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b Meiji Yasuda Life Insurance Company, 2-1-1 Marunouchi Chiyoda-ku, Tokyo ... Finally, individuals may remain asymptomatic for the rest of their lives.
Mathematical Biosciences 190 (2004) 39–69 www.elsevier.com/locate/mbs

A mathematical model for Chagas disease with infection-age-dependent infectivity Hisashi Inaba a

a,*

, Hisashi Sekine

b

Department of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba Meguro-ku, Tokyo 153-8914, Japan b Meiji Yasuda Life Insurance Company, 2-1-1 Marunouchi Chiyoda-ku, Tokyo 100-0005, Japan Received 26 November 2002; received in revised form 28 January 2004; accepted 25 February 2004 Available online 9 April 2004

Abstract In this paper we develop a mathematical model for Chagas disease with infection-age-dependent infectivity. The effects of vector and blood transfusion transmission are considered, and the infected population is structured by the infection age (the time elapsed from infection). The authors identify the basic reproduction ratio R0 and show that the disease can invade into the susceptible population and unique endemic steady state exists if R0 > 1, whereas the disease dies out if R0 is small enough. We show that depending on parameters, backward bifurcation of endemic steady state can occur, so even if R0 < 1, there could exist endemic steady states. We also discuss local and global stability of steady states. Ó 2004 Elsevier Inc. All rights reserved. Keywords: Chagas disease; Epidemic model; Backward bifurcation; Infection-age-dependent infectivity

1. Introduction Chagas disease or American trypanosomiasis is a vector transmitted, endemic disease in Central and South America. It is the second to malaria in the continent in the number of people infected and at risk. An estimated 16–18 million person are infected with Trypanosoma cruzi, the causative agent of Chagas disease, and 100 million people are at risk of infection, neither adequate drugs nor a vaccine is available. Chagas disease has three stages. The acute stage follows the invasion of the bloodstream by the protozoan parasite Trypanosoma cruzi. This stage lasts from 1 to 2 months, and infected *

Corresponding author. Fax: +81-3 5465 8343. E-mail address: [email protected] (H. Inaba).

0025-5564/$ - see front matter Ó 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.mbs.2004.02.004

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H. Inaba, H. Sekine / Mathematical Biosciences 190 (2004) 39–69

individuals may or may not show symptoms of the disease. After the acute phase, the infected individual enters the chronic stage which has a variable duration that goes from 10 to 20 years. At its end, the disease may follow three different paths. Individuals may develop megasyndromes; others may present myocarditis which is the terminal form with highest mortality in the group of 20 to 50 years of age. Finally, individuals may remain asymptomatic for the rest of their lives. Individuals in this group may live an ordinary life although some may die of ‘sudden death’ associated with heart failure produced by the parasite. The disease is transmitted by hematophagous arthroped (Homoptera: Reduviidae), and also transmitted by blood transfusion. Based on data for 1993–94, prevalence rate among blood donors in Bolivia reached the level beyond 10%, but in other countries of Central and South America, prevalence rates were at most several percent [21]. Thus the horizontal transmission risk is not so high in general, but we could not neglect this factor in some cases. So far some authors have developed mathematical models for Chagas disease. Busenberg and Vargas [5] have developed an SI model that allows host population growth, on the other hand, Velasco-Hern andez [24,25] assumed that demographic reproduction of host population is in its replacement level, but his model can distinguish the acute stage and the chronic stage. Since their models are formulated by system of ordinary differential equations, transition rates between disease status are constant. Cohen and G€ urtler [6] have developed a model for household transmission taking into account the existence of reservoirs (domestic animals) of the protozoan parasite. In this paper, we formulate a structured population model for the spread of Chagas disease in a demographically steady state host population. Different from those authors mentioned above, in order to reflect the dependence of disease progress on the duration since infection (disease age), we assume that the infected population is structured by the disease age, and the infection rate and the removed rate depends on the disease age.

2. The basic model We divide the host population under consideration into two groups: SðtÞ (uninfected, but susceptible) and IðtÞ (infected, acutely or chronically ill hosts) where t denotes time. In contrast to the Velasco-Hern andez’s model, we do not divide the infected population into two compartments, the acute stage and the chronic stage. Instead, we introduce infection-age s that is time since the moment of being infected. Then the infected host population I is stratified by infection-age as Z 1 iðt; sÞ ds; IðtÞ ¼ 0

where iðt; Þ denotes the infection-age density at time t. Then the total size of host population at risk is given by Z 1 iðt; sÞ ds: ð2:1Þ T ðtÞ ¼ SðtÞ þ 0

In fact, the acute stage is rather short (from 1 to 2 months) in comparison with the length of the chronic stage (from 10 to 20 years), so this simplification would not affect seriously the long term

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dynamics of Chagas disease. On the other hand, the advantage of our modeling is that the epidemiological difference between the acute stage and the chronic stage could be reflected in the infection-age dependent infectivity, and we can take into account the infection-age dependent removal rate. In rigorous sense there would be possibility that the removed population is alive and bitten again by vectors, or multiple infectious bites on infected people may affect the disease progression and the ultimate fate. But for simplicity, we assume that the removed population will no longer be involved in the transmission process, and multiple infectious bites on infected hosts does not play a role in the progress of the disease. Let MðtÞ denote the number of susceptible vectors at time t, V ðtÞ the number of infected vectors at time t and U ðtÞ :¼ MðtÞ þ V ðtÞ be the total number of vectors. Let b1 (b2 ) be the birth rate of host population (vectors), l1 (l2 ) the death rate of host population (vectors). For our purpose, it is reasonable to assume that lj > 0, bj > 0 (j ¼ 1; 2) and l1  l2 . If we let a be the number of bites per vector per unit time and c be the proportion of infected bites that give rise to infection, then the V vector makes acV infectious bites of which a fraction S=T are on susceptible hosts. That is, the number of new infection of hosts per unit time by vector transmission is given by acV ðS=T Þ. In reality, the number of bites per vector per unit time would depend on the ratio of the number of host population to the number of vectors, but here we assume that a is a constant for simplicity. Next let k be the average number of blood transfusion per infected host per unit time and hðsÞ be the probability that a blood transfusion from infected hosts with infection-age s infects the susceptible host. That is, here we assume that blood donors are randomly chosen from the total population, and so there is no screening and the recipients of blood donations are donating blood themselves at the same rate as anybody else. This is an unrealistic assumption, but we will use it for simplicity. Then the number of new infection of hosts per unit time by blood transfusion from infected host with infection-age s is given by ðS=T Þk  hðsÞiðt; sÞ. Hence the force of infection (the probability per unit of time for a susceptible to become infected) for the host population, denoted by k1 ðtÞ, is given by   Z 1 1 hðsÞiðt; sÞ ds ; ð2:2Þ aV ðtÞ þ k1 ðtÞ ¼ T ðtÞ 0 where a :¼ ac and hðsÞ :¼ k  hðsÞ. By using similar argument, we know that the force of infection for the vector population, denoted by k2 ðtÞ, is given by Z 1 1 k2 ðtÞ ¼ bðsÞiðt; sÞ ds; ð2:3Þ T ðtÞ 0  and bðsÞ  denotes the proportion of bites to infected hosts with infection-age s where bðsÞ :¼ abðsÞ that give rise to infection in vector. Finally we assume that infected hosts are removed from the infected status with an infection-age dependent rate cðsÞ. Under the above assumption, our basic model for Chagas disease can be formulated as follows: dSðtÞ ¼ b1  k1 ðtÞSðtÞ  l1 SðtÞ; dt

ð2:4aÞ

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oiðt; sÞ oiðt; sÞ þ ¼ ðl1 þ cðsÞÞiðt; sÞ; ot os

ð2:4bÞ

iðt; 0Þ ¼ BðtÞ;

ð2:4cÞ

dMðtÞ ¼ b2  CðtÞ  l2 MðtÞ; dt

ð2:4dÞ

dV ðtÞ ð2:4eÞ ¼ CðtÞ  l2 V ðtÞ; dt where BðtÞ denotes the number of newly infected hosts per unit time and CðtÞ the the number of newly infected vectors per unit time given by BðtÞ ¼ k1 ðtÞSðtÞ;

CðtÞ ¼ k2 ðtÞMðtÞ:

ð2:5Þ

In the following, we assume that c, b and h are non-negative, bounded integrable functions, and the initial conditions are given by Sð0Þ ¼ S0 > 0; ið0; sÞ ¼ i0 ðsÞ 2 L1þ ðRþ Þ; Mð0Þ ¼ M0 > 0; V ð0Þ ¼ V0 > 0: Note that if we seek for classical solutions of the above differential equation system, we assume the consistency condition as   Z 1 S0 i0 ð0Þ ¼ Bð0Þ aV0 þ hðsÞi0 ðsÞ ds : S0 þ ki0 kL1 0 On the other hand, as we see below if we once convert the above system to an integral equation system (which is essentially equivalent with the differential equation system as the epidemic modeling) or if we adopt a semigroup setting to seek the solution in a weak sense (see [19]), we do not need to mind the consistency. If we add two equations (2.4d) and (2.4e) for vector population, we obtain a single equation for the total vector population U :¼ M þ V as dU ðtÞ ð2:6Þ ¼ b2  l2 U ðtÞ: dt It is easily seen that U  :¼ b2 =l2 is a globally stable steady state. Then in our modeling, we can assume without loss of generality that UðtÞ equals to the constant b2 =l2 for all t P 0 provided that M0 þ V0 ¼ U  ¼ b2 =l2 . Of course, this is an assumption for model simplification, in fact the seasonal variation of the density of vector would play an important role in the spread of Chagas disease. Next by integrating (2.4b) along the characteristic line, we have  Bðt  sÞCðsÞ; t  s > 0; iðt; sÞ ¼ ð2:7Þ CðsÞ ; s  t > 0; i0 ðs  tÞ CðstÞ where we have used the notations:   Z s cðrÞ dr ; CðsÞ :¼ exp  l1 s  0

which is the survival rate at duration s in the infected stage.

