ANALYSIS OF AN AGE-DEPENDENT SI EPIDEMIC MODEL WITH DISEASE-INDUCED MORTALITY AND PROPORTIONATE MIXING ASSUMPTION: THE CASE OF VERTICALLY TRANSMITTED DISEASES M. EL-DOMA Received 3 January 23 and in revised form 8 February 24 An SI epidemic model for a vertically as well as horizontally transmitted disease is investigated when the fertility, natural mortality, and disease-induced mortality rates depend on age and the force of infection corresponds to a special form of intercohort transmission called proportionate mixing. We determine the steady states and obtain explicitly computable threshold conditions, and then perform stability analysis. 1. Introduction In this paper, we study an age-structured SI epidemic model, where age is assumed to be the chronological age, that is, the time since birth. The disease is fatal and horizontally as well as vertically transmitted. Horizontal transmission is the passing of infection through direct or indirect contact with infected individuals, for example, malaria is a horizontally transmitted disease. Vertical transmission is the passing of infection from parents to newborn or unborn offspring, for example, AIDS, Chagas, and hepatitis B are vertically as well as horizontally transmitted diseases. Vertical transmission plays an important role in maintaining some diseases, for example, see [5, 6. In [14, several examples of vertically transmitted diseases are given, and [5 is devoted to the study of the models and dynamics of vertically transmitted diseases. In this paper, we study an SI age-structured epidemic model with vertical transmission and disease-induced mortality rate. We determine the steady states, prove thresholds results, and then perform stability analysis. For the present model, we show that there is a parameter Rα, where αa isthe disease-induced mortality rate, which determines the existence of a unique endemic steady state if Rα < 1 <R = R. Actually, in this case, a trivial steady state is also a possible steady state, and if R < 1, then there is only a trivial steady state. The endemic steady state, under suitable conditions, is locally asymptotically stable whenever it exists; and the trivial steady state is globally stable if R < 1 and unstable if Rα < 1 <R. We also show that if qrα > 1, where q is the vertical transmission parameter, see Section 2 for definitions, then the only steady state for the model is the trivial steady Copyright 24 Journal of Applied Mathematics 24:3 24 235 253 2 Mathematics Subject Classification: 45K5, 45M1, 92D3, 92D25 URL: http://dx.doi.org/1.1155/s111757x43118x
236 An age-structured SI epidemic model with fatalities state. Note that if q =, that is, the case of no vertical transmission, then if Rα > 1, the only steady state is the trivial steady state. However, if Rα > 1, but qrα < 1, then an endemic steady state as well as a trivial steady state are possible steady states; the endemic steady state may not be unique in this case due to lack of monotonicity. In addition, we show that if we impose some conditions on the epidemiological and demographic parameters of the model, then it is possible to show that the model could give rise to a continuum of nontrivial endemic steady states. In this respect, this model behaves like the SIR model studied in [8. Also, by other assumptions, we could obtain that the total population consists of infectives only or susceptibles only. The stability of some of these steady states are determined. In [4, an SIR age-structured epidemic model with disease-induced mortality rate independent of age, and without vertical transmission, is considered. The analysis was carried out when the death, birth, and recovery rates are constants independent of age. Similar threshold conditions for the existence of the unique endemic equilibrium, when Rα < 1 <R, as in this paper are obtained in the special case q =. Furthermore, numerical investigations indicated that the endemic equilibrium is locally asymptotically stable, this is, in agreement with our results in this paper. In [12, 17, a McKendrick-Von Foerster type equation for an SI age-structured epidemic model with disease-induced mortality, but without assuming vertical transmission, is studied. It is a simpler version of the SIR age-structured epidemic model studied in [4. The steady states are determined and some stability results are given. The present paper generalizes the results of these two papers by including the case of vertical transmission. In [18, an SI age-structured epidemic model with vertical transmission as well as horizontal transmission is studied when the disease-induced mortality rate is constant, and a fraction of the offspring of infected mothers die of AIDS effectively at birth and the remaining fraction survive. The model is for HIV/AIDS, assuming HIV infection always leads to AIDS. Analytical results as well as numerical