Page 1

A schistosomiasis model with an age-structure in human hosts

and its application to treatment strategiesq

Pei Zhang, Zhilan Feng*, Fabio Milner

Mathematics Department, Purdue University, West Lafayette, IN 47907, USA

Received 15 January 2006; received in revised form 13 June 2006; accepted 28 June 2006

Available online 4 August 2006

Abstract

We study a system of partial differential equations which models the disease transmission dynamics of

schistosomiasis. The model incorporates both the definitive human hosts and the intermediate snail hosts.

The human hosts have an age-dependent infection rate and the snail hosts have an infection-age-dependent

cercaria releasing rate. The parasite reproduction number R is computed and is shown to determine the

disease dynamics. Stability results are obtained via both analytic and numerical studies. Results of the mod-

el are used to discuss age-targeted drug treatment strategies for humans. Sensitivity and uncertainty anal-

ysis is conducted to determine the role of various parameters on the variation of R. The effects of various

drug treatment programs on disease control are compared in terms of both R and the mean parasite load

within the human hosts.

? 2006 Elsevier Inc. All rights reserved.

Keywords: Schistosomiasis; Multiple hosts; Age-structured model; Stability; Sensitivity analysis; Age-targeted

treatment

0025-5564/$ - see front matter ? 2006 Elsevier Inc. All rights reserved.

doi:10.1016/j.mbs.2006.06.006

qThis work was supported in part by NSF Grant DMS-0314575 and by James S. McDonnell Foundation 21st

Century Science Initiative.

*Corresponding author. Tel.: +1 765 494 1915; fax: +1 765 494 0548.

E-mail addresses: peizhang@math.purdue.edu (P. Zhang), zfeng@math.purdue.edu (Z. Feng), milner@math.

purdue.edu (F. Milner).

www.elsevier.com/locate/mbs

Mathematical Biosciences 205 (2007) 83–107

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1. Introduction

The spread and persistence of schistosomiasis have always been among the more complex host–

parasite processes to model mathematically, because of the several different forms that the para-

sites take while infecting two separate hosts (definitive human hosts and intermediate snail hosts)

during their life cycle. For schistosome (and other helminth) parasites, the number of parasites

infecting an individual host (i.e., the intensity of the infection) plays an important role in deter-

mining the outcome of an infection. Thus, transmission rates, pathogenicity, and development

of host immunity are all typically assumed to depend upon intensity. These features preclude

modeling the system through traditional (microparasite) SIR models. Macroparasite models are

considered more appropriate as they take into consideration the distribution of parasites within

human hosts [22]. A number of mathematical models have been developed for schistosomiasis

using a variety of approaches. The earliest models of schistosomiasis consider the population sizes

of both humans and snails to be constant [21,23]. Several recent studies have addressed the

dynamics of schistosomiasis and other helminth (e.g., onchocercasis) infection of humans [1–5,

9,10,15,16,19,20,22,25,33]. These models have made contributions to the understanding of the

transmission dynamics of schistosomes. However, most existing models for schistosomiasis do

not include explicitly the dynamics of the intermediate snail host.

Anotherimportantfeatureassociatedwithschistosomiasisinfectionistheage-dependentprevalence

inhumans[7].Epidemiologicalstudieshaveshownthatchildrenofschoolageusuallyexhibitthehigh-

est prevalence of schistosome infections ([27,30] whereas adults exhibit some of the more serious con-

sequences of infection [17,32]. Various age-targeted treatments have been adopted in different

populationsandmathematicalmodelshavebeenusedtoassessthecost-effectivenessofthediseasecon-

trol programs [6,7]). However, the models also do not consider explicit snail population dynamics.

We have previously studied mathematical models that include mass chemotherapy in human

hosts with explicit snail dynamics [11,12] and schistosome mating biology [34]. However, none

of these models include an age-structure in humans. Results in these studies suggest that while

the incorporation of schistosome mating biology does not alter the model behaviors dramatically,

the inclusion of an explicit interaction with the snail population will have important impact on the

disease dynamics. For example, the mean parasite load of human hosts does not depend linearly

on the transmission rate from snail to human as simpler models (simpler snail dynamics) predict,

but rather on the square root of this transmission rate. The density dependence considered in snail

dynamics may produce a bifurcation at which the unique endemic equilibrium changes its stability

and stable periodic solutions exist.

In this paper, we develop a new mathematical model which includes both an age-structure of

the human population and explicit snail population dynamics. The model is studied both analyt-

ically and numerically in terms of steady states and their stability as well as possible bifurcations.

