Parasitism and host patch selection: A model using aggregation methods

Modelling Vol. 27, No. 4, pp. 73-80, 1998 Copyright@1998 Elsevier Science Ltd Printed in Great Britain. Ail rightsreserved 0895-7177198 il9.00 + 0.00

Mathl. Comput.

PII: SO895-7177(98)00007-7

Parasitism and Host Patch Selection: A Model Using Aggregation Methods S. MORAND CNRS UMR No5555 Centre de Biologie et d’Ecologie Tropicale, Universite Perpignan av. Villeneuve, 66860 Perpignan Cedex, FYance morandclluniv-perp.f r P. AUGER AND J.-L. CHASSB CNRS UMR No5558 Universite Claude Bernard Lyon 1 43 Bd. du 11 Novembre 1918, 69622 Villeurbanne Cedex, France paugerQbiomserv.univ-lyonl.fr Abstract-There is a growing interest in studying the effects of parasites on the modification and evolution of hosts’ behaviour. In this paper, we deal with a case of parasitism affecting the spatial pattern of host distribution. We develop a simple model with two patches, one host and one parasite. Parasites live in Patch 1, hosts live in the two patches and migrate from one patch to the other. We study the case of a migration independent of parasite density and the case of a migration dependent on density. In the two cases, we make the assumption that the choice of patch is fast, whereas the growth of populations are slow. So we use aggregation methods which are particularly adapted for systems exhibiting different times scales. The aggregated model obtained in the case of a density independent migration is a classical predator-prey model. The case of a density dependent migration aggregated model is very different and a nonstandard one, and exhibits an interesting result. Under certain conditions, parasites always become extinct in the case of a density independent migration, whereas the adaptation of hosts (density dependent migration) allows to stabilize the host-parasite system. This first application of the aggregation methods to epidemiology is very promising because these methods allow us to deal with more real assumptions about the behavioural interplay between hosts and parasites. Keywords-Host-parasite

model, Patches migrations,

Aggregation

methods.

INTRODUCTION The role of parasites and pathogens in regulating animal populations is now well recognised [l-3]. A great number of mathematical models have been carried out and various aspects of epidemiology have been investigated [4-61. There is also a growing interest in studying the effects of parasitism on the modification and evolution of animal behaviour [7]. The effects of parasites on the behaviour of their hosts are well documented. Holmes and Bethel [8] indicated different ways by which parasites may modify the infected members of a prey population: disorientation, altered responses. The behavior of those prey infected by parasites is modified so as to make them more easily preyed upon their predator hosts. Numerous examples show that parasites could manipulate the intermediate host in order to facilitate its ingestion by the prey [9]. However, the behavioural consequences of being parasitized have rarely been integrated into epidemiological models [lO,ll]. This lack is particularly evident for parasites affecting the spatial distribution of animals. The lack of modelling that integrates behaviour and epidemiology is probably due to

73

S. MORAND et al.

74

the difficulty of dealing with complex models. In this paper, we use aggregation methods [12-161 to deal with parasitism affecting spatial pattern of host distribution. Aggregation methods have been developed in ecological modelling and look for conditions necessary to obtain a reduced set of differential equations governing aggregated

variables.

These methods

are particularly

useful for systems

scales associated to the dynamics of variables. Here, we develop a simple model based on empirical aim is to obtain

an aggregated

results

of Poulin

model which could take into account

exhibiting

different

and Fitzgerald

time

[17]. Our

the effects of parasites.

THE MODEL Density Independent Migration We develop assume

a simple

model with two patches,

one host and one parasite.

In this section,

we

the following.

1. Parasites live and infect hosts in Patch 1. 2. Hosts migrate with constant migration rates (per unit time and per individual), pendent of hosts and parasite densities. 3. The choice of patch is fast, whereas the growth process are slow. Three equations describe the changes changes of parasites in Patch 1 (pi):

of hosts in Patch

of host population 1 (ni)

i.e., inde-

and the infections

and in Patch

2 (ns),

and the

(lb) &l

Edt =

E(WPl

- /JlPl),

where E CC 1 is a small parameter. Ki and Ks are, respectively, the host carrying capacities in Patches 1 and 2, ~1 and ~2, the growth rates of host in Patch 1 and Patch 2. a refers to the rate of infection and ,UI to the mortality of parasites. Migration rate of hosts (i) from Patch 1 to Patch 2 and k is the migration rate of hosts from Patch 2 to Patch 1. These two parameters are assumed to be constant and independent of the host and parasite densities. We choose the total

densities

to represent

the system

globally,

i.e., the aggregated

variables

are p=p1,

(2)