examples are given, and in a simple example, it has been shown that R > 1 is a requirement for the infection to develop; this is in agreement with our results in this paper. An extension to the model in [18 toan SIR age-structured epidemic model to model HIV/AIDS, assuming HIV infection does not necessarily lead to AIDS, is given in [19. The asymptotic behavior of the solution is explored and several numerical examples are given. In [1, refinements of the models in [18, 19 are discussed. We note that several recent papers have dealt with age-structured epidemic models with vertical transmission, but without disease-induced mortality, for example, see [8, 1, 11, 13. We observe that in such models, there is a restrictive condition on the epidemiological and demographic parameters of the model for the existence of the endemic equilibrium; for example, a typical condition would be, some quantity equals one, whereas for the models in [12, 17 and this paper, we see that there is some parameter range of values for which the endemic equilibrium exists. We also note that the model we investigate in this paper is based on some restrictive assumptions: we assume that the latent period is negligibly short and the infectivity is independent of the duration of the infection. As has been stated in [1, 3, 18, 19, this
M. El-Doma 237 simplification is a preliminary to numerical works for a more complicated and realistic model. For discussions of other types of models and assumptions, see [2, 7, 15, 16, 21, 22. The organization of this paper is as follows: in Section 2, wedescribethemodeland obtain the model equations; in Section 3, we reduce the model equations to several subsystems; in Section 4, we determine the steady states; and in Section 5, we perform stability analysis. 2. The model In this section, we consider an age-structured population of variable size exposed to a fatal communicable disease. The disease is both vertically and horizontally transmitted. We assume the following. 1 sa,tandia,t, respectively, denote the age-density for susceptibles and infectives of age a at time t. Then 2 a1 sa,tda = total number of susceptibles at time t of ages between a 1 and a 2 and 2 a1 ia,tda = total number of infectives at time t of ages between a 1 and a 2. We assume that the total population consists entirely of susceptibles and infectives. 2 The horizontal transmission of the disease occurs according to the following proportionate mixing assumption see Dietz and Schenzle [9: k 1 asa,t k 2 a ia, tda,wherek 1 a andk 2 a are bounded, nonnegative, continuous functions of a, and k 2 a is not identically zero. The term k 1 a k 2 a ia,tda is called force of infection and we let λt = k 2 aia,tda. 3 The fertility rate βa is nonnegativeand continuous,with compact support [,A A>. The number of births of susceptibles per unit time is given by s,t= βa[sa, t +1 qia,tda, q,1, where q is the probability of vertically transmitting the disease; i,t = q βaia,tda, 2.1 that is, allnewbornsfromsusceptiblesare susceptible, but a fraction q of newborns from infected parents are infective, that is, they acquire the disease via birth vertical transmission. 4 The natural death rate µa is the same for susceptibles and infectives and µaisa nonnegative, continuous function and there exists a [, suchthat µa > µ> a>a, µ a 2 >µ a1 a 2 >a 1 >a. 2.2 5 The disease-induced death rate αa is a nonnegative, continuous function of a [,. 6 The initial age distributions sa,= s aandia,= i a are continuous, nonnegative, and integrable functions of a [,. These assumptions lead to the following system of nonlinear integro-partial differential equations with nonlocal boundary conditions, which describes the dynamics of the
238 An age-structured SI epidemic model with fatalities transmission of the disease: sa,t + sa,t + µasa,t = k 1 asa,tλt, a>, t>, a t ia,t + ia,t + µaia,t = k 1 asa,tλt αaia,t, a>, t>, a t s,t = βa [ sa,t+1 qia,t da, t, i,t = q βaia,tda, t, λt = k 2 aia,tda, t, sa,= s a, ia,= i a, a. 2.3 We note that problem 2.3 is an SI epidemic model; the same model but with q = the case of no vertical transmission is dealt with in [12, 17. Also in [1, problem 2.3 is considered with α =. 3. Reduction of the model In this section, we develop some preliminary formal analysis of problem 2.3. We define pa,tby pa,t = sa,t+ia,t. 3.1 Then from 2.3, by adding the equations, we find that pa,t satisfiesthe following: pa,t a + pa,t + [ µa+αa pa,t = αasa,t, a>, t>, t p,t = Bt = βapa,tda, t, pa,= p a = s a+i a, a. 3.2 Also, from 2.3, sa,t and ia,t satisfy the following systems of equations: ia,t a sa,t a s,t = + ia,t t + sa,t + µasa,t = k 1 asa,tλt, a>, t>, t βa [ sa,t+1 qia,t da = Bt i,t, t, sa,= s a, a, + µaia,t = k 1 asa,tλt αaia,t, a>, t>, i,t = q βaia,tda, t, ia,= i a, a, λt = k 2 aia,tda, t. So, it is clear that 3.2, 3.3, and 3.4 are equivalent to the original problem 2.3. 3.3 3.4
