COMMUNICATIONS ON doi:10.3934/cpaa.2011.10.1537PURE AND APPLIED ANALYSISVolume 10, Number 5, September 2011 pp. 1537–1548

FLUCTUATION AND EXTINCTION DYNAMICS IN

HOST-MICROPARASITE SYSTEMS

Kaifa Wang

Department of Medical Device and EquipmentSchool of Biomedical Engineering and Medical Imaging

Third Military Medical University, Chongqing 400038, P. R. China.

Yang Kuang

School of Mathematical and Statistical Sciences

Arizona State University, Tempe, AZ 85287-1804, USA

Abstract. Although experimental and observational studies have shown that

microparasites can induce the deterministic reduction, fluctuation and extinc-

tion scenarios for its host population, most existing host-parasite interactionmodels fail to produce such rich dynamical behaviors simultaneously. We ex-

plore the effects of explicit dynamics of parasites under logistic host growth anddifferent infection rate function. Our results show that the explicit dynamics

of parasites and standard incidence function can induce the host density fluc-

tuation and extinction scenario in the case of logistic host growth.

1. Introduction. Parasites, particularly in microparasites that have direct repro-duction within their host ([1, 2]), such as viruses, bacteria, or unicellular eukaryotes,have been shown to reduce host density and even to induce host population extinc-tion in some cases (e.g., [9] and references cited therein). For example, Daphniapopulations can exhibit fluctuation throughout the growing season ([8]) and theoscillation behaviors was also observed in chronic HBV or HCV carriers ([5, 7, 20]).

In an effort to understand the parasite induced host extinction, Ebert et al. ([9])formulated the following plausible but ad hoc epidemiological microparasite model

dx

dt= r(x+ θy)[1− k(x+ y)]− dx− βxy,

dy

dt= βxy − (d+ a)y,

(1)

where x, y represent the densities of uninfected and infected hosts respectively; r isthe maximum per capita birth rate of uninfected hosts; θ ≤ 1 is the relative fecun-dity of an infected hosts; 1/k measures the carrying capacity for host population.Uninfected hosts are die at a density-dependent rate of, dx, and become infectedwith a rate of βxy. Infected hosts are produced at the rate of βxy and die at adensity-dependent rate of (d+a)y that is higher than the rate d at which susceptiblehosts die. The parameter a measures the virulence of the parasite, which is defined

2000 Mathematics Subject Classification. Primary: 92D25, 92D30; Secondary: 34D23.

Key words and phrases. Host-microparasite model, stability, reduction, fluctuation, extinction.The first author is supported by the National Natural Science Fund of P. R. China (Nos.

30770555 and 10571143) and the Natural Science Foundation Project of CQ CSTC (2010BB5020).The second author is supported in part by DMS-0436341 and DMS/NIGMS-0342388.

1537

1538 KAIFA WANG AND YANG KUANG

as the contribution of the parasite to the mortality of the host. Model (1) predictsthe existence of a globally attractive positive steady state ([13]).

After a careful examination of the infection rate in model (1), Hwang and Kuang([13]) replaced the mass action incidence function with a standard incidence func-tion, which is described by βxy/(x + y) and has been employed in some host-microparasite interaction models ([10, 11, 12, 14, 18]). A comprehensive survey onthis modification can be found in [13]. With this simple modification, model (1)becomes

dx

dt= r(x+ θy)[1− k(x+ y)]− dx− β x

x+ yy,

dy

dt= β

x

x+ yy − (d+ a)y,

(2)

where β measures the maximum number of infections an infective host can cause ina unit of time. In [13], a complete mathematical investigation of the revised model(2) shows that the host extinction and reduction dynamics do occur but oscillatorybehaviors can not happen.

Note that models (1) and (2) have not included the dynamics of free parasitesexplicitly. The reason is that there is a plausible assumption that the amount offree parasites is simply proportional to the number of infected hosts because thedynamics of the parasite is usually substantially faster than that of the infectedhost ([3, 13, 14, 21, 22, 25]). Hence, in average, the number of infected hosts y canbe considered also a measure of free parasites.

However, it is not clear when such a plausible assumption is actually reasonable.Hence we would like to consider the effect of an explicit parasite dynamics in both(1) and (2).