We also use techniques based on Latin hypercube sampling to identify quantitatively the most

influential parameters in affecting the magnitude of threshold conditions through the uncertainty

and sensitivity analysis. Results from this model are applied to the study of disease control via

age-targeted treatment of humans. Two criteria are used to assess the effect of various age-depen-

dent control programs. The first one is to use the overall mean parasite load (defined by the ratio

of the total number of parasites to the total number of humans) and the mean parasite load within

each age group, and the second one is to look at the reduction in the reproductive number R.

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P. Zhang et al. / Mathematical Biosciences 205 (2007) 83–107

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Our paper is organized as follows. Section 2 introduces the age-dependent model; Section 3 car-

ries out the local and global stability analysis of the steady states and shows that periodic solu-

tions can arise via a Hopf bifurcation; applications of the model results to the assessment of

age-dependent treatment strategies are given in Section 4; and in Section 5 we summarize our

findings.

2. The model formulation

In the literature, there are two different approaches to modeling macroparasite infections. The

first approach combines the classical Lotka/Sharpe–McKendrick model for a population struc-

tured by the age of hosts. In this approach, one counts the hosts of age a carrying i parasites which

leads to an infinite sequence of renewal equations numbered by i, and this infinite system is then

reduced to a single equation by the use of the generating function [16,19]. The other approach uses

special interaction coefficients to account a posteriori for the observed clumping of parasites in

human hosts. This approach leads to an infinite ODE system which is then reduced to a two-di-

mensional system for the total number of hosts and the total number of parasites [2]. The first

approach provides more information including how many hosts of age a carry i parasites while

the second approach gives only the numbers of hosts and parasites.

In this paper, we take a mixed approach. We structure the human host population according to

age and adult parasite population according to the age of the host which carries the particular

parasite. Hence, with this state space, we know how many humans there are of a given age,

and for humans of a given age how many parasites are carried on average, while the parasite dis-

tribution among the hosts of a given age is not tracked.

Let N(t), P(t), and C(t) denote the numbers of human hosts, adult parasites, and free-living

cercariae, respectively. Assume that individuals are born uninfected. Based on the models in

Anderson and May [2] and Dobson [10] we have considered an age-independent model (see

[11]) in which the equation for the total number of adult parasites P and the human hosts N have

the form:

dN

dt¼ Kh? lhN ? aP;

dP

dt¼ bCN ? ðlhþ lpþ a þr

hÞP ? a

1 þ k

k

??P2

N;

ð1Þ

where b is the instantaneous rate of infection of human hosts by one cercaria, lhis the per capita

natural death rate of human hosts, lpis the per capita death rate of adult parasites, a is the dis-

ease-induced death rate of humans per parasite; r is the treatment rate of human hosts, k is the

clumping parameter of the negative binomial distribution of parasites. C(t) is determined by the

number of infected snails whose equation is given later in the system for snails. All parameters and

variables with their definitions are listed in Table 1.

To incorporate an age structure of humans into the equations in (1), we let n(t,a), p(t,a) denote

the density functions of human hosts of age a at time t and parasites within human hosts of age a,

respectively. We remark that a denotes the age of humans (not of parasites). Hence,

P. Zhang et al. / Mathematical Biosciences 205 (2007) 83–107

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NðtÞ ¼

Z1

0

nðt;aÞda

and PðtÞ ¼

Z1

0

pðt;aÞda

are the total number of humans and the total number of adult parasites, respectively. P/N repre-

sents the mean parasite load within humans. Using a similar argument as before we can derive the

equations for n(t,a) and p(t,a) and get:

o

otnðt;aÞ þo

o

otpðt;aÞ þo

oanðt;aÞ ¼ ?lhðaÞnðt;aÞ ? apðt;aÞ;

oapðt;aÞ ¼ bðaÞCnðt;aÞ ? dpðaÞpðt;aÞ ? a1 þ k

k

PðtÞ

NðtÞpðt;aÞ;

ð2Þ

where

dpðaÞ ¼ lhðaÞ þ lpþ a þ rðaÞ:

ð3Þ

The boundary conditions are n(t,0) = Kh(birth rate of human hosts), p(t,0) = 0 (humans are born

uninfected), and initial conditions are n(0,a) = n0(a), p(0,a) = p0(a) for some given functions n0

and p0. Notice that, in the special case when the parameter functions b(a), lh(a) and r(a) are con-

stants, the equations in (2) reduce to those in (1) by integration.