4. The steady states M. El-Doma 239 In this section, we look at the steady-state solution of problem 2.3. A steady state s a, i a, λ,andb must satisfy the following equations: ds a + µas a = λ k 1 as a, a>, da s = βa [ s a+1 qi a da = B i, di a da + [ µa+αa i a = λ k 1 as a, a>, i = q λ = βai ada, 4.1 4.2 k 2 ai ada. 4.3 Anticipating our future needs, we define the threshold parameter Rαby Rα = where ha and other related functions are defined as follows: hada, 4.4 ha = βaπ 2 a, 4.5 π 2 a = πaexp ατdτ, 4.6 πa = exp µτdτ. 4.7 Note that Rα is the expected number of offspring of an infected individual at birth over a life time. Also note that R, usually denoted by R, is the net reproduction rate. Thus, R is the expected number of offspring produced in a life time by an individual in the absence of the disease. For example, if R > 1, then the total population is growing and becomes unbounded when t. In this paper, we will see that, in some situations, R may be greater than one but the disease controls the population size and thus, keeps the size bounded. By solving 4.1, we obtain that s a satisfies s a = s πaexp λ k 1 τdτ. 4.8 By using 4.8in4.2, we obtain that i a satisfies i a = i π 2 a+λ s π 2 a where F υ σisgivenby σ F υ σ = υσexp F k1 σdσ, 4.9 [ ατ λ k 1 τ dτ. 4.1
24 An age-structured SI epidemic model with fatalities Using 4.2and4.9, we find that where T is defined as follows: i T = qλ B haf k1 σdσ da, 4.11 T = 1 qrα+qλ haf k1 σdσ da. 4.12 From 4.3and4.9, weobtainthat λ [1 s f af k1 σdσda = i f ada, 4.13 where f aisdefinedasfollows: f a = k 2 aπ 2 a. 4.14 In the following proposition, we show that if λ =, then the steady state of problem 2.3 is either the disease-free equilibrium or the trivial equilibrium. Proposition 4.1. Suppose that λ =. Then the steady is either the trivial equilibrium s a =, i a =, or the disease-free equilibrium s a = B πa, i a =, R = 1. 4.15 Proof. If λ =, then from 4.13 andassumption2insection 2, i = ; and from 4.9, i a =. Therefore, from 4.1and4.8, s a = B πa. From 4.1, we find that B [1 R = ; and so if R = 1andB, we obtain the disease-free equilibrium 4.15; otherwise, we obtain B =, and accordingly, from 4.1 and 4.8, we obtain the trivial equilibrium s a =, i a =. This completes the proof of the proposition. We note that it is easy to see that 4.15solvesproblem2.3. We notice that from 4.11and4.12, and for a nontrivial equilibrium B, T = is equivalent to the following equations: qrα = 1, 4.16 qλ haf k1 σdσ =. 4.17 In the following proposition, we show that in the special case T = andr = 1, the steady state of problem 2.3 gives rise to a continuum of endemic equilibriums. Proposition 4.2. Suppose that the following two conditions hold: 1 R = q = 1, 2 the support of both k 1 a and αa lietorightofthesupportofβa. Then problem 2.3 gives rise to a continuum of endemic equilibriums. Proof. From assumptions 1 and 2, we obtain that Rα = 1, and therefore, 4.16 is satisfied; also by assumption 2, 4.17 is satisfied, and therefore, T =. From 4.11,
M. El-Doma 241 we see that i is undetermined. Using 4.9, 4.1, and 4.17, we obtain that s is also undetermined. Using 4.13, we see that for a fixed i,b andafixed s [,B, the left-hand side of 4.13 is an increasing function of λ and approaches + as λ, and equals zero if λ =. Accordingly, for each i,b and s [,B, we obtain λ > as a solution of 4.13 which gives rise to an endemic equilibrium. We note that if i =, then from 4.13, either λ = and thus, the steady state is given by Proposition 4.1,orλ > ; and the latter exists if and only if the following condition holds: s f aj k1 σdσda > 1, where J k1 σisdefinedasfollows: σ J υ σ = υσexp ατdτ. 4.18 This completes the proof of the proposition. We note that Proposition 4.2 proves that the SI model of this paper and the SIR model studied in [8 behave in a similar fashion in this special case. In the following proposition, we determine the steady state of problem 2.3 inthe special case T = andr > 1. Proposition 4.3. Suppose that the following three conditions hold: 1 R > 1, 2 qrα = 1, and3 the support of k 1 a lies to the right of the support of βa. Then the steady state of problem 2.3 is either the trivial equilibrium or is given by i a = B π 2 a, s a =, Rα = 1 = q. 4.19 Proof. From B = βa[s a+i ada and 4.8, 4.9, and assumption 3, we obtain that So, if Rα R, then we obtain the following: Using 4.1, we obtain that s satisfies B [R 1 = i [ R Rα. 4.2 i = B [ R 1 [. 4.21 R Rα s = B [ 1 Rα [ R Rα. 4.22 From 4.22, we deduce that Rα 1 since s, and hence Rα = 1 = q by assumption 2. Accordingly, s = and thus, i = B from 4.21; and hence we obtain 4.19from4.9. Now, if Rα = R,thenfrom4.2, we obtain that B = since R > 1; and hence the only steady state, in this case, is the trivial equilibrium s a = i a =. This completes the proof of the proposition. We note that it is easy to see that 4.19solvesproblem2.3.