The organization of this paper is as follows. In section 2, under the assumptionof mass action incidence function, the effect of the parasite dynamics is studied,and the results show that the host density reduction and fluctuation scenarios canbe addressed, but it fails to explain host deterministic extinction phenomena. Insection 3, the model, which combines the factors of standard incidence functionand parasite dynamics, is discussed and all scenarios, including host deterministicreduction, extinction and fluctuation, can be observed simultaneously. Finally, adiscussion and some biological implications of our findings are given in section 4.

2. Effects of parasite dynamics with mass action incidence function. Whenmass action incidence function is considered but the explicit dynamics of parasiteis ignored, model (1) is proposed in [9] to understand the parasite induced hostextinction. However, for model (1), the host-extinct equilibrium E0 = (0, 0) is al-ways unstable. Furthermore, it is easy to see that the basic reproduction number ofparasites is R0 = β(r− d)/rk(a+ d). As stated in [13], using the auxiliary functionD = 1/xy as Dulac function, we have

Theorem 2.1. In model (1), the disease-free equilibrium E1 = ((r − d)/rk, 0)is globally asymptotically stable if R0 < 1 and the unique infection equilibriumE∗ = (x∗, y∗) is globally asymptotically stable if R0 > 1. Here,

x∗ =a+ d

β, y∗ =

−B1 +√B2

1 −B2

2rkθ.

where

B1 = rk(1 + θ)x∗ +βc

ux∗ − rθ, B2 = 4rkθ[rkx∗ − (r − d)]x∗.

DYNAMICS IN HOST-MICROPARASITE SYSTEMS 1539

Incorporating the parasite dynamics in model (1), we havedx

dt= r(x+ θy)[1− k(x+ y)]− dx− βxv,

dy

dt= βxv − (d+ a)y,

dv

dt= cy − uv.

(3)

where we assume that the free parasites are released from infected hosts at the rateof cy and die at the rate of uv. Observe that when there is no parasites, the growthof hosts obeys the following simple logistic equation

dx

dt= (r − d)x− rkx2 = (r − d)x(1− x

K) (4)

where K = (r− d)/(rk). Therefore, in the rest of this paper, we assume that r > dto ensure that the host is viable when there is no parasite.

The objective of this section is to perform a qualitative analysis of system (3).We begin this by computing the basic reproduction number of parasites in (3).Recall that the basic reproduction number is the mean number of secondary caseswhich is caused by a typical infected individual in a totally susceptible populationin its lifetime in the absence of any control policies. For an infected host, in itsmean lifetime of T = 1/(d+a), it can generate in average P (T ) amount of parasiteswhere P (T ) = cT . A parasite has an average lifetime of 1/u and hence can ata maximum produce I(T ) amount of infections, where I(T ) = βK/u. Therefore,according to the definition of the basic reproduction number, we can see that thebasic reproduction number of parasites in (3) is

R0 = P (T )I(T ) =βc(r − d)

rku(a+ d).

It is easy to see that the host-extinct equilibrium E0 = (0, 0, 0) and the disease-free equilibrium E1 = (K, 0, 0) always exist for (3). If R0 > 1, in addition tothe host-extinct and disease-free equilibria, there is an unique infection equilibriumE∗ = (x∗, y∗, v∗) for (3). Here,

x∗ =u(a+ d)

βc,

y∗ =−B1 +

√B2

1 −B2

2rkθ,

v∗ =cy∗

u,

(5)

where

B1 = rk(1 + θ)x∗ +βc

ux∗ − rθ, B2 = 4rkθ[rkx∗ − (r − d)]x∗.

Note that rkx∗ − (r − d) = r−dR0

(1−R0).

Now, we consider the stability of the equilibria of model (3). The Jacobian matrixJ of (3) at X = (x, y, v) is

J =

r − d− 2krx− kr(1 + θ)y − βv rθ − kr(1 + θ)x− 2krθy −βxβv −(a+ d) βx0 c −u

(6)

1540 KAIFA WANG AND YANG KUANG

Clearly, the characteristic equations associated with the Jacobian matrix (6) at theequilibria E0 and E1 are

HE0(ω) = [ω − (r − d)](ω + a+ d)(ω + u) = 0,

HE1(ω) = [ω + (r − d)][ω2 + (a+ d+ u)ω + u(a+ d)(1−R0)] = 0,

respectively. Thus, E0 is always unstable, E1 is locally asymptotically stable ifR0 < 1 and unstable if R0 > 1.