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Then, by formal integration of (2.4a), (2.4d) and (2.4e), we obtain the following expressions: 8   b1 b1 > þ S  SðtÞ ¼ el1 t  ð‘1  BÞðtÞ; > 0 > l1 l1 <   ð2:8Þ MðtÞ ¼ lb22 þ M0  lb22 el2 t  ð‘2  CÞðtÞ; > > > : V ðtÞ ¼ V0 el2 t þ ð‘2  CÞðtÞ; where ‘j ðsÞ :¼ expðlj sÞ ðj ¼ 1; 2Þ and the asterisk denotes the convolution operation defined by Z t ðf  gÞðtÞ ¼ f ðt  rÞgðrÞ dr: 0

From the above argument, we know that SðtÞ, T ðtÞ, MðtÞ and V ðtÞ are determined if once B and C are given. Thus for any fixed t0 > 0, we can define operators Sðf Þ, T ðf Þ, Mðf Þ and V ðf Þ for f ðÞ 2 Cð½0; t0 ; Rþ Þ (the set of non-negative continuous functions on ½0; t0 ) as   b1 b1 l1 t  ð‘1  f ÞðtÞ; ð2:9aÞ e Sðf ÞðtÞ :¼ þ S0  l1 l1 T ðf ÞðtÞ :¼ Sðf ÞðtÞ þ ðC  f ÞðtÞ þ F0 ðtÞ;   b2 b2 l2 t  ð‘2  f ÞðtÞ; e Mðf ÞðtÞ :¼ þ M0  l2 l2

ð2:9bÞ

V ðf ÞðtÞ :¼ V0 el2 t þ ð‘2  f ÞðtÞ;

ð2:9dÞ

ð2:9cÞ

where F0 is a given function depending on the initial data as Z 1 CðsÞ F0 ðtÞ :¼ i0 ðs  tÞ ds: Cðs  tÞ t Note that from the expression (2.7), T ðBÞðtÞ denotes the total size of risk population with the number of newly infected hosts per unit time BðÞ 2 Cð½0; t0 ; Rþ Þ. It follows from (2.2), (2.3) and (2.7), we obtain that   Z 1 Z t SðtÞ CðsÞ BðtÞ ¼ hðsÞCðsÞBðt  sÞ ds þ hðsÞ aV ðtÞ þ i0 ðs  tÞ ds ; T ðtÞ Cðs  tÞ 0 t Z t  Z 1 SðtÞ CðsÞ CðtÞ ¼ bðsÞCðsÞBðt  sÞ ds þ bðsÞ i0 ðs  tÞ ds : T ðtÞ Cðs  tÞ 0 t By using the above operators given by (2.9), we arrive at the following system of integral equations: BðtÞ ¼

SðBÞðtÞ ½aV ðCÞðtÞ þ ðhC  BÞðtÞ þ F1 ðtÞ; T ðBÞðtÞ

ð2:10aÞ

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CðtÞ ¼

MðCÞðtÞ ½ðbC  BÞðtÞ þ F2 ðtÞ; T ðBÞðtÞ

where Fj are known functions given by Z 1 CðsÞ F1 ðtÞ :¼ hðsÞ i0 ðs  tÞ ds; Cðs  tÞ t Z 1 CðsÞ i0 ðs  tÞ ds: bðsÞ F2 ðtÞ :¼ Cðs  tÞ t

ð2:10bÞ

ð2:11aÞ ð2:11bÞ

Although we omit the proof here, by applying standard fixed-point argument to (2.10), we can easily see that the model is well-posed, that is, the system (2.10) has a unique non-negative solution with respect to the non-negative initial conditions for any t0 > 0. For more technical detail, the reader may refer to [11], [20, Chapter VI, Section 2] and [23]. As is mentioned above, the semigroup approach to (2.4) is also possible, by which the linearized stability principle used in the following section can be proved [19,22].

3. The basic reproduction ratio In this section, we discuss the disease initial invasion phase to show the important role of the basic reproduction ratio. The basic reproduction ratio, 1 denoted by R0 , is defined as the expected number of secondary cases produced in a completely susceptible population by a typical infected individual during its entire period of infectiousness ([8] and [9, Chapter 5]). Here it should be remarked that in traditional references for vector transmitted diseases, the basic reproduction ratio is defined as the average number of secondary cases in the host population arising from a single primary case in the host population via vector cases ([1, Chapter 14] and [2, p. 100]). In this sense, R0 is the growth factor for the two-generation unit (host-vectorhost), so it is also called as the two-generation factor [8, p. 367]. On the other hand, according to the definition by Diekmann et al. [8], R0 is the asymptotic pergeneration growth factor for the norm of the population vector whose components are the densities of hosts and vectors. Therefore, R0 is calculated as the spectral radius of the next-generation operator, which transforms the population vector of primary cases into the population vector of secondary cases. First let us calculate the basic reproduction ratio defined as the per-generation growth factor. In the initial invasion phase, the system is assumed to be in the disease-free steady state given by   b1 b2     ; 0; ; 0 : ðS ; i ; M ; V Þ ¼ l1 l2 In order to reduce the heterogeneity due to the disease age, we focus on the renewal process of the population vector composed of new cases, so let us define the population vector as 1 In tradition, R0 is often called as the basic reproduction rate or the reproductive rate, but we follow the advice by Diekmann et al. [8] to call it number or ratio, since it does not have a dimension.

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PðtÞ :¼ ðBðtÞ; CðtÞÞT (T denotes the transpose of the vector). At the disease-free steady state, we can assume that SðtÞ ¼ T ðtÞ ¼ b1 =l1 , MðtÞ=T ðtÞ ¼ b2 l1 =ðb1 l2 Þ, then it follows from (2.2), (2.3) and (2.5) that Z 1 BðtÞ ¼ aV ðtÞ þ hðsÞiðt; sÞ ds; ð3:1aÞ 0

CðtÞ ¼

b2 l1 b1 l2

Z

1

ð3:1bÞ

bðsÞiðt; sÞ ds: 0

From (2.4e) and (2.7), we have expressions 8 R t l ðtsÞ l t > < RV ðtÞ ¼ V0 e 2 þ 0 e 2 RCðsÞ ds; 1 t hðsÞiðt; sÞ ds ¼ F1 ðtÞ þ 0 hðsÞCðsÞBðt  sÞ ds; 0 > : R 1 bðsÞiðt; sÞ ds ¼ F ðtÞ þ R t bðsÞCðsÞBðt  sÞ ds: 2 0 0

ð3:2Þ

Therefore we arrive at an integral equation system as Z t PðsÞPðt  sÞ ds; PðtÞ ¼ FðtÞ þ

ð3:3Þ

0

where FðtÞ and PðsÞ defined by   F1 ðtÞ þ aV0 el2 t FðtÞ :¼ ; b2 l1 F ðtÞ b1 l2

2

 PðsÞ :¼

hðsÞCðsÞ b2 l1 bðsÞCðsÞ b1 l2

 a‘2 ðsÞ : 0

Then the next-generation operator, denoted by KðSÞ, can be calculated as   Z 1 hh; Ci a=l2 PðsÞ ds ¼ b2 l1 hb; Ci : KðSÞ ¼ 0

ð3:4Þ

b1 l2

0

In fact, if ðf; gÞT is a given vector whose elements denote the densities of newly infected hosts and vectors, then      hh; Cif þ ða=l2 Þg hh; Ci a=l2 f ¼ b2 l1 b2 l1 hb; Ci 0 hb; Cif g b1 l2 b1 l2 is the vector whose elements are the densities of secondary infected hosts and vectors produced by ðf; gÞT during its entire period of infectiousness. Hence, we can calculate the per-generation growth factor as the spectral radius, denoted by rðKðSÞÞ, of the next-generation operator KðSÞ, 1

R0 ¼ rðKðSÞÞ ¼ lim kKðSÞn kn ; n!1

ð3:5Þ

where k  k denotes the operator norm. From the Perron–Frobenius theory for positive matrix, we know that rðKðSÞÞ is the unique positive dominant eigenvalue of KðSÞ if KðSÞ is primitive 2 [4, Chapter 2]. In our case, the next-generation matrix is primitive, we can calculate R0 as

2

A non-negative matrix A is called primitive if there exists a natural number m such that Am is positive.