242 An age-structured SI epidemic model with fatalities In the next proposition, we determine the steady state of problem 2.3 in the special case T andqrα = 1. Proposition 4.4. Suppose that qrα = 1 and 4.17 does not hold. Then the steady state of problem 2.3 is either the trivial solution or is given by 4.19. Proof. Since 4.17 does not hold, then T and thus, we can use 4.11 toobtainthat i = B ; which implies that s = and hence s a =. By using 4.9, we obtain that i a = B π 2 a. From 4.1, we obtain that = s = 1 qb Rα = Rα 1B. Accordingly, either Rα = 1, and therefore, q = 1; and thus, we obtain 4.19 or B = ; and thus, the steady state is the trivial equilibrium. This completes the proof of the proposition. Now, we consider the case T and therefore, we can use 4.11toobtainthati satisfies i = qλ B T Using 4.23and4.1, we obtain that s satisfies haf k1 σdσda. 4.23 s = B [ 1 qrα. 4.24 T Thus, using 4.23, 4.24, 4.9, 4.8, and 4.3, we obtain the following: λ [ 1 B T { q f ada haf k1 σdσ da 1 qrα } f af k1 σdσ da =. 4.25 In the following theorem, we describe the steady state of problem 2.3 in the case Rα < 1 <R. We note that R > 1 describes a situation in which the population size would grow unboundedly, provided the disease is absent, and the theorem describes a situation in which the presence of the disease keeps the population size bounded by mortality. Theorem 4.5. 1 Suppose that Rα < 1 <R. Then the steady state of problem 2.3iseither the trivial equilibrium, or is given by λ >, which is the unique solution of the following characteristic equation: 1 = Rα+q [ 1 Rα +1 q βaπaexp λ k 1 τdτ da haf α σdσda. 4.26 2 Suppose that R < 1. Then the trivial equilibrium s a = i a = is the only steady state.
M. El-Doma 243 Proof. To prove 1, note that since Rα < 1, then by 4.12, T>. Therefore, using B = βa[s a+i ada,and4.8, 4.9, 4.23, and 4.24, we obtain that { B 1 Rα q [ 1 Rα 1 q haf α σdσda βaπaexp λ } =. k 1 τdτ da 4.27 Accordingly, if the steady state is not trivial B, then λ satisfies 4.26. Now, if the support of k 1 a lies to the right of the support of βa, that is, βaπaexp λ a k 1 τdτda = R,thenfrom4.26, we obtain that R 1[1 qrα =, which is not possible since R > 1andRα < 1 by assumption 1. Therefore, 4.27 givesb = ; and therefore, in the case of nonfertile infectibles, the only steady state is the trivial equilibrium. If this special case does not occur, then we observe that the right-hand side of 4.26 is a decreasing function of λ and has a value equal to [1 qrαr + qrα > 1, if λ =, and tends to Rα < 1asλ. Therefore, there exists a unique λ > which solves 4.26 and gives rise to a unique endemic equilibrium via 4.25, 4.12, 4.94.23, 4.24, and 4.8. This proves 1. On the other hand, if R < 1, then this implies that Rα < 1, and therefore, from 4.12, T>. Thus, 4.27 holds; and accordingly as before, if the support of k 1 a lies to the right of the support of βa, then similar arguments to that given in the proof of 1 show that the only steady state, in this special case, is the trivial equilibrium. If this special case does not occur, then as before, the right-hand side of 4.26 is a decreasing function of λ, assumes the value [1 qrαr + qrα < 1, if λ =, and tends to Rα < 1asλ. Therefore, 4.26 and4.27 giveb =, and thus, the only steady state for problem 2.3, if R < 1, is the trivial equilibrium. This completes the proof of 2 and therefore, the proof of Theorem 4.5 is completed. In the following proposition, we determine the steady state of problem 2.3 in the special case of nonfertile infectibles and qrα < 1. Proposition 4.6. Suppose that qrα < 1 and the support of k 1 a lietotherightofthe support of βa. Then the steady state of problem 2.3 is the trivial equilibrium if R 1; and if R = 1, then the steady state is either the one given by Proposition 4.1 or a continuum of endemic equilibriums. Proof. From 4.9 and4.2, we obtain i [1 qrα = and thus, i = since qrα < 1. Hence from 4.9and4.13, we obtain the following: i a = λ s π 2 a F k1 σdσ, 4.28 λ [1 s f af k1 σdσ da =. 4.29 Now, from 4.8, 4.1, and the assumption about the support of k 1 a, we obtain that s [1 R =. 4.3