Using (4), we have x→ K = (r − d)/kr as t→ +∞. Consider asdy

dt= β(

r − dkr

+ ε)v − (a+ d)y,

dv

dt= cy − uv.

(7)

Note that (7) is a linear system and its characteristic equation at the origin is

H(ω) = ω2 + (a+ d+ u)ω + u(a+ d)(1−R0)− βcε.

Since R0 < 1 and ε can be chosen arbitrarily small, we take 0 < ε < u(a + d)(1 −R0)/βc such that u(a+ d)(1−R0)− βcε > 0. Thus, (y, v)→ (0, 0) as t→ +∞ for(7). By a standard comparison theorem of ordinary differential equations, we have(y, v) → (0, 0) as t → +∞ for (3). Consequently, x → (r − d)/kr as t → +∞ for(3).

Therefor, we we have the following result for model (3).

Theorem 2.2. In model (3), E0 is always unstable, E1 is globally asymptoticallystable if R0 < 1 and unstable if R0 > 1.

Next, we consider the stability of E∗ of (3). With the aid of Mathematica, wehave the following characteristic equations associated with the Jacobian matrix (6)at the equilibria E∗

HE∗(ω) = B0ω3 +B1ω

2 +B2ω +B3 = 0, (8)

where

B0 = 1, B1 = a+d+u+A1, B2 = A1(a+d+u)−A2βv∗, B3 = (a+d−A2)βuv∗.

Here,

A1 = krx∗ +rθy∗(1− ky∗)

x∗> 0, A2 = rθ − kr(1 + θ)x∗ − 2krθy∗.

Using the expression in (5), we have

a+ d−A2 =

√[a+ d− rθ +

kru(1 + θ)(a+ d)

βc]2 + (

2kru(a+ d)

βc)2(R0 − 1) > 0.

Thus, B1 > 0, B3 > 0. Let

∆2 =

∣∣∣∣ B1 B0

B3 B2

∣∣∣∣ . (9)

Then, by Routh-Hurwitz criterion, we have the following local stability result.

Theorem 2.3. When R0 > 1 is valid, E∗ is locally asymptotically stable if ∆2 > 0and unstable if ∆2 < 0.

DYNAMICS IN HOST-MICROPARASITE SYSTEMS 1541

It is easy to see that system (3) is uniformly persistent if R0 > 1, the proof ofwhich is rather standard and is the same as Theorem 6 in [23].

Due to the complexity of the expression of ∆2 in (9), we will explore the dy-namics of (3) through numerical simulations based on the hepatitis B virus (HBV)infection. A baseline range of values for most parameters can be determined fromempirical data already in the literature (Table 1). Figure 1A shows that the infectedequilibrium E∗ is stable, Figure 1B indicates that E∗ is unstable and a stable pe-riodic solution appears. Comparing the value of β or R0 in Figure 1A and that inFigure 1B, we conclude that the large rate of virion infection of hepatocytes or basicreproduction number of virion could induce the fluctuation scenario. According tothe normal range of β in Table 1, we observe that the fluctuation dynamics in HBVinfection is rather robust for model (3), see Figure 2.

Table 1. Parameters (Par.) and values, andreferences (Ref.) for HBV infection.Par. Meaning Value Ref.r Maximum hepatocyte growth rate ≤ 1.0 day−1 [6, 17]1/k Hepatocyte carrying capacity 2× 1011 cells [19]β Rate of virion infection of hepatocytes 3.6× 10−5-1.8× 10−3

cells virion−1 day−1 [24]d Normal death rate for hepatocytes 0.0039 day−1 [4, 16]a+ d Infected hepatocyte death rate 0.0693-0.00693 day−1 [19]c Virion production per infected cell 200-1000

virions cell−1 day−1 [24]u Free virion half life 0.693 day−1 [24]

In fact, note that θ ≤ 1 is the relative fecundity of an infected hosts and it isusual small or zero (see [8]). If the infected hosts stops reproducing (i.e., θ = 0) andthey form a small part of the total host population, then one may exclude themfrom the carrying capacity of the susceptible host growth dynamics in (3). Thisresults in

dx

dt= rx(1− kx)− dx− βxv,

dy

dt= βxv − (d+ a)y,

dv

dt= cy − uv.