For the snail dynamics, we can use similar equations introduced in Feng et al. [12], which takes

into consideration realistic features including an infection-age-dependent cercariae release rate of

infected snails and a density-dependent snail birth rate due to the infertility of infected snails:

Table 1

Definition of parameters in models (1), (2), (4) and (5)

SymbolDefinition

Kh

lh(a)

r(a)

lp

a

dp(a)

b(a)

k

b(S,I)

b1(S,I)

b2(S,I)

c1, c2

ls

ds

c

q

n

r(s)

Recruitment rate of human hosts

Per capita natural death rate of human hosts of age a

Treatment rate of human hosts of age a

Per capita death rate of adult parasites

Disease-induced death rate of human hosts by one cercaria

lh(a) + lp+ a + r(a): ‘‘total’’ death rate of adult parasites

Instantaneous rate of infection of human hosts of age a by one cercaria

Clumping parameter of the parasite distribution within humans

birth function of uninfected snails

Ks: Constant birth rate of snails

c1S/(c2+ S + I): density-dependent birth function of snails

Saturation and scaling constant, respectively

Per capita natural death rate of snails

Disease-induced death rate of infection of snails by parasites

Per capita rate of effective egg-production (or miracidia) of adult parasites

Per capita infection rate of snails by miracidia

qc: per capita infection rate of snails by adult parasites

Rate at which infected snails of infection-age s release cercariae

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P. Zhang et al. / Mathematical Biosciences 205 (2007) 83–107

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d

dtS ¼ bðS;IÞ ? lsS ? qMS;

o

otxðt;sÞ þo

xðt;0Þ ¼ qMS;

CðtÞ ¼

0

osxðt;sÞ ¼ ?ðlsþ dsÞxðt;sÞ;

xð0;sÞ ¼ x0ðsÞ;

Z1

S denotes the number of uninfected snails. I denotes the total number of infected snails given by

IðtÞ ¼R1

miracidia die quickly if they cannot find a snail to infect, the total number of miracidia at time t is

assumed to be proportional to the number of adult parasites) and c is the per capita effective egg-

production rate of adult parasites. Other parameters are: b(S,I) is the birth function of uninfected

snails, lsis the per capita natural death rate of snails, dsis the disease-induced death rate of snails,

q is the per capita rate of infection of snails, and r(s) denotes the rate at which infected snails of

infection-age s release cercariae.

Two forms of the birth function b(S,I) will be considered as we did in Feng et al. [12]. One

example will be a Michaelis–Menten type: b(S,I) = c1S/(c2+ S + I), which assumes that infected

snails do not reproduce and that the snail population is bounded; c1and c2are, respectively, the

scaling and saturation constants.

We remark that the infertility of infected snails and the periodicity in the cercaria production

are two of the special features of schistosomiasis, and that it has been shown that the inclusion of

snail dynamics may produce different dynamics (see [12]). Hence, we include both population

dynamics explicitly in the model even though the time scales of the human dynamics (50–70 years)

and the snail dynamics (2–4 years) may be different.

The combination of (2) and (4) leads to the following integro-differential initial-boundary value

problem that governs the disease dynamics:

8

>

>

>

The large number of parameters involved in system (5) can be significantly reduced as many of the

parameters can be lumped together. However, we choose to keep these parameters since they are

more readily related to field data. We shall use techniques based on Latin hypercube sampling (see

[26]) to identify quantitatively the most influential parameter(s) in affecting the magnitude of

threshold conditions through the uncertainty and sensitivity analysis.

rðsÞxðt;sÞds:

ð4Þ

0xðt;sÞds. Here, s denotes the time since infection, i.e., infection-age, and x(t,s) denotes

the infection-age density of snails at time t. M = cP is the number of free-living miracidia (since

o

otnðt;aÞ þo

o

otpðt;aÞ þo

d

oanðt;aÞ ¼ ?lhðaÞnðt;aÞ ? apðt;aÞ;

oapðt;aÞ ¼ bðaÞCnðt;aÞ ? dpðaÞpðt;aÞ ? a1þk

dtSðtÞ ¼ bðS;IÞ ? lsSðtÞ ? qcSðtÞR1

PðtÞ ¼R1

nð0;aÞ ¼ n0ðaÞ;

k

PðtÞ

NðtÞpðt;aÞ;

0pðt;aÞda;

o

otxðt;sÞ þo

osxðt;sÞ ¼ ?ðlsþ dsÞxðt;sÞ;

0pðt;aÞda;

nðt;0Þ ¼ Kh;

pð0;aÞ ¼ p0ðaÞ;

CðtÞ ¼R1

0rðsÞxðt;sÞds;

xðt;0Þ ¼ qcSðtÞR1

IðtÞ ¼R1

Ið0Þ ¼ I0;

0xðt;sÞds;

pðt;0Þ ¼ 0;

0pðt;aÞda;

Sð0Þ ¼ S0;

xð0;sÞ ¼ x0ðsÞ:

>

>

>

>

>

>

>

>

>

>

>

>

>

>

>

:

>

<

>

>

>

>

>

ð5Þ

P. Zhang et al. / Mathematical Biosciences 205 (2007) 83–107

87

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3. Analysis

In this section, we compute the reproductive number R and show that it determines the exis-

tence and stability of possible equilibrium points. For mathematical convenience, we consider a

different formulation of the system (5) as shown below.