244 An age-structured SI epidemic model with fatalities So, if R 1, then from 4.3, we deduce that s =, and hence 4.28and4.8give the trivial equilibrium. Otherwise, if R = 1, then 4.3 gives that s isundetermined. Accordingly, from 4.29, we see that either λ =, and therefore, Proposition 4.1 gives the steady state, or 1 = s f af k1 σdσ da. 4.31 We notice that the right-hand side of 4.31 is a decreasing function of λ with a value greater than one provided we choose s > [ f aj k1 σdσ da 1 whenλ = and approaches zero as λ. Therefore, there exists λ > as a solution of 4.29 which gives rise to an endemic equilibrium. Thus, we obtain a continuum of endemic equilibriums. This completes the proof of the proposition. In the following lemma, we prove that if qrα > 1, then the only steady state of problem 2.3 is the trivial equilibrium. Note that if q =, that is, the case of no vertical transmission, then from 4.26, we can see that if Rα > 1, then the only steady state of problem 2.3, in this special case, is the trivial equilibrium. Lemma 4.7. Suppose that qrα > 1. Then the only steady state of problem 2.3 is the trivial equilibrium. Proof. We note that from 4.9and4.2, we obtain that i [ 1 qrα = qλ s haf k1 σdσda. 4.32 Thus, 4.32 gives that i = since i andqrα > 1. Also from 4.32, we obtain that qλ B haf k1 σdσ da =. 4.33 Now, from 4.8, 4.9, 4.33,and 4.1,we obtainthat B [1 βaπaexp λ k 1 τdτ da =. 4.34 Therefore, from 4.34, we see that either the steady state is trivial B = or 1 = βaπaexp λ k 1 τdτ da. 4.35 By integrating 4.12 by parts and using 4.35, we conclude that T. Accordingly, 4.33 gives the result since B must be equal to zero; otherwise, T is negative. Note that T>givess = by4.24 since s. This completes the proof of the lemma.
M. El-Doma 245 In the following theorem, we prove that problem 2.3 has an endemic equilibrium, when qrα < 1 <R.NotethatLemma 4.7 proved that if qrα > 1, then the only steady state for problem 2.3 is the trivial equilibrium. Theorem 4.8. Suppose that qrα < 1 <R. Then the steady state of problem 2.3 is either the trivial equilibrium or is given by λ > which is a solution of the following characteristic equation: 1 = qrα+ [ 1 qrα + λ 1 q βaπaexp λ k 1 τdτ da haf k1 σdσ da. 4.36 Proof. We note that T>since qrα < 1, and therefore, using 4.26, we obtain 4.36 by integration by parts. Now, if the support of k 1 a lies to the right of the support of βa, then similar arguments to that given in the proof of Theorem 4.5 show that the only steady state, in this special case, is the trivial equilibrium. If this special case does not occur; then the right-hand side of 4.36 assumes the value [1 qrαr + qrα > 1, if λ =, and approaches qrα < 1asλ. Therefore, 4.36 hasasolutionλ >, which gives rise to an endemic equilibrium via 4.23, 4.24, 4.12, 4.9, and 4.8. This completes the proofofthetheorem. We note that this endemic equilibrium may not be unique since the right-hand side of 4.36 may not be monotone. 5. Stability of the steady states In this section, we study the stability of the steady states for problem 2.3asgivenbythe results in Section 4. NotethatasinSection 4, wewillcontinuetodefinef υ, f, h, andj υ, respectively, by 4.1, 4.14, 4.5, and 4.18 throughoutthissection. By integrating problem 3.3 along characteristics lines t a = constant, we find that s a te t [µa t+τ+k 1a t+τλτdτ, a>t, sa,t = [ Bt a i,t a πae k 1τλt a+τdτ, a<t. By integrating problem 3.2 along characteristics lines t a = constant, we find that 5.1 pa,t = p a texp + t exp Bt aπ 2 a+ t [ µa t+τ+αa t+τ dτ t [ µa t+τ+αa t + τ dτ αa t+σsa t+σ,σdσ, σ [ exp µτ+ατ dτ ασsσ,t a+σdσ, σ a>t, a<t. 5.2