(10)

For (10), same as Theorem 7 in [23], we can prove the following result based on theTheorem 1.2 in [26],

Theorem 2.4. Assume R0 > 1 and ∆2 < 0 in (9) hold true. Then system (10)has an orbitally asymptotically stable periodic solution.

3. Effects of parasite dynamics with standard incidence function. Whenstandard incidence function is considered but explicit parasite dynamics is ignored,Hwang and Kuang [13] gave a complete global study of the model (2). However, amain biological limitation of (2) is the lack of oscillatory dynamics. In this section,we add an explicit parasite dynamics to the model (2) and present a systematic

1542 KAIFA WANG AND YANG KUANG

0 500 1000 1500 2000 2500 30000

1

2

3

4

5

6x 10

8

Time, t, (days)

Uninfected hepatocytes, x(t)

0 500 1000 1500 2000 2500 30000

0.5

1

1.5

2x 10

10

A

B

Figure 1. Simulations of two infection outcomes with differentvalues of β for model (3). Here r = 1.0, θ = 0.0001, 1/k =2 × 1011, d = 0.0039, a = 0.005, c = 200, u = 0.693 and the ini-tial condition is (x0, y0, v0) = (1000, 100, 10). β = 1.8 × 10−14 in(A) and β = 1.8 × 10−13 in (B). Note that R0 = 116.282, ∆2 =0.00272 > 0 and E∗ = (1.7133× 109, 9.7148× 1010, 2.8037× 1013)is stable in (A). R0 = 1162.82, ∆2 = −0.00052 < 0 and E∗ =(1.7133× 108, 1.7641× 1010, 5.0912× 1012) is unstable in (B).

0 200 400 600 800 10000

2

4

6

8

10

12

14

16

18x 10

9

Basic reproduction number of virion, R0

Uninfected hepatocytes, x(t)

Figure 2. Bifurcation diagram showing the dynamics of HBV in-fection for increasing basic reproduction number of virion. Withmodel (3) integrated over [0, 3000], the maximum value (solidline) and the minimum value (dash line) of the last 500 itera-tions are plotted. The parameters are r = 1.0, θ = 0.0001, 1/k =2 × 1011, d = 0.0039, a = 0.005, c = 200, u = 0.693 and the initialcondition is (x0, y0, v0) = (1000, 100, 10).

DYNAMICS IN HOST-MICROPARASITE SYSTEMS 1543

computational exploration to the resulting model which takes the form of

dx

dt= r(x+ θy)[1− k(x+ y)]− dx− β xv

x+ y,

dy

dt= β

xv

x+ y− (d+ a)y,

dv

dt= cy − uv.

(11)

It is easy to see that the parasite basic reproduction number for (11) is

R0 =βc

u(a+ d).

For (11), the host-extinct equilibrium E0 = (0, 0, 0) and the disease-free equilibriumE1 = ((r − d)/rk, 0, 0) always exist. Under the assumption of rθ > a+ d, (11) hasan unique infection equilibrium E∗ = (x∗, y∗, v∗) if R0 > 1. When rθ < a + d, E∗

exists only if 1 < R0 < R∗. Here,

x∗ =a+ r − rθ − (a+ d− rθ)R0

krR0[1 + θ(R0 − 1)],

y∗ = (R0 − 1)x∗,

v∗ =cy∗

u,

R∗ =a+ r − rθa+ d− rθ

.

(12)

Note that θ ≤ 1 is usual very small. We assume below that rθ < a+ d.Except at E0, the Jacobian matrix J of (11) at X = (x, y, v) is

J =

r−d−2krx−kr(1+θ)y− βyv(x+y)2 rθ−kr(1+θ)x−2krθy+ βxv

(x+y)2 − βxx+y

βyv(x+y)2 −(a+ d)− βxv

(x+y)2βxx+y

0 c −u

(13)

Clearly, J is undefined at E0. The characteristic equations for the equilibrium E1

is

HE1(ω) = [ω + (r − d)][ω2 + (a+ d+ u)ω + u(a+ d)(1−R0)] = 0. (14)

Thus, all roots of (14) are negative if R0 < 1 and at least one eigenvalue becomespositive if R0 > 1.