3.1. Reformulation of the system (5)

For ease of notation, we introduce a new variable B:

Z1

Solving the x equation along characteristic lines we get

(

BðtÞ ¼ qcSðtÞ

0

pðt;aÞda ¼ qcSðtÞPðtÞ:

xðt;sÞ ¼

e?ðlsþdsÞsBðt ? sÞ;

e?ðlsþdsÞsx0ðs ? tÞ;

t P s;

t < s:

Then I(t) and C(t) can be expressed in terms of B(t):

Zt

IðtÞ ¼

0

CðtÞ ¼

0

rðsÞe?ðlsþdsÞsBðt ? sÞds þ F1ðtÞ;

Zt

e?ðlsþdsÞsBðt ? sÞds þ F2ðtÞ;

where

F1ðtÞ ¼

Z1

Z1

t

rðsÞe?ðlsþdsÞsx0ðs ? tÞds;

F2ðtÞ ¼

t

e?ðlsþdsÞsx0ðs ? tÞds:

Using the new notation we can rewrite the system (5) as

o

otnðt;aÞ þo

o

otpðt;aÞ þo

oanðt;aÞ ¼ ?lhðaÞnðt;aÞ ? apðt;aÞ;

oapðt;aÞ ¼ bðaÞnðt;aÞ

Zt

0

rðsÞe?ðlsþdsÞsBðt ? sÞds

? dpðaÞpðt;aÞ ? ak0PðtÞ

NðtÞpðt;aÞ þ bðaÞnðt;aÞF1ðtÞ;

d

dtSðtÞ ¼ bðS;IÞ ? lsSðtÞ ? BðtÞ;

BðtÞ ¼ nPðtÞSðtÞ;

where

ð6Þ

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P. Zhang et al. / Mathematical Biosciences 205 (2007) 83–107

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n ¼ qc;

k0¼k þ 1

k

:

The existence and uniqueness of the system (6) can be shown using the standard method. Notice

that F1(t) ! 0 as t ! 1. The limiting system of (6) is:

o

otnðt;aÞ þo

o

otpðt;aÞ þo

d

dtSðtÞ ¼ bðS;IÞ ? lsSðtÞ ? BðtÞ;

BðtÞ ¼ nPðtÞSðtÞ;

where

Z1

and

oanðt;aÞ ¼ ?lhðaÞnðt;aÞ ? apðt;aÞ;

oapðt;aÞ ¼ bðaÞnðt;aÞðK ? BÞðtÞ ? dpðaÞpðt;aÞ ? ak0PðtÞ

NðtÞpðt;aÞ;

ð7Þ

ðK ? BÞðtÞ ¼

0

KðsÞBðt ? sÞds

KðsÞ ¼ rðsÞe?ðlsþdsÞs:

In the rest of this section, we will consider the limiting system (7).

3.2. The reproductive number of the parasite

As is done in Feng et al. [12], we consider two forms of the snail birth function b(S,I).

Case 1: b1(S,I) = Ks(constant birth rate).

The system (7) always has the parasite-free equilibrium

E0¼ ðn?ðaÞ;p?ðaÞ;S?;B?Þ ¼ ðKhpðaÞ;0;Ks=ls;0Þ:

As in most epidemiology models, the stability of E0is dependent of the basic reproductive number

(ratio) of the parasites, or the reproductive ratio of the parasites if the host population is under

some influence of a control/prevention program, which is the case our models consider. We define

the reproductive ratio of the parasites as

Z1

where

pðwÞ ¼ e?Rw

represents the survival probability of a person up to age a, dp(h) = lh(h) + lp+ a + r(h) is the

sum of death rates of the parasites within a human host of age h, and

Z1

R ¼

0

bðwÞKKhpðwÞ

Z1

0

nKs

ls

e?Rwþu

w

dpðhÞdhdudw;

ð8Þ

0lhðhÞdh

K ¼

0

rðsÞe?ðlsþdsÞsds

P. Zhang et al. / Mathematical Biosciences 205 (2007) 83–107

89

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represents the total number of cercariae released by an infected snail during its life of infection.