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sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# " 1 4ab2 l1 hb; Ci hh; Ci þ hh; Ci þ : R0 ¼ 2 b1 l22

ð3:6Þ

Next let us consider the two-generation factor. 3 In (3.3), if we insert the expression of CðtÞ into the integral equation of BðtÞ, we obtain a single equation as Z t hðsÞCðsÞBðt  sÞ ds þ F1 ðtÞ þ aV0 el2 t BðtÞ ¼ 0 Z s  Z ab2 l1 t l2 ðtsÞ þ e bðsÞCðsÞBðs  sÞ ds þ F2 ðsÞ ds: ð3:7Þ b1 l2 0 0 By changing the order of integration and changing variables in (3.7), we have Z t Z t Z s Z s l2 ðtsÞ l2 ðtsÞ e bðsÞCðsÞBðs  sÞ ds ds ¼ e bðs  sÞCðs  sÞBðsÞ ds ds 0 0 Z0 t Z 0t ¼ BðsÞ ds el2 ðtsÞ bðs  sÞCðs  sÞ ds 0 s Z t Z ts ¼ BðsÞ ds el2 ðtszÞ bðzÞCðzÞ dz 0  Z0 t  Z s l2 ðszÞ ¼ e bðzÞCðzÞ dz Bðt  sÞ ds: 0

0

Therefore the system (3.3) can be reduced to a scalar Volterra integral equation describing the renewal process of newly infected host population: Z t WðsÞBðt  sÞ ds; ð3:8Þ BðtÞ ¼ GðtÞ þ 0

where

Z ab2 l1 t l2 ðtsÞ þ e F2 ðsÞ ds; GðtÞ :¼ F1 ðtÞ þ aV0 e b1 l2 0 Z ab2 l1 s l2 ðszÞ e bðzÞCðzÞ dz: WðsÞ :¼ hðsÞCðsÞ þ b1 l2 0 l2 t

Then it is easily seen that the following estimate holds: jGðtÞj 6 khk1 ki0 kL1 eðl1 þcÞt þ ajV0 jel2 t þ kbk1 ki0 kL1

eðl1 þcÞt  el2 t ; l2  l1  c

ð3:9Þ

where k  k1 denotes L1 norm, k  kL1 denotes L1 norm and c :¼ inf s P 0 cðsÞ. Moreover we have kWkL1 ¼ hh; Ci þ

ab2 l1 hb; Ci; b1 l22

where we have used the notation as hf ; gi :¼

ð3:10Þ R1 0

f ðxÞgðxÞ dx.

3 The pure vector transmission case with constant transmission rates is already considered in [16], where R0 is used to denote the two-generation factor.

H. Inaba, H. Sekine / Mathematical Biosciences 190 (2004) 39–69

47

Since it follows from (3.9) that limt!1 GðtÞ ¼ 0, we know that if the resolvent kernel of (3.8) is integrable, we have limt!1 BðtÞ ¼ 0, that is, the solution of (3.8) is asymptotically stable. The Paley–Wiener theorem tells us that the resolvent kernel of (3.8) is integrable if and only if Z 1 ezs WðsÞ ds 6¼ 0 ð3:11Þ 1 0

for all z 2 fz 2 C : Rz P 0g [10,13]. From the standard argument about the renewal equation with positive kernel like (3.8), the condition (3.11) holds if and only if Z 1 kWkL1 ¼ WðsÞ ds < 1: ð3:12Þ 0

That is, (3.12) is a necessary and sufficient condition for the asymptotic stability of the solution BðtÞ. In (3.10), hh; Ci is the basic reproduction ratio for the blood transfusion, that is, the number of secondary cases in the host population produced by an infected host via blood transfusion during its entire period of infectiousness. On the other hand, ðb2 =l2 Þðl1 =b1 Þhb; Ci denotes the number of infected vectors produced by an infected host during its entire period of infectiousness and a=l2 denotes the number of infectious bites per an infected vector during its entire period of infectiousness, that is, the second term of the right-hand side of (3.10) is the two-generation factor for the vector transmission. Thus we could expect that kWkL1 acts as the reproduction ratio from host to host. In fact, we can see that Proposition 3.1. Let R0 ¼ rðKðSÞÞ. If R0 > 1, then kWkL1 > 1; if R0 ¼ 1, then kWkL1 ¼ 1; if R0 < 1, then kWkL1 < 1. Proof. Let f ðxÞ ¼ 0 be the characteristic equation of the next-generation operator given by (3.4). Then we have f ðxÞ :¼ detðxL  KðSÞÞ ¼ x2  hh; Cix 

ab2 l1 hb; Ci : b1 l22

Observe that f ð0Þ < 0;

f ð1Þ ¼ 1  kWkL1 ;

f ðR0 Þ ¼ 0:

Then from graphic consideration, it is easy to see that R0 > 1 if and only if kWkL1 > 1, and so on. This completes our proof. h From the above proposition, instead of rðKðSÞÞ we can equivalently use kWkL1 as the threshold criterion. Since the expression (3.10) is much more convenient for our purpose, we will treat kWkL1 as the basic reproduction ratio R0 for Chagas disease. That is, in the following, R0 always refer to kWkL1 . Finally let us combine R0 with the dynamical behavior of the basic system (2.4). It is easy to see that the linearization of the basic system at the disease-free steady state leads to the following system:

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H. Inaba, H. Sekine / Mathematical Biosciences 190 (2004) 39–69

oiðt; sÞ oiðt; sÞ þ ¼ ðl1 þ cðsÞÞiðt; sÞ; ot os Z 1 iðt; 0Þ ¼ aV ðtÞ þ hðsÞiðt; sÞ ds;

ð3:13aÞ ð3:13bÞ

0

dV ðtÞ b2 l1 ¼ dt b1 l2

Z

1

bðsÞiðt; sÞ ds  l2 V ðtÞ;

ð3:13cÞ

0

where we omit the equations for SðtÞ and MðtÞ, since they are determined from iðt; sÞ and V ðtÞ. From the previous argument, it is clear that the linearized system (3.13) can be reduced to the integral equation system (3.3) or the equation (3.8). Thus the stability result for the renewal equation (3.8) shows that if R0 < 1, the zero solution (trivial steady state) of the linearized system (3.13) is asymptotically stable, and it is unstable if R0 > 1. Moreover since it can be proved that the principle of linearized stability holds for PDE system as (2.4) by using the semigroup approach (see Appendix of [19]), the stability and instability of the zero solution in the linearized system (3.13) implies the local stability and instability of the disease-free steady state in the original system (2.4). That is, we can conclude that: Proposition 3.2. Let R0 ¼ kWkL1 . If R0 < 1, the disease-free steady state of the basic system (2.4) is locally asymptotically stable, and if R0 > 1 it is unstable. In general, the threshold criterion in epidemiology states that the disease can invade into the totally susceptible host population if R0 > 1, whereas it cannot if R0 < 1. For our system (2.4), the above proposition tells that the threshold criteria holds if the size of infected invading population is small enough. On the other hand, as we see below, the condition R0 < 1 may be not necessarily sufficient to guarantee the global stability for the disease-free steady state. It follows from (2.1) and (2.4) that Z 1 dT ðtÞ ¼ b1  l1 T ðtÞ  cðsÞiðt; sÞ ds P b1  ðl1 þ cÞT ðtÞ; dt 0 where c :¼ sups P 0 cðsÞ. Then we obtain ! b1 b1 eðl1 þcÞt : T ðtÞ P þ T ð0Þ  l1 þ c l1 þ c

ð3:14Þ

Hence if T ð0Þ P b1 =ðl1 þ cÞ, it follows that T ðtÞ P

b1 l1 þ c

8t P 0:

ð3:15Þ

Even if T ð0Þ < b1 =ðl1 þ cÞ, it follows from (3.14) that the condition T ðtÞ P b1 =ðl1 þ cÞ will be satisfied as time goes to infinity, so without loss of generality, we can assume that T ð0Þ P b1 =ðl1 þ cÞ holds in advance. In this case, we have the following global stability result:

H. Inaba, H. Sekine / Mathematical Biosciences 190 (2004) 39–69

49

Proposition 3.3. Let R be the number defined by R :¼ hh; Ci þ

ab2 ðl1 þ cÞ hb; Ci: b1 l22

ð3:16Þ

Then if T ð0Þ P b1 =ðl1 þ cÞ and R < 1, the disease-free steady state is globally asymptotically stable. Proof. From the above argument, if we assume that T ð0Þ P b1 =ðl1 þ cÞ, then it follows from (2.10b) that CðtÞ 6

ðl1 þ cÞb2 ½ðbC  BÞðtÞ þ F2 ðtÞ: l2 b1

ð3:17Þ

On the other hand, it follows from (2.10a) that BðtÞ 6 aV ðCÞðtÞ þ ðhC  BÞðtÞ þ F1 ðtÞ:

ð3:18Þ

By substituting (3.17) into (3.18) and using (2.9d), we obtain the following inequality as ! ! ðl1 þ cÞab2 hC þ ð‘2  bCÞ  B ðtÞ; ð3:19Þ BðtÞ 6 F3 ðtÞ þ l2 b1 where F3 ðtÞ :¼ F1 ðtÞ þ aV0 el2 t þ

ðl1 þ cÞab2 ð‘2  F2 ÞðtÞ: l2 b1

Under the assumption that l1 < l2 , it is easy to show that there exists a number g > 0 such that jF3 ðtÞj 6 gel1 t :

ð3:20Þ

In fact, we can observe from (2.11) that  0 k 1 el1 t ; jF1 ðtÞj 6  hki0 kL1 el1 t ; jF2 ðtÞj 6 bki L  where b :¼ sups P 0 bðsÞ. Moreover we have Z t Z t l2 ðtsÞ  0 k 1 el1 s ds jð‘2  F2 ÞðtÞj 6 e jF2 ðsÞj ds 6 el2 ðtsÞ bki L 0  0 ðl l Þt   0k 1 2 1 bki L  0k 1 1  e el1 t : el1 t 6 ¼ bki L l2  l1 l2  l1

ð3:21Þ

Therefore we arrive at the following estimate: " #  0k 1  ðl þ c Þab bki 2 1 L el1 t : jF3 ðtÞj 6  hki0 kL1 þ aV0 þ l2 b1 l2  l1 Then we know that there exists a g > 0 such that (3.20) holds. Finally let BðtÞ be the solution of integral equation as Z t BðtÞ ¼ F3 ðtÞ þ KðsÞBðt  sÞ ds; 0

50

H. Inaba, H. Sekine / Mathematical Biosciences 190 (2004) 39–69

where KðsÞ :¼ hðsÞCðsÞ þ

ðl1 þ cÞab2 ð‘2  bCÞðsÞ; l2 b1

then Rwe have that BðtÞ 6 BðtÞ. Hence again using the Paley–Wiener theorem, we know that if 1 R ¼ 0 KðsÞ ds < 1, then it follows that limt!1 BðtÞ ! 0, hence BðtÞ ! 0 as t ! 1. This completes our proof. h Of course, R < 1 is a sufficient condition to the threshold condition R0 < 1, since R > R0 . In other words, we can say that if the basic reproduction ratio is sufficiently smaller than the unity, the disease will be naturally eradicated from the host population regardless of its prevalence.

