on infected population are compared in Figure 5. According to the gradient ofI, we rank the effect of parameters as follows:e2> f1> f2> e1. It is also found that almost all increases ineiorfiwill increase the total number of infections up excepte1. In the first row of Figure 5,e1is labeled on the Y-axis,Iwill decrease withe1rising. Meanwhile we probe the effecst ofeiandfion each groupI1andI2. We find that the parameters play the same role on each group onIbut different onI2ase1andf1vary. Two groups,I1andI2, receive the opposite effects frome1andf1. Specifically,whene1descends orf1rises, the number ofI1increases butI2show a decline which is shown in Figure 6. Interestingly,direction of gradient change in density completely turns around from an intuitive view. High-density diseased area forI1is low-density diseased area forI2,similarly,the low-density diseased area forI1is high-density diseased area forI2.

Figure 5 The graphs show the total number of infected population with respect to different values of ei or fi. We use I to represent the total number, which is equal to the sum of I1 and I2. Expect feedback parameters, other parameters are taken from Table 2. (A) e1 and e2 with f1 = f2 = 0.1; (B) e1 and f1 with e2 = f2 = 0.1; (C)e1 and f2 with e2 = f1 = 0.1; (D) f1 and f2 with e1 = e2 = 0.1; (E) e2 and f1 with e1 =f2 =0.1; (F) e2 and f2 with e1 =f1 =0.1.

Figure 6 The graphs show the number of each group I1 in (A) and I2 in (B) with different value of e1 and f1, corresponding to the total number I shown in Figure 5(B)in which e2 and f2 are equal to 0.1. The red area in (A) turns blue in (B), blue area in(A) turns red in (B).

There are some deductions. It may be a suitable method to modulate feedback parameters with obvious effects likee2andf1, when we intend to control the total infected population.However, when we want to restrict one of infected groups, both proper parameters and appropriate values must be deliberately selected because the influence over another group can not be neglected.

The Figure 7 represents the region of feedback parameters, in whichR0≤1 or the disease remains eradicated. The Figure 7 also establishes some thresholds of feedback parameters to the eradication of disease from all considered groups.

Figure 7 The graph shows the region (shadowed)for R0 ≤1,where f1 =f2 =5×10?10 and numerical values of all other parameters are given in Table 2.

6 Conclusion

Over the last few decades,the spread of different infectious diseases(both incurable diseases like HIV/AIDS and major diseases like Cholera) are posing continuous threatening to public health. More than fifty thousands men,women and children are dying everyday due to different kind of infectious diseases. At regular time periods, different strategies are being suggested to control the transmission of a particular disease, however, success and failure of such control strategies depend on various factors e.g., heterogeneity of host population. Multi-group approach of mathematical formulation of a particular infectious disease is one of different ways to incorporate the associated heterogeneity in the epidemic system. In this paper, a multi-group SEIR epidemic model system (2.1)incorporating infection age and feedback variables has been studied. The proposed model system (2.1) describes the transmission dynamics of the disease in a heterogeneous host population and via heterogeneity, the irregularity of infectiveness of infectious individuals have been incorporated. Both infection age and feedback have important influence on transmission dynamics of infectious diseases. The main contributions of our study are the following:

(a) The feedback strategy to control the infectious disease, is introduced into an SEIR multi-group epidemic model in which the effects of infection age are considered.

(b)The global stability of endemic equilibrium using some important graph theoretic results to Lyapunov function method has been established.

(c) The numerical simulations of a 2-group example show the influences of feedback and infection age on the dynamics of our proposed model.

Basic reproduction numberR0has been computed using the spectral radius of next generation matrix. It is found that the global behaviour of proposed multi-group model system is completely determined viaR0. More precisely,we show that ifR0≤1,then the disease dies out from all groups that means DFE is globally asymptotically stable. Further,we have also proved that ifR0>1, then the disease becomes endemic in all the groups. In this way,forR0>1,the global asymptotic stability of endemic equilibrium has been established using graph theoretical approach to the method of Lyapunov function. Further, via numerical simulations of a 2-group model system incorporating variable infectivity (infection age), we establish that the initial dynamics of system are very sensitive to the shape and timing of the first prevalence peak, but the long-term dynamics shows the same qualitative behaviour. That means the steady state of each infected individual approaches to almost the same endemic equilibrium. Thieme et al.[29] elaborated that undamped oscillations may also occur if the infectivity is at sufficiently higher level but our results show that the damped oscillations also occur in the initial dynamics of infected individuals. Therefore, this result is the answer of question in special case of work[29]. We have also found that the feedback not only changes the level of endemicity of disease but also can play a major role in the eradication of the disease from all considered groups. This particular result is consistent with result obtained in [43]. The feedback parameters have been ranked in order of effect on total number of infected population. This can provide a strategy to modulate the infected population: Priority to alter the parameter with strong effect is a simple way for a greater change in infected population. The region of feedback parameters has been described where the disease dies out from all groups. Furthermore, it is found that some of feedback parameters have dissimilar effects on different groups. When the overall infected population is adjusted by the control parameters, each group also needs to be considered.This requires special attention in practice. We have also quantified thresholds of infection age with respect to change of the dynamical behaviour of infected individuals and thresholds of the feedback in respect of eradication of infectious disease.

Our findings would necessarily contribute for more deeper understanding of role of feedback in the dynamics of infected individuals with infection age. This particular study may also provide important information for future modeling efforts in predicting future epidemics and establishing control strategies. In this way,the findings of this paper may be valuable for health policymakers who work on various types of suitable policies for controlling respective infectious diseases. Moreover, it may be interesting and more reasonable to further investigate of our proposed model by incorporating the death rate and removal rate of infected individuals and taking into account the function of infection age. The threshold dynamics of infected individuals can be investigated and that may change the global stability of equilibria into oscillations. The model system may undergo a Hopf bifurcation. We leave these ideas for future studies.