For convenience of interpretation, we consider the secondary number of infected snails (instead

of parasites) produced by a typical infected snail during its entire period of infection. The quantity

bðwÞKKhpðwÞ

represents the average number of human hosts of age w becoming infected by one infected snail

during its entire period of infection. e?Rwþu

host of age w + u who was infected at age w, and the average number of snails the parasite (u time

units after the host was infected, or parasite of age u) is capable of infecting is nKs=lse?Rwþu

Thus, the total number of snails infected by a typical parasite during its entire life time is

Z1

It follows that the number of secondary infected snails due to a human host of age w is

Z1

Integrating over all ages w 2 [0,1) we obtain the overall average number of secondary infected

snails, which is the number R given in (8).

Case 2: b2(S,I) = c1S/(c2+ S + I) (density-dependent birth rate).

Here, c1and c2are the scaling and saturation constants. In this case the reproductive number is

Z1

where

?S ¼c1

ls

w

dpðhÞdhis the survival probability of a parasite in a human

w

dpðhÞdh.

0

nKs

ls

e?Rwþu

w

dpðhÞdhdu:

bðwÞKKhpðwÞ

0

nKs

ls

e?Rwþu

w

dpðhÞdhdu:

ð9Þ

R0¼

0

bðwÞKKhpðwÞ

Z1

0

n?Se?Rwþu

w

dpðhÞdhdudw;

ð10Þ

? c2> 0

ð11Þ

is the carrying capacity of the snails in the absence of parasites, and p(w) is the same survival func-

tion as given before.

3.3. Steady states and their stability in the case of b1(S,I)

In this section, we consider the case of constant snail recruitment rate, i.e., b1(S,I) = Ks. As is in

most epidemiology models, the reproductive ratio R calculated above provides threshold condi-

tions that determines whether the parasites will go extinct or will persist in the host population.

The next result shows that parasite population will go extinct if R is below 1.

Result 1. The disease-free steady state E0of the system (7) is locally asymptotically stable (l.a.s.) if

R < 1 and unstable if R > 1.

Proof. Linearize the system (7) about the E0and consider exponential solutions of the form

pðt;aÞ ¼ p1ðaÞektþ oðe2ktÞ;

BðtÞ ¼ B1ektþ oðe2ktÞ:

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P. Zhang et al. / Mathematical Biosciences 205 (2007) 83–107

Page 9

Then the linear parts of the p and B equations are of the form

dp1ðaÞ

da

B1¼ nKs

¼ bðaÞKhpðaÞB1^KðkÞ ? dpðaÞ þ k

Z1

where^fðkÞ denotes the Laplace transform of f(s), i.e.,

Z1

Solving the p1equation in (12) and noticing that p1(0) = 0 we get

Za

Substituting the above expression for p1(a) in the B1equation in (12) we have

Z1

By changing the order of integration, introducing s = a ? u, and dividing both sides by B1(since

B15 0) in (13) we get the characteristic equation

Z1

Let G(k) denote the right hand side of (14). Then, at k = 0,

Z1

Clearly, Gð0Þ ¼ R. It is easy to see G0(k) < 0, limk!1G(k) = 0, limk!?1G(k) = 1. It follows that

the equation G(k) = 1 has a unique real root k*< 0 if G(0) < 1 (or R < 1), and k*> 0 if G(0) > 1

(or R > 1). Let k = x + iy be an arbitrary complex solution to G(k) = 1. Then

1 ¼ GðkÞ ¼ jGðx þ iyÞj 6 GðxÞ;

which implies that k*> x. It follows that the parasite-free steady state is l.a.s. if R < 1, and unsta-

ble if R > 1. When R > 1 our next result shows that an endemic steady state exists.

??p1ðaÞ;

ls

0

p1ðaÞda;

ð12Þ

^fðkÞ ¼

0

e?ksfðsÞds:

p1ðaÞ ¼

0

bðuÞ^KðkÞB1KhpðuÞe?Ra

uðdpðsÞþkÞdsdu:

B1¼ nKs

ls

0

Za

0

bðuÞ^KðkÞB1KhpðuÞe?Ra

uðdpðsÞþkÞdsduda:

ð13Þ

1 ¼

0

Z1

0

bðuÞ^KðkÞKhpðuÞnKs

ls

e?Ruþs

u

ðdpðsÞþkÞdsdsdu:

ð14Þ

Gð0Þ ¼

0

bðuÞKKhpðuÞ

Z1

0

nKs

ls

e?Ruþs

u

dpðsÞdsdsdu:

ð15Þ

h

Result 2. When R > 1 the system (7) has an unique endemic steady state and it is locally asymptot-

ically stable.