4. Endemic steady states Let us denote S  , i ðsÞ, M  and V  as the stationary states of SðtÞ, iðt; sÞ, MðtÞ and V ðtÞ respectively, and let kj ðj ¼ 1; 2Þ be the force of infection corresponding to the stationary state. Then we have 8 0 ¼ b1  k1 S   l1 S  ; > > > > di ðsÞ  > > < ds ¼ ðl1 þ cðsÞÞi ðsÞ; ð4:1Þ i ð0Þ ¼ k1 S  ; > >   >  > 0 ¼ b2  k2 M  l2 M ; > > : 0 ¼ k2 M   l2 V  : Then it is easy to obtain the expression for steady states as follows:   b1 b1 k1 CðsÞ   ; ; ðS ; i ðsÞÞ ¼ l1 þ k1 l1 þ k1   b2 k2 b2   ; ðM ; V Þ ¼ ; l2 þ k2 l2 ðl2 þ k2 Þ T  ¼ ð1 þ ek1 Þ

b1 ; l1 þ k1

ð4:2aÞ ð4:2bÞ ð4:2cÞ

where e :¼ h1; Ci is the total duration in the infected status of host population, and the force of infection at the steady state are given by    l1 þ k1 ak2 b2 b1   þ k1 hh; Ci ; ð4:3aÞ k1 ¼ b1 ð1 þ ek1 Þ l2 l2 þ k2 l1 þ k1 k2 ¼

k1 hb; Ci : 1 þ ek1

ð4:3bÞ

H. Inaba, H. Sekine / Mathematical Biosciences 190 (2004) 39–69

From (4.3a) and (4.3b), we can derive an equation satisfied by k1 as   k1 ab2 ðl1 þ k1 Þhb; Ci þ hh; Ci : k1 ¼ 1 þ ek1 l2 b1 k1 ðhb; Ci þ l2 eÞ þ l2 If k1 ¼ 0, we obtain the disease-free steady state as   b1 b2     ; 0; ; 0 : ðS ; i ; M ; V Þ ¼ l1 l2

51

ð4:3cÞ

ð4:3dÞ

On the other hand, if there exists non-trivial steady state, the corresponding force of infection 6 0 must satisfy the characteristic equation as k1 ¼   1 ab2 ðl1 þ k1 Þhb; Ci  þ hh; Ci ¼ 1: ð4:4Þ Rðk1 Þ :¼ 1 þ ek1 l2 b1 k1 ðhb; Ci þ el2 Þ þ l2 By using (4.2), Rðk1 Þ can be expressed as follows:   S a M   hb; Ci þ hh; Ci : Rðk1 Þ ¼  l2 T  T This decomposition shows that for a given force of infection k, RðkÞ is the sum of the average number of newly infected hosts infected by infectious vectors produced by an infected host during its infective period and the average number of newly infected hosts produced by horizontal transmission from an infected individual during its infective period. Then Eq. (4.4) shows that at equilibrium, the effective reproductive number of the disease is Rðk1 Þ ¼ 1, that is, at the endemic steady state, each infection will on average produce exactly one secondary infection [1, Chapter 2]. As is often observed for common childhood diseases, if the effective reproductive number is decreasing with respect to k, there exists at most one endemic steady state, so the bifurcation of endemic steady state from the disease-free steady state is always supercritical. However, for our model we have another possibility. Again observe that RðkÞ is written as RðkÞ ¼ R1 ðkÞ þ R2 ðkÞ; where ab2 hb; Ci R1 ðkÞ :¼ l2 b1 1 þ ek R2 ðkÞ :¼

! l2 l1  hb;Ciþel 1 2 þ ; hb; Ci þ el2 kðhb; Ci þ el2 Þ þ l2

hh; Ci ; 1 þ ek

where R1 is the effective reproductive number for vector transmission and R2 is the effective reproductive number for blood transfusion. Then it is easy to see that R2 ðkÞ is a decreasing function, but R1 ðkÞ is also a decreasing function if E :¼ l1 

l2 P 0: hb; Ci þ el2

ð4:5Þ

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H. Inaba, H. Sekine / Mathematical Biosciences 190 (2004) 39–69

Then we can conclude that if the condition (4.5) holds, the equation RðkÞ ¼ 1 has at most one positive solution, since RðkÞ is a monotone decreasing function. Then it is easy to see that the following threshold property holds: Proposition 4.1. Suppose that E P 0. Then if R0 6 1, there is no endemic steady state and if R0 > 1, there is a unique endemic steady state. Corollary 4.2. The bifurcation of endemic steady state at R0 ¼ 1 is supercritical if cðsÞ, s 2 ½0; s0 , is small enough for sufficiently large s0 . In particular, the backward bifurcation does not occur if c  0. Proof. Observe that l2 l1 E ¼ l1  ¼ l1  : hb; Ci þ el2 l1 hb; Ci=l2 þ el1 If cðsÞ 6  for 0 6 s 6 s0 , we have l1 ð1  eðl1 þÞs0 Þ þ 1 P el1 P l1 þ 

Z

1

l1 s

e

Rs s0

cðrÞdr

ds:

s0

The right-hand side of the above inequality goes to one if  ! 0 and s0 ! 1, then we have E > 0. This completes our proof. h On the other hand, if E < 0, then it would be possible that RðkÞ has a unimodal pattern, hence RðkÞ ¼ 1 could have two positive roots when R0 < 1, that is, the bifurcation of the endemic steady state at R0 ¼ 1 could be subcritical. Biologically speaking, E < 0 means that as the prevalence rises, the susceptible vectors more and more encounter the infected hosts and the infected vectors may be more efficiently produced by an infected host, since we assume that a (the number of bites per vector per unit time) is constant. If the additional effort of the vector to find hosts is negligible as the size of host population decreases, our assumption would be reasonable. On the other hand, if a depends on the size of host population, for example, it is more and more difficult for vectors to bite host population as the size of hosts decreases, our results will be altered. This latter case will be more appropriate in a sparsely populated area. Now let us go back to our model analysis. More precisely analyzing the behavior of the reproduction number RðkÞ, we can formulate conditions for subcritical bifurcation to occur. Proposition 4.3. Let us define numbers G and D as G :¼ el22 A1 þ A3 EðA2 þ el2 Þ; D :¼ e2 ðA1 l2 þ A3 EÞ2  eA1 G ¼ A1 A3 Eehb; Ci þ A23 E2 e2 ; where E is given by (4.5) and Aj (j ¼ 1; 2; 3) are all positive numbers given by A1 :¼

ab2 hb; Ci þ hh; Ci; l2 b1 hb; Ci þ l2 e

A2 :¼ hb; Ci þ l2 e;

H. Inaba, H. Sekine / Mathematical Biosciences 190 (2004) 39–69

A3 :¼

53

ab2 hb; Ci: l2 b1

If G P 0, then there exists pffiffiffiffi a unique endemic steady state if and only if R0 > 1. If G < 0, then n :¼ ððA1 l2 þ A3 EÞe þ DÞ=ðeA1 A2 Þ > 0, R0 < RðnÞ and the following holds: 1. 2. 3. 4.

If If If If

1 6 R0 , there exists only one endemic steady state. R0 < 1 < RðnÞ, then there exists two endemic steady state. RðnÞ ¼ 1, then there exists only one endemic steady state. RðnÞ < 1, then there is no endemic steady state.

Proof. Observe that R0 ðkÞ ¼ 

e ð1 þ ekÞ

 A1 þ 2

 A3 E 1 A2 A3 E f ðkÞ  ¼ ; A2 k þ l2 1 þ ek ðA2 k þ l2 Þ2 ð1 þ ekÞ2 ðA2 k þ l2 Þ2

where f ðkÞ :¼ eA1 A22 k2 þ 2eA2 ðA1 l2 þ A3 EÞk þ G: Then we know that if f ðkÞ P 0 for k P 0, RðkÞ is monotone decreasing function and Rð1Þ ¼ 0, hence RðkÞ ¼ 1 has only one positive root if and only if Rð0Þ ¼ R0 > 1. It is easy to see that f ðkÞ has a positive root if and only if f ð0Þ ¼ G < 0, since eA1 A22 > 0 and l2 A1 þ A3 E ¼ l2 hh; Ci þ

ab2 l1 hb; Ci > 0: b1 l2

Then if G P 0, f ðkÞ P 0 for all k P 0, then RðkÞ is monotone decreasing function. On the other hand, if G < 0, we have D > 0 and R0 ðkÞ ¼ 

eA1 A2 ðk  nÞðk  gÞ ð1 þ ekÞ2 ðA2 k þ l2 Þ2

where n > 0 and ðA1 l2 þ A3 EÞe  g :¼ eA1 A2

pffiffiffiffi D

;

:

Then we know that RðkÞ attains its maximum for k P 0 at k ¼ n and R0 ¼ Rð0Þ < RðnÞ. Therefore if R0 < 1, RðkÞ ¼ 1 has two positive root if RðnÞ > 1, it has only one positive root if RðnÞ ¼ 1 and it has no positive root if RðnÞ < 1. This completes our proof. h Note that if we assume that h ¼ 0 and b and c are constant, we have   ab2 bðl2 þ bÞ l2 c l1  G¼ : l2 þ b l2 b1 ðl1 þ cÞ2 In this case, the backward bifurcation condition G < 0 is equivalent to a simple condition l1  l2 c=ðl2 þ bÞ < 0, which is already given in [16].