In order to obtain the global stability of E1, it is sufficient to prove that (y, v)→(0, 0) as t→ +∞. Take an auxiliary system of (11) as

dy

dt= βv − (a+ d)y,

dv

dt= cy − uv.

(15)

Note that (15) is a linear system and its characteristic equation at the origin is

H(ω) = ω2 + (a+ d+ u)ω + u(a+ d)(1−R0).

Thus, (y, v) → (0, 0) as t → +∞ for (15) if R0 < 1. The positivity of solutions of(11) together with a standard comparison theorem of ordinary differential equations,we have (y, v) → (0, 0) as t → +∞ for (11). Consequently, x → (r − d)/kr ast→ +∞ for (11). Thus, we have proven the following result for model (11).

Theorem 3.1. E1 is globally asymptotically stable if R0 < 1 and unstable if R0 > 1.

1544 KAIFA WANG AND YANG KUANG

Consider now the stability of E∗ for (11). Let

A1 = a+ d− rθ, A2 = a+ r − rθ, A3 = a+ d, A4 = r − d.

With the aid of Mathematica, using the expression in (12), we have the char-acteristic equations associated with the Jacobian matrix (13) at the equilibria E∗

is

HE∗(ω) = B0ω3 +B1ω

2 +B2ω +B3 = 0, (16)

where

B0 = 1, B1(R0) =f1(R0)

uR0[1 + θ(R0 − 1)],

B2(R0) =f2(R0)

uR20[1 + θ(R0 − 1)]

, B3(R0) =f3(R0)

R20[1 + θ(R0 − 1)]

,

and

∆2(R0) =

∣∣∣∣ B1 B0

B3 B2

∣∣∣∣ =f4(R0)

u2R30[1 + θ(R0 − 1)]2

. (17)

Here,f1(R0) = C10R

20 + C11R0 + C12,

f2(R0) = C20R30 + C21R

20 + C22R0 + C23,

f3(R0) = C30R30 + C31R

20 + C32R0 + C33,

f4(R0) = C40R50 + C41R

40 + C42R

30 + C43R

20 + C44R0 + C45.

(18)

The expressions of Cij in (18) can be found in the appendix. After some algebracalculations, we have

f1(1) = u(a+ r + u) > 0,

f3(1) = (r − d)(u(a+ d)− βc) = 0, but f′

3(1) = 2u(r − d)(a+ d) > 0,f4(1) = (r − d)(a+ r + u)(a+ d+ u) > 0.

Since function f1(R0), f3(R0) and f4(R0) are continuous, there exist R1 > 1, R2 > 1and R3 > 1 such that

B1 > 0 if 1 < R0 < R1,B3 > 0 if 1 < R0 < R2,∆2 > 0 if 1 < R0 < R3,

respectively. Thus, by Routh-Hurwitz criterion, we have

Theorem 3.2. Suppose that R0 > 1 and rθ < a+ d. Then E∗ is locally asymptot-ically stable if 1 < R0 < min{R1, R2, R3, R

∗}.

Clearly the Jacobian matrix at E∗ is complicated and model (11) is not differ-entiable at E0. In order to gain a global insights into the dynamics of model (11),we carried out a systematic computational exploration of (11). Figure 3 illustratessome of the typical model outcomes. Again, the values for the model parameterscome from Table 1.

Figure 3A shows that the infected equilibrium E∗ is stable, Figure 3B indicatesthat E∗ is unstable and a stable periodic solution appears, Figure 3C indicates thatthe host-extinction equilibrium E0 is stable. Comparing the value of β or R0 inFigure 3, we conclude that the large rate of virion infection of hepatocytes or basicreproduction number of virion could induce the fluctuation or extinction scenarios.The outcome of HBV infection will become increasingly more serious as the basicreproduction number of virion is increased and the final result is inevitably an acuteliver failure (ALF) or an acute liver necrosis (ALN) ([15]), see also Figure 4.