246 An age-structured SI epidemic model with fatalities From 3.2and5.2, wefindthat t Bt = habt ada+ + βaexp t t +σ a βa+σexp +σ a [ µτ+ατ dτ αasa,t σdadσ [ µa t + τ+αa t + τ dτ p a tda. By integrating problem 3.4 along characteristics lines t a = constant, we find that 5.3 ia,t t [ i a texp µa t + τ+αa t + τ dτ t t [ = + exp µa t+τ+αa t+τ dτ k 1 a t+σsa t+σ,σλσdσ, σ [ i,t aπ 2 a+ exp µτ+ατ dτ k 1 σsσ,t a+σλt a+σdσ, σ a>t, a<t. 5.4 From 3.4and5.4, we findthat λt = t + t k 2 a + σexp k 2 ai a texp +σ a t [ µτ+ατ dτ k 1 asa,t σλt σdadσ Setting i = Vt and using 3.4and5.4, we obtain that t Vt = q havt ada + q + q t t βa + σexp βai a texp [ t µa t + τ+αa t + τ da+ f ai,t ada. 5.5 +σ a [ µτ+ατ dτ k 1 asa,t σλt σdadσ [ µa t + τ+αa t + τ dτ da. 5.6 We note that, by assumptions 3, 4, and 6 in Section 2, and the dominated convergence theorem, t βap a texp t [ µa t + τ+αa t + τ dτ da, as t. 5.7
Also, by the same reasoning as above, q t t k 2 ai a texp βai a texp t t M. El-Doma 247 [ µa t + τ+αa t + τ dτ da, as t, [ µa t + τ+αa t + τ dτ da, as t. 5.8 Consequently, Bt, λt, and Vt satisfy the following limiting equations see Miller [2: Bt= λt= + habt ada ha+σ [ Bt a σ Vt a σ J α aexp f avt ada+ Vt=q havt ada+q f a+σj k1 a [ Bt a σ Vt a σ exp k 1 τλt a σ +τdτ dadσ, k 1 τλt a+σ +τdτ λt σdadσ, ha+σ [ Bt a σ Vt a σ J k1 aexp k 1 τλt a σ +τdτ λt σdadσ. 5.9 Now, we linearize the system of 5.9 by considering perturbations wt, ηt,and ζt defined by wt = λt λ, ηt = Bt B, ζt = Vt V. 5.1 Now, if we define wt xt = ηt, 5.11 ζt then the linearization of 5.9can be rewritten as follows: xt = where Aσisgivenby σ Y 11 λ f σ F k1 ada [ σ Aσ = Y 12 hσ 1+ F α ada σ λ qhσ F k1 ada Y 13 Aσxt σdσ, 5.12 [ σ f σ 1 λ F k1 ada σ hσ F α ada, 5.13 [ σ qhσ 1 λ F k1 ada
248 An age-structured SI epidemic model with fatalities where Y 11 = B V [ σ F k1 a f a + σda λ f a + σf k1 a + τdadτ, Y 12 = V B σ ha + σf α a + τk 1 adadτ, Y 13 = q B V [ σ ha + σf k1 ada λ ha + σk 1 af k1 a + τdadτ. 5.14 In the following theorem, we show that the trivial equilibrium s a = i a = is unstable if R > 1 and locally asymptotically stable if R < 1. Theorem 5.1. The trivial equilibrium B = is unstable if R > 1 and locally asymptotically stable if R < 1. Proof. We note that the characteristic equation for the system 5.12isgiveninAppendix A. If B =, then λ = V =, whence the characteristic equation A.1 becomes[1 q e σz hσdσ{1 e σz hσdσ σ e σz hσj α adadσ}=, and therefore, [ [ 1 q e σz hσdσ 1 e σz βσπσdσ =. 5.15 Setting z = x + iy and if we suppose that 1 = e σx+iy βσπσdσ, thenweobtain that 1 = e σx βσπσcosσydσ, 5.16 = e σx βσπσsinσydσ. 5.17 If R > 1, then from 5.16, we see that if we set y = and define a function gxby gx = e σx βσπσdσ, 5.18 then gx is a decreasing function for x>, gx asx,andg = R > 1. Therefore, there exists x > suchthatgx = 1. Accordingly, the trivial equilibrium is unstable. If R < 1, then 5.16 cannot be satisfied for x. Also, by similar argument, we see that 1 q e σz hσdσ = cannot be satisfied for x. Therefore, the trivial equilibrium is locally asymptotically stable if R < 1. This completes the proof of the theorem.