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H. Inaba, H. Sekine / Mathematical Biosciences 190 (2004) 39–69

The above argument shows that the reason of occurring a backward bifurcation is the unimodal pattern of the effective reproductive number as a function of the force of mortality, but it is still unclear whether there is a parameter set realizing the backward bifurcation, since R0 and G cannot be given independently. In order to see this possibility, let us introduce another characterization of endemic steady states, since to calculate RðnÞ is not easy task. From (4.4) we know that if k1 is the force of infection corresponding to the endemic steady state, it must be a positive solution of the quadratic equation as  2   Q Q2 þ 1 ð4:6Þ Px2 þ Qx þ ð1  R0 Þ ¼ P x þ  R0 ¼ 0; 2P 4P where

  hb; Ci þe ; P :¼ e l2

  hb; Ci ab2 hb; Ci : Q :¼ e þ ð1  hh; CiÞ þe  l2 l22 b1 Though we omit the proof, it is easy to see from graphic consideration that we can rephrase Proposition 4.3 as follows: Proposition 4.4. If Q P 0, there exists a unique endemic steady state if and only if R0 > 1. If Q < 0, the following holds: 1. 2. 3. 4.

If If If If

R0 P 1, there exists a unique endemic steady state. R0 < 1 and H :¼ Q2  4P ð1  R0 Þ > 0, there exist two endemic steady states. R0 < 1 and H ¼ 0, there is a unique endemic steady state. R0 < 1 and H < 0, there is no endemic steady state.

From Proposition 4.4, we know that Q plays an important rule because it determines whether the bifurcation of steady states at R0 ¼ 1 is subcritical or supercritical. In particular, the condition Q < 0 is necessary for the occurrence of backward bifurcation. If we define Rc :¼ 1  Q2 =4P , then H ¼ 4P ðR0  Rc Þ. If H > 0 and R0 < 1, the force of infection corresponding to two endemic steady states are calculated as pffiffiffiffi H   ; ð4:7Þ k1 ¼ kc  2P where kc :¼ 

Q l2 Q ¼ 2P 2eðhb; Ci þ el2 Þ

is the force of infection corresponding to the unique endemic steady state if R0 ¼ Rc . The situation is sketched in Fig. 1. It is clear that if the transmission rate by blood transfusion h is larger, Q tends to become negative. The quantity Q is decomposed into two parts:

H. Inaba, H. Sekine / Mathematical Biosciences 190 (2004) 39–69

55

λ*

s

(a)

s

0

R0

u

1

λ*

s

λ*c u

(b)

0

s

Rc

1

u

R0

Fig. 1. (a) Bifurcation diagram for the forward (supercritical) bifurcation, Q P 0. (b) Bifurcation diagram for the backward (subcritical) bifurcation, Q < 0.

Q ¼ Q1 þ Q2 ; where



 1 ab2  hb; Ci; Q1 :¼ 2e þ l2 l22 b1   hb; Ci Q2 ¼ hh; Ci þe : l2

Q2 is a contribution from the transmission by blood transfusion, and it is always negative or zero. On the other hand, the sign of Q1 mainly depends on the quantity 1 ab2 1 a l1 b2 1  ¼  ; l2 l22 b1 l2 l2 l2 b1 l1 where 1=l1 is the life span of the host population, 1=l2 the life span of the vector population, a=l2 is the number of infectious bites per an infected vector during its entire period of infectiousness,

56

H. Inaba, H. Sekine / Mathematical Biosciences 190 (2004) 39–69

and l1 b2 =l2 b1 is the ratio of the total size of vector population to the total size of host population. Since l2 is much larger than l1 , it may be possible that for realistic value of parameters, Q1 is negative. In such a case, the endemic steady state could appear, even if the basic reproduction number is less than one. In Proposition 4.4, it is still not very clear to us whether the backward bifurcation really can occur or not, since we do not know whether there exists a parameter set which guarantees the condition R0 < 1, Q < 0 and H > 0 simultaneously. Then in order to illustrate a possibility of backward bifurcation, we rewrite (4.4) as follows:    ab2 ðl1 þ k1 Þhb0 ; Ci þ hh0 ; Ci ¼ 1; ð4:8Þ 1 þ ek1 l2 b1 k1 ðhb0 ; Ci þ el2 Þ þ l2 where  is a bifurcation parameter and b0 ðsÞ, h0 ðsÞ and other parameters are normalized as 1¼

ab2 l1 hb ; Ci þ hh0 ; Ci: b1 l22 0

ð4:9Þ

The above situation could be realized if we assume that the rates of infection between susceptibles (humans or vectors) and infected human population are mainly controlled by the infectivity level of human infecteds, hence both b ¼ b0 and h ¼ h0 could be proportionally changing. In such a case,  is no other than the basic reproduction ratio R0 . Now let us define a function Uð; k1 Þ by    ab2 ðl1 þ k1 Þhb0 ; Ci  Uð; k1 Þ :¼ þ hh0 ; Ci  1: ð4:10Þ 1 þ ek1 l2 b1 k1 ðhb0 ; Ci þ el2 Þ þ l2 From our assumption (4.8) and (4.9), we have Uð1; 0Þ ¼ 0, hence from the Implicit Function Theorem the equation U ¼ 0 defines locally a function k1 ¼ k1 ðÞ with k1 ð1Þ ¼ 0 such that Uð; k1 ðÞÞ ¼ 0 if oU ab2 ¼ e  E 3 hb0 ; Ciðhb0 ; Ci þ el2 Þ 6¼ 0; ð4:11Þ  ok1 ð;k Þ¼ð1;0Þ l2 b1 1

where E is given in (4.5) and we have used the condition (4.9). If we assume (4.11), it also follows from the Implicit Function Theorem that dk1 U ð1; 0Þ 1 ¼ ¼ ; d U  ð1; 0Þ U  ð1; 0Þ ¼1

k1

ð4:12Þ

k1

where U ¼ oU=o and Uk1 ¼ oU=ok1 . If dk1 ð1Þ=d > 0, we know that the bifurcation at  ¼ 1 is forward, while dk1 ð1Þ=d < 0, the bifurcation at  ¼ 1 is backward. Then we obtain the following conclusion: Proposition 4.5. Suppose that the assumptions (4.8) and (4.9) hold. If Uk1 ð1; 0Þ < 0, the bifurcation at  ¼ 1 is forward, and if Uk1 ð1; 0Þ > 0, the bifurcation at  ¼ 1 is backward. It is not difficult to find parameter values satisfying (4.9) and Uk1 ð1; 0Þ > 0. For example, if we set

H. Inaba, H. Sekine / Mathematical Biosciences 190 (2004) 39–69

e ¼ 20;

a ¼ 0:1;

b¼h¼

1 ; 40

l1 b2 ¼ 10; l2 b1

l1 ¼

1 ; 70

57

l2 ¼ 1;

then (4.9) is satisfied and Uk1 ð1; 0Þ > 0 holds. It seems that those parameter values would not be far from reality. Of course, if possible, we should more carefully examine by using real data whether the multiple endemic steady states could occur in the real.

5. Local stability of endemic steady states In this section, we consider the local stability of the endemic steady state. To this end, let us introduce new variables as SðtÞ ¼ S  þ X ðtÞ; MðtÞ ¼ M  þ Y ðtÞ; k1 ðtÞ ¼ k1 þ fðtÞ;

iðt; sÞ ¼ i ðsÞ þ jðt; sÞ; V ðtÞ ¼ V  þ ZðtÞ; k2 ðtÞ ¼ k2 þ gðtÞ;

T ðtÞ ¼ T  þ W ðtÞ; where ðS  ; i ðsÞ; M  ; V  Þ denotes the endemic steady state and ðk1 ; k2 ; T  Þ denotes the value of ðk1 ; k2 ; T Þ at the endemic steady state. Then we can derive the linearized equation at the endemic steady state ðS  ; i ; M  ; V  Þ as follows: dX ðtÞ ¼ ðk1 þ l1 ÞX ðtÞ  S  fðtÞ; dt

ð5:1aÞ

ojðt; sÞ ojðt; sÞ þ ¼ ðl1 þ cðsÞÞjðt; sÞ; ot os

ð5:1bÞ

jðt; 0Þ ¼ k1 X ðtÞ þ S  fðtÞ;

ð5:1cÞ

dY ðtÞ ¼ ðk2 þ l2 ÞY ðtÞ  M  gðtÞ; dt

ð5:1dÞ

dZðtÞ ¼ k2 Y ðtÞ þ M  gðtÞ  l2 ZðtÞ; dt

ð5:1eÞ

fðtÞ ¼

1 ðaZðtÞ þ hh; jðt; Þi  k1 W ðtÞÞ; T

ð5:1fÞ

gðtÞ ¼

1 ðhb; jðt; Þi  k2 W ðtÞÞ; T

ð5:1gÞ

dW ðtÞ ¼ l1 W ðtÞ  hc; jðt; Þi: dt

ð5:1hÞ

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H. Inaba, H. Sekine / Mathematical Biosciences 190 (2004) 39–69