Proof. Recall that a represents the disease-induced human death rate by a single parasite. Thus, a

is very small. For the analytic proof of this result we let a = 0. For the case of a > 0 the result is

confirmed through numerical simulations. Let E*= (n*(a),p*(a),S*,B*) denote the endemic steady

state with positive components, and let N?¼R1

0n?ðaÞda and P?¼R1

0p?ðaÞda. Let a = 0. Then

n*(a) = Khp(a), and E*satisfies

P. Zhang et al. / Mathematical Biosciences 205 (2007) 83–107

91

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d

dap?ðaÞ ¼ bðaÞKhpðaÞKB?? dpðaÞp?ðaÞ;

0 ¼ Ks? lsS?? B?;

B?¼ nP?S?:

Solving for p*(a) we get

Za

Substituting this into the B*equation in (16) we get

Z1

By changing the order of integration, introducing s = a ? u, and dividing both sides by B*we get

the following equation for S*:

Z1

For E*to be biologically feasible, we need to require S*2 (0, Ks/ls). Since H(0) = 0 and

HðKs=lsÞ ¼ R > 1, the monotonicity of H(S*) implies that H(S*) = 1 has a unique root S*in

(0,Ks/ls). In fact, S*= (Ks/ls)(1/R). We can then get B*and p*(a) using the second equation in

(16) and the equation (17), respectively. Thus, we have an unique endemic steady state E*when

R > 1. We proceed to show the stability of E*. Since a = 0, the system (7) becomes

o

otnðt;aÞ þo

o

otpðt;aÞ þo

d

dtSðtÞ ¼ bðS;IÞ ? lsSðtÞ ? BðtÞ;

BðtÞ ¼ nPðtÞSðtÞ:

Note that at E*

?

whereeR ¼ R evaluated at a = 0. Solving the n equation in (20) along the characteristic lines

nðt;aÞ ¼

n0ðaÞ

and substituting this into the p equation in (20) we get

0bðwÞpðwÞe?Ra

qðt;aÞ;

ð16Þ

p?ðaÞ ¼

0

bðuÞKhpðuÞB?Ke?Ra

udpðsÞdsdu:

ð17Þ

B?¼ nS?

0

Za

0

bðuÞKhpðuÞB?Ke?Ra

udpðsÞdsduda:

ð18Þ

1 ¼

0

bðuÞKKhpðuÞ

Z1

0

nS?e?Ruþs

u

dpðsÞdsdsdu ¼: HðS?Þ:

ð19Þ

oanðt;aÞ ¼ ?lhðaÞnðt;aÞ;

oapðt;aÞ ¼ bðaÞnðt;aÞðK ? BÞðtÞ ? dpðaÞpðt;aÞ;

ð20Þ

n?ðaÞ ¼ KhpðaÞ;

P?¼ls

n

eR ? 1

t > a

t 6 a

?

;

S?¼1

eR

Ks

ls

:

ð21Þ

KhpðaÞ;

pðaÞ

pða?tÞ;

(

ð22Þ

pðt;aÞ ¼

KhðK ? BÞðtÞRa

wdPðsÞdsdw;

t > a

t 6 a

(

ð23Þ

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P. Zhang et al. / Mathematical Biosciences 205 (2007) 83–107

Page 11

where

qðt;aÞ ¼ n0ðaÞðK ? BÞðtÞ

pðaÞ

pða ? tÞ

Za

0

bðwÞp1ðwÞe?lpwdw þ pðaÞp1ðaÞe?lpap0ða ? tÞ:

Then

PðtÞ ¼eR

1

nK

ls

KsðK ? BÞðtÞ þ QðtÞ

where QðtÞ ¼R1

PðtÞ ¼eR1

dtSðtÞ ¼ Ks? lsSðtÞ ? nPðtÞSðtÞ:

The linearization of (24) at (P*,S*) (P*and S*are given in (21)) yields the following characteristic

equation

^KðkÞ

K

1 ?