M. El-Doma 249 In the next theorem, we show that the trivial equilibrium B = isgloballystableif R < 1. A proof of this theorem is essentially contained in [5, and therefore, we omit the proof. Theorem 5.2. Suppose that R < 1. Then the trivial equilibrium is globally stable. In the following theorem, we show that the steady state given by Proposition 4.3 is always unstable, whereas the steady state given by Proposition 4.4 is either locally asymptotically stable or unstable, depending on a threshold parameter. Theorem 5.3. 1 The steady state given by Proposition 4.3 is unstable. 2 The steady state given by Proposition 4.4 is locally asymptotically stable if βaπaexp λ a k 1 cdcda < 1,andunstableif βaπaexp λ a k 1 cdcda > 1. Proof. To prove 1, we note that in this case, B = V, and therefore, the characteristic equation A.1 satisfies the following: [ [ σ 1 e σz hσdσ 1 e σz βσπσexp λ k 1 cdc dσ =. 5.19 Whence, by assumption 3 of Proposition 4.3,weobtainthat βaπaexp λ k 1 cdc da = βaπada = R > 1. 5.2 Therefore, this steady state is unstable by similar arguments as given in the proof of Theorem 5.1. This completes the proof of 1. The proof of 2 is similar and therefore is omitted. This completes the proof of the theorem. We note that if we set λ =, then the characteristic equation A.1 satisfies the following: [ {[ 1 e az βaπada 1 q [ 1 B qb e az hada e σz f a+σj k1 adadσ } e σz ha+σj k1 adadσ =. 5.21 e az f ada Then if we set B =, we see that we can recover the stability result for the trivial equilibrium given by Theorem 5.1; but in general, the stability of the disease-free equilibrium given by Proposition 4.1 is not determined due to lack of information about B. Anticipating our future needs, we define a function Db;λ ;B inappendix B. In the following theorem, we show that in the special case q =, the endemic equilibrium with λ > is locally asymptotically stable when Rα < 1 <R.
25 An age-structured SI epidemic model with fatalities Theorem 5.4. Suppose that 1 Rα < 1 <R, 2 ddb;λ ;B /db, 3 β + B f af k1 ada >. Then the unique endemic steady state with λ > is locally asymptotically stable. Proof. We start by noting that the imaginary part of the characteristic equation A.1can be rewritten in the following form: e xb sin ybd b;λ ;B db =. 5.22 We also note that D;λ ;B = β + B f af k1 ada > by assumption 3. We can check that Db;λ ;B asb since βb has compact support; and µa is nonnegative and eventually positive and increasing. Therefore, by assumption 2, Db;λ ;B decreases to zero as b and e xb Db;λ ;B decreasestozeroforx. Accordingly, e xb sin ybd b;λ ;B db, if y. 5.23 If y =, then the real part of the characteristic equation A.1 satisfies [ 1 B e xσ f a + σf k1 adadσ σ + λ B [ 1 e xσ hσdσ + λ B [ [ σ σ e xσ f a + σf k1 a + τk 1 adadτ dσ X σ e xσ hσf α adadσ e xσ f σf k1 adadσ e xσ ha + σf α a + τk 1 adadτ dσ =. 5.24 Now, using 4.26 and4.25, we get that 5.24 cannot be satisfied when x. Accordingly, the steady state with λ > is locally asymptotically stable. This completes the proofofthetheorem. In the general case q, we can show that the endemic equilibrium is locally asymptotically stable as in Theorem 5.4. But observe that the function Db;λ ;B ;V, which corresponds to the function Db;λ ;B, in the special case q =, takes several pages to write, and therefore, we omit the details of the proof.
M. El-Doma 251 Appendices A. The characteristic equation for the system 5.12 {1 B V [ e σz f a + σf k1 adadσ σ } λ e σz f a + σf k1 a + τk 1 adadτ dσ {[ σ 1 e σz hσdσ e σz hσf α adadσ [ 1 q + λ q + λ σ { B V [ 1 q σ e σz hσdσ + qλ e σz hσf α adadσ e σz f σf k1 adadσ σ σ e σz hσf k1 adadσ σ } e σz hσf k1 adadσ e σz ha + σf α a + τk 1 adadτ dσ σ e σz hσf k1 adadσ e σz hσdσ + qλ + q B V σ e σz hσf α adadσ [ e σz ha + σf k1 adadσ σ } λ e σz ha + σf k1 a + τk 1 adadτ dσ [ σ + λ e σz f σf k1 adadσ e σz f σdσ {λ q V B σ e σz ha + σf α a + τk 1 adadτdσ σ e σz hσf k1 adadσ + q B V [ 1 e σz hσdσ σ e σz hσf α adadσ [ e σz ha + σf k1 adadσ σ } λ e σz ha + σf k1 a + τk 1 adadτ dσ =, where F υ σ, f a, and ha are given, respectively, by 4.1, 4.14, and 4.5. A.1