By formal integration, we can arrive at the following expressions: Z t  ðk1 þlÞt  X ðtÞ ¼ X0 e S eðk1 þlÞðtsÞ fðsÞ ds;

ð5:2aÞ

0

 jðt; sÞ ¼

Y ðtÞ ¼ Y0 e

jðt  s; 0ÞCðsÞ; CðsÞ ; j0 ðs  tÞ CðstÞ

ðk2 þl2 Þt

M



Z

0 6 s 6 t; s > t; t



eðk2 þl2 ÞðtsÞ gðsÞ ds;

ð5:2bÞ

ð5:2cÞ

0

ZðtÞ ¼ ðY0 þ Z0 Þel2 t  Y ðtÞ; Z t W ðtÞ ¼ W0 el1 t  el1 ðtsÞ hc; jðs; Þi ds;

ð5:2dÞ ð5:2eÞ

0

where ðX0 ; j0 ðsÞ; Y0 ; Z0 Þ is a given initial data and Z 1 W0 :¼ X0 þ j0 ðsÞ ds: 0

After a long calculation, from (5.1f) and (5.2b), we obtain Z t  S fðtÞ ¼ F1 ðtÞ þ K1 ðsÞjðt  s; 0Þ ds;

ð5:3Þ

0

where F1 ðsÞ and K1 ðsÞ are given by

Z Z aS  aS  M  t ðk þl2 ÞðtsÞ 1 l2 t ðk2 þl2 Þt 2 fðY þ Z Þe  Y e g þ e bðsÞ 0 0 0 T ðT  Þ2 0 s  Z CðsÞ aS  M  k2 t ðk þl2 ÞðtsÞ j0 ðs  sÞ ds ds  e 2  W0 el1 s ds  2 Cðs  sÞ ðT  Þ 0  Z s Z 1 Z CðsÞ S 1 CðsÞ l1 ðszÞ  e cðsÞ hðsÞ j0 ðs  zÞ ds dz ds þ  j0 ðs  sÞ ds ds T t Cðs  zÞ Cðs  sÞ 0 z   Z 1 Z t S  k1 CðsÞ l1 t l1 ðtsÞ   W0 e e cðsÞ ð5:4Þ j0 ðs  sÞ ds ds ; þ T Cðs  sÞ 0 s Z S S  k1 s l1 ðsxÞ e cðxÞCðxÞ dx K1 ðsÞ : ¼  hðsÞCðsÞ þ  T T 0 Z s  Z s Z s M Sa  ðk2 þl2 ÞðsxÞ ðk2 þl2 ÞðssÞ l1 ðszÞ þ e bðxÞCðxÞ dx þ k e e cðzÞCðzÞ dz ds : 2 ðT  Þ2 0 0 0

F1 ðtÞ : ¼

ð5:5Þ

On the other hand, it follows from (5.2a) that we have Z t  k1 X ðtÞ ¼ F2 ðtÞ þ K2 ðsÞjðt  s; 0Þ ds; 0

ð5:6Þ

H. Inaba, H. Sekine / Mathematical Biosciences 190 (2004) 39–69

59

where 

F2 ðtÞ :¼ k1 X0 eðk1 þl1 Þt  k1

Z

t



eðk1 þl1 ÞðtsÞ F1 ðsÞ ds;

ð5:7Þ

0

K2 ðsÞ :¼

k1

Z

s



eðk1 þl1 ÞðssÞ K1 ðsÞ ds:

ð5:8Þ

0

Let BðtÞ :¼ jðt; 0Þ be the number of newly produced infecteds per unit time. Then it follows from (5.1c), (5.3) and (5.6) that we arrive at the renewal integral equation as Z t KðsÞBðt  sÞ ds; ð5:9Þ BðtÞ ¼ GðtÞ þ 0

where GðtÞ :¼ F1 ðtÞ þ F2 ðtÞ and KðsÞ :¼ K1 ðsÞ þ K2 ðsÞ ¼ K1 ðsÞ 

k1

Z

s



eðk1 þl1 ÞðssÞ K1 ðsÞ ds:

ð5:10Þ

0

Then it is easy to see that the integral kernel KðsÞ is integrable on ½0; 1Þ, and the following holds, though we omit the proof: Lemma 5.1. If l2 > l1 , then there exists a number MF > 0 and MK > 0 such that jF ðtÞj 6 MF el1 t ;

jKðsÞj 6 MK el1 s :

ð5:11Þ

Again it follows from the Paley–Wiener Theorem that limt!1 BðtÞ ¼ 0 if and only if the characteristic equation Z 1 z þ l1 ^ ^ K1 ðzÞ ¼ 1; z 2 C; ezs KðsÞ ds ¼ ð5:12Þ KðzÞ :¼ z þ k1 þ l1 0 ^ and K ^ 1 denote the Laplace transformation of K has no root in the right half plane of C, where K and K1 respectively. From the principle of linearized stability for the evolution equation as (2.4) (see [7], [19, Appendix], [22, Corollary 4.3] and [26, Theorem 4.13]), we can state the following: Proposition 5.2. If the characteristic equation (5.12) has no root in fz 2 C : Rz P 0g, then the endemic steady state ðS  ; i ðsÞ; M  ; V  Þ is locally asymptotically stable, whereas it is unstable if (5.12) has a root with positive real part. In general, we could expect that subcritically bifurcating solutions will be unstable and supercritically bifurcating solution will be stable (The Factorization Theorem, [15, II.8]). First we show this conjecture is valid for small endemic steady state which bifurcates from the disease-free steady state. We again assume that (4.8) and (4.9) hold, so the force of infection k1 corresponding to the endemic steady state is locally given by k1 ¼ k1 ðÞ at the neighborhood of  ¼ 1. Now we define a function F ðz; Þ, z 2 C, as follows:

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H. Inaba, H. Sekine / Mathematical Biosciences 190 (2004) 39–69

Wðz; Þ :¼

z þ l1 ^ 1 ðz; Þ  1; K z þ k1 ðÞ þ l1

ð5:13Þ

^ 1 ðz; Þ is defined as the Laplace transform of K1 ðsÞ with k ¼ k ðÞ, bðsÞ ¼ b0 ðsÞ and where K 1 1 hðsÞ ¼ h0 ðsÞ. Then the equation W ¼ 0 is the characteristic equation at the endemic steady state with the force of infection k1 ðÞ. Now we should remember that S  , T  and M  depend on k1 , so they also depend on . From our assumption (4.9) and the Implicit Function Theorem, it follows that Wð0; 1Þ ¼ 0, hence the equation W ¼ 0 defines locally a function z ¼ zðÞ such that zð1Þ ¼ 0 and WðzðÞ; Þ ¼ 0 if oW 6¼ 0: ð5:14Þ oz ðz;Þ¼ð0;1Þ

The above condition (5.14) is satisfied because Z 1 oW ¼ sK1 ðsÞjk ¼0 ds < 0: 1 oz 0

ðz;Þ¼ð0;1Þ

Therefore we can calculate z0 ð1Þ as follows: z0 ð1Þ ¼ 

W ð0; 1Þ W ð0; 1Þ ¼ R1 ; Wz ð0; 1Þ sK1 ðsÞjk ¼0 ds 0

ð5:15Þ

1

where W ¼ oW=o is calculated as follows: oW dk1 ð1Þ oU ¼ 2 ð1; 0Þ þ 1 ¼ 1: o ðz;Þ¼ð0;1Þ d ok1

ð5:16Þ

(5.16) can be proved as follows: First we observe from (4.3b) that k2 is also a function of  and it is easy to see that dk2 ð1Þ dk1 ð1Þ ¼ hb; Ci: d d

ð5:17Þ

By differentiating W with respect to , we obtain oW dk1 ð1Þ 1 o ^ þ K1 ðz; Þ : ð0; 1Þ ¼  o d l1 o ðz;Þ¼ð0;1Þ Observe that ^ 1 ðzÞ oK o

ðz;Þ¼ð0;1Þ

¼

Z

1

e 0

zs

oK1 ðsÞ ds o ðz;Þ¼ð0;1Þ

  dk1 ð1Þ 1 ab2 ðl  l1 l2 e  hb; Cil1 Þ þ 1;  2e þ þ 2hb; Ci ¼ d l1 b1 l32 2

where we have used (4.9), (5.17) and the following integrals:

ð5:18Þ

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Z

1

Z

0

Z

1

1 1 hc; Ci ¼  e; l1 l1

el2 ðsxÞ bðxÞCðxÞ dx ds ¼

1 hb; Ci; l2

s 0

1

Z

0

Z

el1 ðsxÞ cðxÞCðxÞ dx ds ¼ 0

Z

0

Z

s

s

Z

0 1

Z

ðs  xÞel2 ðsxÞ bðxÞCðxÞ dx ds ¼

0 s

e 0

s

0

l2 ðsxÞ

Z

61

1 hb; Ci; l22

s

el1 ðszÞ cðzÞCðzÞ dz ds ds ¼ 0

1  l1 e : l1 l2

Therefore, from (4.11), (4.12), (5.17) and (5.18) we can conclude that   oW dk ð1Þ ab2 dk1 ð1Þ oU ð0; 1Þ ¼ 2 1 e þ hb; Ci Eðhb; Ci þ el Þ þ 1 ¼ 2 ð1; 0Þ þ 1 ¼ 1: 2 o d d ok1 b1 l32 Then we conclude that (5.16) holds. Now it follows from (5.15) and (5.16) that z0 ð1Þ ¼ 