DYNAMICS IN HOST-MICROPARASITE SYSTEMS 1545

0 500 1000 1500 2000 2500 30000

0.5

1

1.5

2x 10

11

0 500 1000 1500 2000 2500 30000

5

10

15

x 108

Uninfected hepatocytes, x(t)

0 500 1000 1500 2000 2500 30000

10

20

30

40

50

Time, t, (days)

A

B

C

Figure 3. Simulations of three different infected outcomes underdifferent value of β for model (11). Here r = 1.0, θ = 0.0001, 1/k =2 × 1011, d = 0.0039, a = 0.005, c = 200, u = 0.693 and the initialcondition is (x0, y0, v0) = (1000, 100, 10). β = 1.2 × 10−4 in (A),β = 2.8 × 10−3 in (B) and β = 4.0 × 10−3 in (C). Note thatR0 = 3.8912 and E∗ = (4.9875× 1010, 1.442× 1011, 4.1616× 1013)is stable in (A). R0 = 90.7956 and E∗ = (4.4951 × 108, 4.0364 ×1010, 1.1649 × 1013) is unstable in (B). R0 = 129.708 and E0 isstable in (C).

In fact, if we suppose that the infected hosts has no fecundity (i.e., θ = 0) andthe death rate of susceptible hosts is neglected, (11) is reduced to the model in [12]and a rigorous proof of the globally asymptotically stable of the equilibrium E0

was given by a change of variable technique when the basic reproduction number ofvirion R0 is sufficient large. Furthermore, an attracting limit cycle is also observedby numerical simulation. Thus, host deterministic reduction, extinction and fluctu-ation are the robust phenomena in the model with explicit parasite dynamics andstandard incidence function.

4. Discussion. Epidemiological models are often used to explain empirical resultswhere microparasites reduce the density or lead to the extinction or fluctuation oftheir host populations ([9]). In this paper, we perform an analysis on the effectof explicit dynamics of parasites with mass action incidence function or standardincidence function. We see that a combination of explicit dynamics of parasitesand standard incidence function can indeed generate the often observed reduction,

1546 KAIFA WANG AND YANG KUANG

0 50 100 1500

0.5

1

1.5

2

2.5

3x 10

9

Basic reproduction number of virion, R0

Uninfected hepatocytes, x(t)

Figure 4. Bifurcation diagram showing the dynamics of HBV in-fection for increasing basic reproduction number of virion. Withmodel (11) integrated over [0, 3000], the maximum value (solidline) and the minimum value (dash line) of the last 500 itera-tions are plotted. The parameters are r = 1.0, θ = 0.0001, 1/k =2 × 1011, d = 0.0039, a = 0.005, c = 200, u = 0.693 and the initialcondition is (x0, y0, v0) = (1000, 100, 10).

fluctuation and extinction of the host population for different parasite basic repro-duction numbers.

We summarize the main effects on host population of all aspects in Table 2.Clearly, the popular quasi-steady-state assumption that the amount of free para-sites is proportional to the number of infected hosts, is not reasonable under logistichost growth, since the host density fluctuation scenario will be lost. Hence, we sus-pect that one of the possibly many causes of deterministic oscillations of host is thenatural explicit parasite dynamics interacting with logistic host growth dynamics.Furthermore, we observe that the main effect of the standard incidence functionis its ability of inducing the host extinction scenario with logistic host growth. Inconclusion, we believe that a combination of logistic host growth, standard inci-dence function and explicit dynamics of the parasite forms a more balanced modelframework for describing basic host-parasite interactions than most other existingmodels.Table 2. Summarize the main effects on host population of all studied modelsModel Factors Effects on host population Reference

(1) (a)+(b) reduction [9, 13](3) (a)+(B) reduction and fluctuation [23] and this paper(2) (A)+(b) reduction and extinction [13](11) (A)+(B) reduction, fluctuation and extinction [12] and this paper

Here, (a) mass action incidence function; (b) implicit parasite dynamics;

(A) standard incidence infection function; (B) explicit parasite dynamics.

Acknowledgments. We would like to thank the referees for some helpful sugges-tions.

DYNAMICS IN HOST-MICROPARASITE SYSTEMS 1547

Appendix. Expressions of Cij in (18).