where k is an eigenvalue and^fðkÞ denotes the Laplace transform of f. We need to show that (25)

has no roots with a non-negative real part wheneR > 1. Suppose not. Then (25) has a root

x þ iy þ lseR þ ls

Introducing the notation

Z1

Z1

Noticing that

R1

R1

R1

KðK ? KcÞ

½K ? Kc?2þ ðKsÞ2? 1 þ

tqðt;aÞda. Noticing that Q(t) ! 0 as t ! 1, we have the following limiting sys-

tem for P(t) and S(t) (see (20))

ls

KsðK ? ðPSÞÞðtÞ;

d

K

ð24Þ

k þ lseR þ ls

1

^KðkÞ

K

ðeR ? 1Þ ¼ 0;

ð25Þ

k = x + iy for which x P 0 and

^Kðx þ iyÞ

K ?^Kðx þ iyÞðeR ? 1Þ ¼ 0:

ð26Þ

Kc¼

0

KðtÞe?xtcosytdt;

Ks¼

0

KðtÞe?xtsinytdt:

^Kðx þ iyÞ

K ?^Kðx þ iyÞ¼

0KðtÞe?xt?iytdt

0KðtÞe?xt?iytdt

0KðtÞdt

R1

R1

R1

0KðtÞdt ?R1

0KðtÞdt ?R1

0KðtÞð1 ? e?xtcosytÞdt þ iR1

¼

0KðtÞe?xt?iytdt? 1

0KðtÞdt

¼

0KðtÞe?xtsinytdt? 1

?

½K ? Kc?2þ ðKsÞ2

¼

KKs

!

i

P. Zhang et al. / Mathematical Biosciences 205 (2007) 83–107

93

Page 12

and separating the real and imaginary parts of the left hand side of (26) we get

?

y ? ls

From the second equation in (27), we have

½K ? Kc?2þ ðKsÞ2¼y

and substituting this into the first equation in (27), we have

x þ lseR þ ls ðK ? KcÞy

which simplifies to

x þ yK ? Kc

Ks

x þ lseR þ ls

½K?Kc?2þðKsÞ2

KðK?KcÞ

½K?Kc?2þðKsÞ2? 1

eR ? 1

?

eR ? 1

??

¼ 0;

KKs

??

¼ 0:

8

>

>

:

<

ð27Þ

K

ls

1

eR ? 1

1

Ks;

ls

1

eR ? 1

1

Ks? 1

??

ðeR ? 1Þ ¼ 0;

þ ls¼ 0:

ð28Þ

However, from

½K ? Kc?2þ ðKsÞ2¼y

we have

K

ls

1

eR ? 1

1

Ks¼

y

Ks

1

eR ? 1

1

ls

;

y

Ks> 0, which yields

x þ yK ? Kc

Ks

as Kc< K and x P 0. This contradicts with (28). Hence all eigenvalues will have negative real

parts. It follows that for solutions near E*P(t) ! P*and S(t) ! S*as t ! 1. Consequently,

p(t,a) ! p*(a) as t ! 1 [see (17)]. From (22), it is clear that n(t,a) ! n*(a) as t ! 1. Therefore,

E*is l.a.s. if a = 0. This result is confirmed for the case of a > 0 by numerical simulations (see

Fig. 1). Fig. 1 illustrates that the number of parasites P(t) will go to zero if R < 1 [see

Fig. 1(a)] and it will stabilize at a positive level if R > 1 [see Figs. 1(b) and (c)]. The parameter

values used in Fig. 1 are chosen based on [12,13] and they are: k = 0.243, ls= 0.3, lp= 0.2,

q = 0.0005, c = 4, ds= 0.01, a = 10?5(the time unit is year). r(s) is a periodic function of period

eight weeks which is chosen to be piecewise linear in our simulation. b(a) = 2 · 10?4and

r(a) = 0.2 are both constant, and lh(a) is obtained from the census of United States in the year

of 2000. The proof of Result 2 is complete.

h

þ ls> 0

The following result shows that when R < 1 the parasite-free steady state is actually globally

asymptotically stable.

Result 3. If R < 1 then E0is a global attractor, i.e.

lim

t!1ðnðt;aÞ;pðt;aÞ;SðtÞÞ ¼

for all positive solutions of the system (7).

KhpðaÞ;0;Ks

ls

??