W ð0; 1Þ 1 ¼ R1 < 0: Wz ð0; 1Þ sK ðsÞj 1 k ¼0 ds 0

ð5:19Þ

1

(5.19) means that z ¼ zðÞ goes to the right half plane if  < 1, while it goes to the left half plane if  > 1. Since zð1Þ ¼ 0 is the dominant, isolated root of the characteristic equation, it follows from the above observation that sufficiently near the trivial steady state the backwardly bifurcating solution is unstable, while the forwardly bifurcating solution is locally stable. Proposition 5.3. Suppose that (4.8) and (4.9) hold. For sufficiently small endemic steady states bifurcating from the disease-free steady state, the subcritically bifurcating steady states are unstable, while the supercritically bifurcating steady states are locally stable. Proof. First let us observe that W can be decomposed as follows: Wðz; Þ ¼ f^ ðzÞ  1 þ gðz; Þ; ð5:20Þ where f^ denotes the Laplace transform of a function f defined by Z ab2 l1 s l2 ðszÞ e b0 ðzÞCðzÞ dz; f ðsÞ :¼ h0 ðsÞCðsÞ þ b1 l2 0 and gðz; Þ :¼ Wðz; Þ  f^ ðzÞ þ 1. Then we can see that there exists a number minðl1 ; l2 Þ > g > 0 such that for all z 2 C with Rz P  g, g satisfies the estimate as gðz; Þ ¼ Oðj  1jÞ:

ð5:21Þ ^ If  ¼ 1, it follows from (4.9) that the unperturbed characteristic equation Wðz; 1Þ ¼ f ðzÞ  1 has a unique real root z ¼ 0 and it is strictly dominant, that is, any other complex root has a real part strictly less than zero, and for any real f, W ¼ 0 has at most finitely many roots in the half plane as Rz > f. Now we can apply the argument given in [13, Chapter IV, Section 4, p. 71] to conclude

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that as long as j  1j is sufficiently small, there exists a number g > f > 0 such that zðÞ is a unique characteristic root in the half plane Rz P  g and characteristic roots other than the dominant root zðÞ stay in the left half plane Rz < g. Therefore as long as j  1j is sufficiently small, the endemic steady state bifurcating from the disease-free steady state is locally asymptotically stable in case of a forward bifurcation, and it is unstable in case of a backward bifurcation. h Though the above result is only applied to the local solution near the bifurcating point, in fact we can prove the following instability result for the global branch: Proposition 5.4. Suppose that there exists an endemic steady state with the force of infection k1 > 0. If kc > k1 , then the endemic steady state with the force of infection k1 is unstable. Proof. Under the assumption there exists an endemic steady state whose force of infection k1 is less than kc . From (5.5) and (5.8), the characteristic equation is given by z þ l1 ^ ^ KðzÞ ¼ K1 ðzÞ ¼ 1; ð5:22Þ z þ k1 þ l1 and it follows from (4.2) that l hh; Ci þ k1 hc; Ci ab2 k2 hc; Ci þ l1 hb; Ci ^ þ : ð5:23Þ Kð0Þ ¼ 1 ð1 þ ek1 Þðk1 þ l1 Þ b1 ð1 þ ek1 Þ2 ðk2 þ l2 Þ2 ^ ^ ¼ 0, it is easily seen that Since KðzÞ is a positive continuous function for z P 0 and limz!1 KðzÞ ^ ^ (5.22) has a positive root if Kð0Þ > 1. Therefore it is sufficient to show that Kð0Þ > 1 at the en demic steady state with the force of infection k1 . We can define a function Gðk1 Þ by ^  1, because k2 is a function of k1 as is seen in (4.3b). Then we have Gðk1 Þ :¼ Kð0Þ Gðk1 Þ ¼

l1 hh; Ci þ k1 hc; Ci ab2 k2 hc; Ci þ l1 hb; Ci þ  1: ð1 þ ek1 Þðk1 þ l1 Þ b1 ð1 þ ek1 Þ2 ðk2 þ l2 Þ2

By using (4.3b) and hc; Ci ¼ 1  el1 , we can calculate as   k k1 l11  e þ hh; Ci l ab hb; Ci 1 þ l11 2  1   þ Gðk Þ ¼   2  1: k l22 b1 hb;Ci   ð1 þ ek1 Þ 1 þ l11 ð1 þ ek1 Þ 1 þ k1 e þ l2

If we use the fact that k1 is a positive root of (4.4), we obtain   k1 l11  e þ hh; Ci 1 þ ek1  hh; Ci       1 þ Gðk1 Þ ¼ k ð1 þ ek1 Þ 1 þ l11 ð1 þ ek1 Þ 1 þ k1 e þ hb;Ci       l2 hb;Ci hb;Ci   2 e  0 k1  ðk1 Þ l1 e þ l2  2ek1 e þ l2 þ 1R  Q l     1 : ¼ k1 hb;Ci   ð1 þ ek1 Þ 1 þ l1 1 þ k1 e þ l2

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63

Moreover if we use the relation (4.6), then    1 1 hb; Ci : ð1  R0 Þ ¼  k1 Q þ ðk1 Þ2 e e þ l1 l1 l2 Therefore we conclude that         k1 hb;Ci  k1  ðk1 Þ2 l2e1 e þ hb;Ci e þ  2ek  1 þ Q 1 l1 l l2 2      Gðk1 Þ ¼ k ð1 þ ek1 Þ 1 þ l11 1 þ k e þ hb;Ci l2   hb;Ci    2e e þ l2 k1 ðk1  kc Þ    : ¼ ð1 þ ek1 Þ 1 þ k1 e þ hb;Ci l2

ð5:24Þ

^ ¼ 1 þ Gðk1 Þ > 1. This completes our proof. h Therefore we know that if k1 < kc , then Kð0Þ Of course, the above proposition is meaningless if Q P 0, and it follows from (4.7) that kc > k1 implies that there exist two endemic steady states and k1 corresponds to the smaller endemic steady state. That is, in the above proposition we implicitly assume that Q < 0, R0 < 1 and H > 0. Subsequently if for some parameter value the integral kernel K is non-negative in (5.9), it is easily seen from the standard argument for the renewal integral equation that the solution BðtÞ goes to zero as t ! 1 if and only if Z 1 ^ KðsÞ ds ¼ Kð0Þ ¼ 1 þ Gðk1 Þ < 1: 0

Then we obtain the following conditions immediately: Proposition 5.5. Suppose that one of the following inequalities holds: Z s  K1 ðsÞ  k1 eðk1 þl1 ÞðssÞ K1 ðsÞ ds P 0;

ð5:25Þ

0

K1 ðsÞ  k1 el1 s P 0: Then the endemic steady state corresponding to the force of infection stable if k1 > kc , whereas it is unstable if k1 < kc .

ð5:26Þ k1

is locally asymptotically

Proof. If the integral kernel KðsÞ is non-negative in (5.9), it is easily seen from the standard argument for the renewal integral equation that the solution BðtÞ goes to zero as t ! 1 if and only if Z 1 ^ KðsÞ ds ¼ Kð0Þ ¼ 1 þ Gðk1 Þ < 1: 0

Since Gðk1 Þ > 0 if k1 < kc and Gðk1 Þ < 0 if k1 > kc , we can arrive at the conclusion under the condition (5.25). Next observe that the characteristic equation (5.12) is equivalent to ^ 1 ðzÞ  K

k1 ¼ 1: z þ l1

ð5:27Þ

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Moreover (5.27) is written as follows: Z 1 ezs /ðsÞ ds ¼ 1;

ð5:28Þ

0

where /ðsÞ :¼ K1 ðsÞ  k1 el1 s : Then under the condition of (5.26), the dominant characteristic root of (5.22) is located in the left half plane if and only if    ^ ^ 1 ð0Þ  k1 ¼ 1 þ 1 þ k1 Gðk Þ < 1; /ð0Þ ¼K 1 l1 l1 which is equivalent to the condition k1 > kc . This complete our proof. h Again the instability part of the above proposition assumes implicitly that Q < 0, R0 < 1 and H > 0. On the other hand, the stability part can have a sense even if Q P 0. The above conclusion is not so informative as long as we do not know the conditions for parameters under which (5.25) or (5.26) is satisfied. So we give a simple example to guarantee the condition (5.26): 5.6. The condition (5.26) is satisfied for sufficiently small k1 if h :¼ inf hðsÞ > 0 and RProposition 1 cðrÞ dr < 1. 0 Proof. Observe that

Z s 1 k1 hðsÞCðsÞ þ el1 ðsxÞ cðxÞCðxÞ dx 1 þ ek1 1 þ ek1 0 Rs Rs  i el1 s h  cðrÞ dr  cðrÞ dr  0 þ k1 1  e 0 : ¼  hðsÞe 1 þ ek1

K1 ðsÞ P

Moreover it follows that Rs Rs    cðrÞdr  cðrÞdr þ k1 1  e 0 hðsÞe 0 P h‘ðsÞ þ k1 ð1  ‘ðsÞÞ ¼ ðh  k1 Þ‘ðsÞ þ k1 ; where ‘ðsÞ :¼ expð /ðsÞ P el1 s

Rs 0

cðrÞ drÞ and h :¼ inf s P 0 hðsÞ. Then we know that if h > k1 ,

ðh  k1 Þ‘ð1Þ  eðk1 Þ2 : 1 þ ek1

Hence under the condition h > 0 and ‘ð1Þ > 0, it follows that /ðsÞ > 0 if k1 is sufficiently small. h For the case of supercritical bifurcation, we can raise another conditions that guarantee the local stability of the endemic steady state:

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65

Proposition 5.7. Suppose that Q > 0 and R0 > 1. If k1 is small enough and Q > 1=l1 , then the endemic steady state with the force of infection k1 is locally asymptotically stable. Proof. Suppose that the characteristic equation (5.12) has a root z0 with Rz0 P 0. Then we have Z z 0 þ l1 1 16 K1 ðs; k1 Þ ds; ð5:29Þ z þ l þ k 0

1

1

0

K1 ðs; k1 Þ

is given by (5.5). That is, here instead of K1 ðsÞ, we write as K1 ðs; k1 Þ to clear the where dependence on k1 . For k1 > 0, we obtain z0 þ l1 ð5:30Þ z þ l þ k < 1: 0

1

1

Let us define a real function f ðxÞ, x P 0, by

f ðxÞ :¼

Z



1

K1 ðs; xÞ ds ¼



0

x l1



  1  2e e þ hb;Ci Þ xðx  k c l @1   2   A: hb;Ci ð1 þ exÞ 1 þ x e þ l2 0

Then we have f ð0Þ ¼ 1 and f 0 ð0Þ ¼ l11  Q, where Q is given in (4.6). If Q > 1=l1 , then f 0 ð0Þ < 0 and we can conclude that f ðk1 Þ < 1 for sufficiently small k1 > 0. In such a case, it follows from (5.30) that (5.29) does not hold, so we arrive at a contradiction. That is, there is no characteristic root with non-negative real part. This completes our proof. h

6. Blood transfusion transmission In this section, we consider a special case such that the Chagas disease is transmitted only by blood transfusion. As is pointed out in Section 1, we could expect that in many countries the prevalence rate among blood donors is not so high and the accuracy of blood screening has been improved, hence we should not overestimate the risk of horizontal transmission of Chagas disease. Therefore though we here analyze a purely horizontal transmission model for Chagas disease, it should not be understood as a realistic model for Chagas disease. However, from theoretical point of view, it would be interesting to see the difference between vector transmission and horizontal transmission. We also remark that the following model has been already studied by several authors as a model for HIV/AIDS epidemic in drug users or homosexual populations under more general formula for the force of infection [14,23], but our calculation in the previous section will be able to provide some new insights for this type of models. If we neglect the transmission by vector, we can take a ¼ 0 in the model (2.4), so the dynamics of host population is separated from the vector dynamics and we obtain a simplified model as follows: Z dSðtÞ SðtÞ 1 ¼ b1  l1 SðtÞ  hðsÞiðt; sÞ ds; ð6:1aÞ dt T ðtÞ 0

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oiðt; sÞ oiðt; sÞ þ ¼ ðl1 þ cðsÞÞiðt; sÞ; ot os Z SðtÞ 1 iðt; 0Þ ¼ hðsÞiðt; sÞ ds: T ðtÞ 0

ð6:1bÞ ð6:1cÞ

For the above model, it is easy to see that the basic reproduction ratio is given by Z 1 R0 ¼ hh; Ci ¼ hðsÞCðsÞ ds; 0

and the subcritical bifurcation of the endemic steady state does not occur. That is, it is easy to see that the following holds, though we omit the proof: Proposition 6.1. If R0 6 1, there exists only disease-free steady state and it is globally stable. If R0 > 1, there is a unique endemic steady state. From the above observation, we know that the bifurcation at R0 ¼ 1 for the blood transfusion transmission model is always supercritical. Then we can prove the following: Proposition 6.2. If R0 > 1 and jR0  1j is small enough, the endemic steady state is locally asymptotically stable. Proof. We assume that the rate of transmission h can be written as hðsÞ ¼ R0 h0 ðsÞ, where h0 is a normalized transmission rate satisfying hh0 ; Ci ¼ 1. Then it follows from (4.3a) that k1 ¼ e1 ðR0  1Þ: Therefore K1 ðsÞ can be rewritten as follows: Z R0  1 s l1 ðsxÞ K1 ðsÞ ¼ h0 ðsÞCðsÞ þ e cðxÞCðxÞ dx: eR0 0 Next note that the characteristic equation (5.12) can be written as ^ 1 ðzÞ  K

k1 ¼ 1: z þ l1

ð6:2Þ

Let us introduce a new parameter  by  :¼ R0  1. Then we have a characteristic equation with a parameter as ^ 0 ðzÞ þ F ðz; Þ ¼ 1; K

ð6:3Þ

where K0 ðsÞ :¼ hðsÞCðsÞ and Z 1 Z Z s   1 ðzþl1 Þs zs l1 ðsxÞ e e cðxÞCðxÞ dx ds  e ds: F ðz; Þ :¼ eð1 þ Þ 0 e 0 0 In the following, we adopt the argument by Iannelli [13, Chapter 4]. In the half plain Rz > l1 , the function F is continuously differentiable with respect to ðz; Þ, F ðz; 0Þ ¼ 0 and it is easily seen that there exists M > 0 and 0 < g < l1 such that jF ðz; Þj < jjM for Rz > g and  P 0. From the

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67

^ 0 ðzÞ ¼ 1 has a standard argument for Lotka’s renewal equation, the unperturbed equation K dominant real root 0, which is the unique one in the half plain Rz P  g1 for some g1 2 ðg; 0Þ. Let us define a number m such that ^ 0 ðg1 þ iyÞj: m :¼ inf j1  K y2R

Then we can assume that m > 0, since there exists at most finite number of characteristic roots in the neighbourhood of the line Rz ¼ g1 . Moreover, we can take a large L > 0 such that ^ 0 ðzÞj > 1=2 for jzj > L and Rz P  g1 . Then if jj is so small that jj < M 1 ðm ^ ð1=2ÞÞ, we j1  K ^ 0 ðzÞj on the boundary of any domain such that fz : Rz > g1 g \ obtain jF ðz; Þj < j1  K fz : jzj < qg with q > L. By the Rouche Theorem, we can conclude that the characteristic equation (6.3) has one and only one root in the half plane Rz P  g1 , which we denote as zðÞ. From our assumption, zð0Þ ¼ 0 and it follows from the Implicit Function Theorem that Z 1 1 oF ð0; 0Þ dzðÞ 1 o sK0 ðsÞ ds < 0: ¼ ¼ R1 d el1 0 sK0 ðsÞ ds 0 Therefore, the differentiable path zðÞ starting from zð0Þ ¼ 0 goes to the left of the imaginary axis as  increases from zero. Thus we conclude that for sufficiently small  > 0, all the root of (6.3) have negative real part. This completes our proof. h

7. Discussion In this paper, we develop a structured population model for the spread of Chagas disease in a demographically steady state host population. In order to reflect the dependence of disease progress on the duration since infection (disease age), we assume that the infected population is structured by the disease age, and the infection rate and the removed rate depend on the disease age. The effects of vector and blood transfusion transmission are considered. We have calculated the basic reproduction ratio R0 to show that the disease can invade into the susceptible population and unique endemic steady state exists if R0 > 1, whereas the disease dies out if R0 is small enough. We also proved that depending on parameters, the backward bifurcation of endemic steady state can occur, so even if R0 < 1, there could exist endemic steady states. Our analysis shows that under our assumption of vector transmission mechanism, the backward bifurcation occurs due to the regulation of the host population by the disease-induced death rate. The presence of a backward bifurcation has practically important consequences for the control of infectious diseases. If the bifurcation of endemic state at R0 ¼ 1 is forward one, the size of infected population will be approximately proportional to the difference jR0  1j. On the other hand, in a system with a backward bifurcation, the endemic steady state that exists for R0 just above one could have a large infectious population, so the result of R0 rising above one would be a drastic change in the number of infecteds. Conversely, reducing R0 back below one would not eradicate the disease, as long as its reduction is not sufficient. That is, if the disease is already endemic, in order to eradicate the disease, we have to reduce the basic reproduction number so far that it enters the region where the disease-free steady state is globally asymptotically stable and there is no endemic steady state. Though our Proposition 3.3 shows a sufficient condition to

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eradication of the disease, to find a necessary and sufficient condition for the disease eradication is still an open problem. The reader may refer to [12,17–19] for other epidemic models showing the subcritical bifurcation. For the practical prevention purpose, our analysis suggests that to reduce the ratio of the density of vectors to the density of hosts will be most effective to control the vector transmission, because if this ratio is decreasing, the backward bifurcation is more difficult to occur and the basic reproduction ratio becomes smaller. As a disease-age dependent model for Chagas disease, our system is formulated as simple as possible, but our mathematical analysis is still incomplete. In particular, we do not know what will happen in the region that the force of infection is not small, and the system is far from the diseasefree steady state. On the other hand, from realistic point of view, there remains many factors neglected in our basic model which should be considered in future modeling. For example, we have neglected the (chronological) age structure of human hosts, but it is an essential factor to take into account demographic change of host population. Moreover, the population dynamics of vector would be much more complex. Its birth and death process would depend on seasonal variation of environments, and existence of reservoir host animals living around humans would play a very important role to spread the disease. In fact, in the United States, the disease exists almost exclusively as a zoonosis [3]. These animals, as blood source, will contribute to maintain population densities of vectors, but conversely some of these animals, as predator, could suppress the vector population. Animals also serve as vehicles to disperse the vector bugs to other region. To construct and analyze more realistic models for Chagas disease is a future challenge to us.

Acknowledgements We are grateful to the handling editor and two reviewers for their kind comments to our previous manuscript, which were most helpful to improve it. We also thank Katumi Kamioka for his help to make figures.

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