C10 = −uA1 + θ(u2 + βc+ u(A3 −A2)),C11 = u2(1− θ) + βc(1− 2θ) + 2uθA2 + u(1− θ)A3,C12 = −(1− θ)(βc− uA2),C20 = −A1(u2 + βcθ + uA3) + uθ(βc+ uA3 −A2(u+A3)),C21 = βc(1− 3θ)(u−A1) + u2A3(1− θ) + 2uθA2(u+A3) + βcθA4,C22 = βcA1(2− 3θ) + uA2(1− θ)(u+A3) + βc(A4(1− 2θ)− 2u(1− θ)),C23 = −βc(1− θ)(A1 +A4),C30 = −uθA2A3 −A1(βcθ + uA3),C31 = θ(2uA2A3 + βc(3A1 +A2 +A4)),C32 = βcA1(2− 3θ) +A2(uA3(1− θ)− 2βcθ) + βcA4(1− 2θ),C33 = −βc(1− θ)(A1 +A2 +A4),C40 = uA2

1(u2 + βcθ + uA3)− θA1(u4 + 2βcu2 + β2c2θ+uA3(2u2 + βc(1 + θ)) + u2A2

3 − uA2(2u2 + βcθ + 2uA3))+uθ2(βc(βc+ u2) +A3(u3 + 2βcu) + u2A2

3 + uA22(u+A3)

−A2(u3 + 2βcu+A3(βc+ 2u2) + uA23)),

C41 = βcuA21(1− 3θ)−A1(u4(1− θ) + 2βcu2(1− 2θ) + β2c2θ(2− 5θ)

+A3(2u3(1− θ) + βcu(1− 4θ2)) + u2A23(1− θ) + uθA2(4u2

−βc(1− 5θ) + 4uA3) + βcuθA4)− θ(4u2θA22(u+A3)− 2u3A2

3(1− θ)+βc(u(2u2(2θ − 1) + βc(5θ − 2))− βcθA4) + uA3(u(2u2(θ − 1)+βc(7θ − 4))− βcθA4) + uA2(u3(1− 3θ) + 2βcu(1− 4θ)+A3(2u2(1− 3θ) + βc(1− 4θ)) + uA2

3(1− 3θ) + βcθA4)),C42 = βcu(u2(1− 6θ) + βc(1− 8θ) + θ2(5u2 + 9βc)−A2

1(2− 3θ))+u2A3(u(1− θ)(A3 + u(1− θ)) + 2βc(1− 5θ + 4θ2))−2θu2A2

2(1− 3θ)(u+A3 + βcθA4(2βc(1− 2θ) + uA3(2− 3θ))+uθA2(3u2(1− θ) + 4βcu(2− 3θ) +A3(2βc(2− 3θ) + 6u2(1− θ))+3uA2

3(1− θ)− βcA4(1− 4θ))−A1(uA2(2u(1− θ)(u+A3)+5βcθ(1− 2θ)) + βc(βc(1− 8θ + 10θ2)− 2u2(1− θ)−uθA3(5− 6θ) + uA4(1− 2θ))),

C43 = u(1− θ)(βcA21 + 4uθA2

2(u+A3)) + βcA1(βc(3− 12θ + 10θ2)−uA2(1− 9θ + 10θ2) + 2uA3(1− 3θ + 2θ2) + uA4(1− θ))+uA2((θ − 1)(u(θ − 1)(u+A3)2 + βc(4θ − 1)(2u+A3))+2βcθA4(2− 3θ)) + βc(u(1− θ)(u(θ − 1)(2u+ 3A3)+βc(7θ − 3) +A3A4(1− 3θ)) + βcA4(1− 6θ + 6θ2)),

C44 = (θ − 1)(u2A22(θ − 1)(u+A3) + βcA1(βc(3− 5θ) + uA2(5θ − 2)

+uA3(1− θ)) + βcuA2((1− θ)(2u+A3)−A4(1− 4θ))+βc(2βcu(θ − 1) +A4(2βc(1− 2θ) +A3u(1− θ)))),

C45 = βc(βc− uA2)(A1 +A4)(1− θ)2.

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Received August 2009; revised August 2010.

E-mail address: kaifawang@yahoo.com.cn

E-mail address: kuang@asu.edu