94

P. Zhang et al. / Mathematical Biosciences 205 (2007) 83–107

Page 13

Proof. Let

p1ðaÞ ¼ limsup

t!1

pðt;aÞ;

S1¼ limsup

t!1

SðtÞ;

P1¼ limsup

t!1

PðtÞ;

P1¼ liminf

t!1

PðtÞ:

Using the Corollary 2.4 in Thieme [29], we can choose a sequence tn! 1 such thatdSðtnÞ

S(tn) ! S1as n ! 1. Using this sequence and the S equation in (24) we get

0 6 Ks? lsS1? nP1S1:

Since P1P 0, from the above inequality

Ks

lsþ nP16Ks

Then from the P equation in (24) we have

P16 P1S1eRls

SinceeR < 1, the above inequality implies that P1= 0, i.e., limt!1P(t) = 0. Using this and the S

have from (23) that limt! 1p(t,a) = 0 for all a. It is obvious from (22) that limt!1n(t,a) = Khp(a)

for all a. This completes the proof of Result 3.

dt

! 0 and

S16

ls

:

Ks6 P1eR:

ð29Þ

equation in (24) we get limt!1S(t) = Ks/ls. Noticing that limt!1B(t) = limt! 1nP(t)S(t) = 0 we

h

3.4. Steady states and stability for the case of b2(S,I)

In this case, the reproductive number R0is given by (10). The parasite-free steady state is

E0

0¼ ðKhpðaÞ;0;?SÞ;

where?S is given by (11). Similarly to the case of b1(S,I), we have the following global stability of

E00.

Result 4. E00is a global attractor if R0< 1 and it is unstable if R0> 1.

abc

Fig. 1. Plots of the number of adult parasites P(t) vs. time for different R values. In (a) R < 1 and it shows that the

parasite-free steady state is stable. In (b) and (c) R > 1 and the endemic steady state is stable. Moreover, the endemic

level increases as R increases.

P. Zhang et al. / Mathematical Biosciences 205 (2007) 83–107

95

Page 14

Proof. The proof is the same as that for the case of b1(S,I) (see Results 1 and 3).

h

The existence and the stability of the endemic steady state are more difficult to prove than

that for the case constant birth rate of snails. In fact, the following result shows that the

endemic steady state is not always stable in the case of density dependent birth rate of

snails, b2(S,I).

Result 5. (a) The system (7) has a unique endemic equilibrium E}if R0> 1; (b) When the model

parameters (except c1) are chosen in a realistic region, a Hopf bifurcation may occur at a critical point

c1¼ ? c1such that E}is l.a.s. if c1< ? c1and unstable if c1> ? c1in which case stable periodic solutions

exist.

Proof. (a) Again our analytical proof is for the case of a = 0. Our numerical simulations support

the result for the case of a > 0. Let n}ðaÞ, p}ðaÞ, S}and I}denote the steady state values of the

corresponding variables at E}, and let P}¼R1

d

dap}ðaÞ ¼ bðaÞKhpðaÞKnP}S}? dpðaÞp}ðaÞ;

c1S}

c2þ S}þ I}? lsS}? nP}S};

I}¼nP}S}

0p}ðaÞda. Then P}> 0, n}ðaÞ ¼ KhpðaÞ, and

p}ðaÞ;S};I}satisfy the equations

0 ¼

lsþ ds:

ð30Þ

Solving the first equation for p}ðaÞ we have

Za

Integrating both sides from 0 to 1, dividing both sides of the resulting equation by P}, and using

(10) we get 1 ¼ S}eR0=?S or

S}¼

p}ðaÞ ¼ P}S}

0

e?Ra

wdpðsÞdsbðwÞKhpðwÞKndw:

1

eR0?S;

ð31Þ

whereeR0¼ R0evaluated at a = 0 and?S > 0 is given in (11). Using (31) and the last two equations

c1

c2þ

e R0

in (30) we get

1

n

lsþds?SP}þ

1

e R0?S¼ lsþ nP}

which can be written as

a2ðP}Þ2þ a1P}þ a0¼ 0

ð32Þ

with

96

P. Zhang et al. / Mathematical Biosciences 205 (2007) 83–107

Page 15

a0¼ls c2þ1

?

a2¼n2?S

eR0

the equation (32) has a unique positive solution ifeR0> 1 and no positive solutions ifeR06 1. This

eR0?S

eR0þ

??

? c1¼ ls?S

ls

lsþ ds

1

eR0? 1

??

;

a1¼n c2þ

?S

?S

eR0

?

;

1

lsþ ds:

Obviously, a1> 0 and a2> 0. It is also clear that a0< 0 ifeR0> 1 and a0> 0 ifeR0< 1. Therefore,

finishes the proof of the part (a).

Fig. 2. Plots of the number of adult parasites P(t) vs. time. In (a) and (b) R0< 1 and the parasite-free steady state is

stable. In (c) and (d) R0> 1 and R0is not too big. The endemic steady state is stable. In (e) and (f) R0> 6 and stable

periodic solutions exist.

P. Zhang et al. / Mathematical Biosciences 205 (2007) 83–107

97