252 An age-structured SI epidemic model with fatalities B. The function Db;λ ;B D b;λ ;B b = hb+hb F α ada { + B f a + bf k1 ada b b b + λ B { + hb σ f a + σf k1 af α mdmdσ da } hb σ f a + σf k1 adσ da b b σ b b σ b σ b f a + σf k1 a + τk 1 af α mhb σdτ dmdσ da f b σha + σf α a + τk 1 af k1 mdτ dmdσ da hb σ f a + σf k1 a + τk 1 adτ dadσ f a + bf k1 a + τk 1 adadτ }, B.1 where F υ σ, f a, and ha are given, respectively, by 4.1, 4.14, and 4.5. Acknowledgments This paper was completed while the author was visiting The Abdus Salam ICTP, and he wishes to thank the Director Professor M. A. Virasoro for invitation and hospitality at the centre during his stay. The author would like to thank Professor Mimmo Iannelli, Professor Herbert W. Hethcote, Professor Horst R. Thieme, Professor G. C. Wake, Professor K. Dietz, Professor Michel Langlais, and Professor Carlos Castillo-Chavez for giving him references. This research was supported in part by the Third World Academy of Sciences TWAS under Research Grant Agreement no. 98-364 RG/MATHS/AF/AC. References [1 R. M. Anderson and R. M. May, Infectious Diseases of Humans: Dynamics and Control,Oxford University Press, Oxford, 1991. [2, Understanding the AIDS pandemic, Scientific American 266 1992, no. 5, 58 66. [3 R. M. Anderson, R. M. May, and A. R. McLean, Possible demographic consequences of AIDS in developing countries, Nature 322 1988, no. 6161, 228 234. [4 V. Andreasen, Disease regulation of age-structured host populations, Theoret. Population Biol. 36 1989, no. 2, 214 239. [5 S. Busenberg and K. Cooke, Vertically Transmitted Diseases. Models and Dynamics, Biomathematics, vol. 23, Springer-Verlag, Berlin, 1993. [6 S.BusenbergandK.L.Cooke,The population dynamics of two vertically transmitted infections, Theoret. Population Biol. 33 1988, no. 2, 181 198.
M. El-Doma 253 [7 S. N. Busenberg and K. P. Hadeler, Demography and epidemics, Math. Biosci. 11 199, no. 1, 63 74. [8 Y. Cha, M. Iannelli, and F. A. Milner, Existence and uniqueness of endemic states for the agestructured S-I-R epidemic model, Math. Biosci. 15 1998, no. 2, 177 19. [9 K. Dietz and D. Schenzle, Proportionate mixing models for age-dependent infection transmission, J. Math. Biol. 22 1985, no. 1, 117 12. [1 M. El-Doma, Analysis of nonlinear integro-differential equations arising in age-dependent epidemic models, Nonlinear Anal. TMA 11 1987, no. 8, 913 937. [11, Analysis of a general age-dependent vaccination model for a vertically transmitted disease, Nonlinear Times and Digest 2 1995, no. 2, 147 171. [12, Analysis of an age-dependent SI epidemic model with disease-induced mortality and proportionate mixing assumption, Int. J. Appl. Math. 3 2, no. 3, 233 247. [13, Stability analysis of a general age-dependent vaccination model for a vertically transmitted disease under the proportionate mixing assumption, IMA J. Math. Appl. Med. Biol. 17 2, no. 2, 119 136. [14 P. E. Fine, Vectors and vertical transmissions: an epidemiologic perspective, Ann. N. Y. Acad. Sci. 266 1975, 173 194. [15 H. W. Hethcote and J. W. Van Ark, Epidemiological models for heterogeneous populations: proportionate mixing, parameter estimation, and immunization programs, Math. Biosci. 84 1987, no. 1, 85 118. [16 W. M. Liu, H. W. Hethcote, and S. A. Levin, Dynamical behavior of epidemiological models with nonlinear incidence rates, J. Math. Biol. 25 1987, no. 4, 359 38. [17 K. Louie, M. G. Roberts, and G. C. Wake, The regulation of an age-structured population by a fatal disease, IMA J. Math. Appl. Med. Biol. 11 1994, no. 4, 229 244. [18 R. M. May, R. M. Anderson, and A. R. McLean, Possible demographic consequences of HIV/AIDS epidemics. I. Assuming HIV infection always leads to AIDS, Math. Biosci. 9 1988, no. 1-2, 475 55. [19, Possible demographic consequences of HIV/AIDS epidemics. II. Assuming HIV infection does not necessarily lead to AIDS, Mathematical Approaches to Problems in Resource Management and Epidemiology New York, 1987 C. Castillo-Chavez, S. A. Levin, and C. A. Shoemaker, eds., Lecture Notes in Biomath., vol. 81, Springer, Berlin, 1989, pp. 22 248. [2 R. K. Miller, Nonlinear Volterra Integral Equations, W. A. Benjamin, California, 1971. [21 A. Pugliese, Population models for diseases with no recovery, J. Math. Biol. 28 199, no. 1, 65 82. [22 H. R. Thieme, Epidemic and demographic interaction in the spread of potentially fatal diseases in growing populations, Math. Biosci. 111 1992, no. 1, 99 13. M. El-Doma: Faculty of Mathematical Sciences, University of Khartoum, P.O. Box 321, 11115 Khartoum, Sudan E-mail address: meldoma@yahoo.com
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