N = ni + n2. Migration changes the proportions N is invariant for the fast part (migration). (* denotes) is patches, but not their total density N. The fast equilibrium

of hosts on the

i.e., k(N - nl) = EnI,

knz = knl,

(3a)

with solutions kN 72; = k+k v

i

ni =N=Ic

k

and

n;=$,

and

nf vi=-z--z. N

Thus, in the case of density independent migration, proportions vi and v; of hosts on the two patches.

(3b) L

(3c)

k+k

the fast equilibrium

corresponds

to constant

Parasitism

and Host Patch Selection

75

Hence, the aggregated model (see Appendix A) is as follows:

(44 dP dt = P(--CLI + aN),

(4b)

where the parameters are as follows:

r=s+k+E

rlk

I-&

(54

ak

(5b)

a=iz7

Model (4) is a classical prey-predator model that has been studied in many textbooks [18]. There are three equilibrium points (0,O) unstable, (K, 0) stable when K < pi/a and (N* = pi/a, P* = r/a(l - p~/aX)) which belongs to the positive orthant when K > /_~/a. Then, it is stable. Thus, two cases occur. - K > pi/a, (N*, P*) is globally asymptotically stable (the positive orthant is positively invariant). For any initial conditions, hosts and parasites coexist. - K < pi/a, (K,O) is asymptotically stable. Parasites become extinct and hosts tend to the carrying capacity K. In the case of density independent migration, there is a possibility that parasites become extinct. Now, let us consider a parasite density dependent migration of hosts and its effects on the stability of the host-parasite system. Migration

of Hosts Depending

on Parasite

Density

Results of Poulin and Fitzgerald [17] suggested that micro habitat selection by juvenile fish may serve to reduce the risk of parasitism. They demonstrated that fish prefer some kind of habitat, but leave this habitat when parasites are present. In this section, we develop a similar model with two patches, one host and one parasite. This model is still based on the three assumptions of the previous section with changes on Assumption 2 (Figure 1) which becomes the following. Two-hosts prefer to live in Patch 1 when parasites are absent and switch to Patch 2 when parasites are present.

Patch 1

Host carrying capacity Ki Figure 1. Flowchart

Patch 2

Host carrying capacity

K2 of the model of host’s patch selection induced by parasitism.

S. MOFLAND et al.

76

Three equations still describe the changes of hosts in Patch 1 (nr) and in Patch 2 (ns) and the changes of parasites in Patch 1 (pr):

E~=kn2-InlPl+E(71n1(l-~)-anlPl), dm Edt

= Enlpl - kn2+ er2n2

dpl Ez = 4wpl

-

(64 (6b)

,

(6c)

PIPI),

where E << 1 is a small parameter. Parameters have the same significance as in Section 1. However, migration rate of hosts (E) from Patch 1 to Patch 2 is now proportional to the number of parasites (pr), whereas k is the migration rate (constant) of hosts from Patch 2 to Patch 1. The fast equilibrium is now different from the previous section:

km = inIP,

i.e., k(N-

nl)= inIP,

(7)

where solutions are now functions of parasite and host total densities n*1 =

AfL k-tkP

n; _ Vi(P) = y - &

n+ _

and

2

EPN

k+kP’

iiP nI v,*(P) = N = -. k+kP

and

The most important change comes from the fact that now the proportions of hosts on the patches at the fast equilibrium are no more constant and depend upon the parasite density. Hence, the aggregated model (see Appendix A) is as follows: dN -= dt

&

krl + (il.rg - ak) P -

rlk2N K1 (k+kP)

r2E2NP2 - K2(k+iEP)

dP P

(94

(9b)

dt=

The form of this model is very different from the aggregated model in case of constant migrations. This shows how a small change on the fast part can have important effect on the form of the aggregated model. This model (9) is new and nonstandard and we shall study its properties, existence of equilibria, and stability. The positive orthant is positively invariant. The N nullclines are N=O

and

N=KlKz

(krl + (h-2 - ak) P) (k + iP) (rlK2k2 + ~zK~j1~P~)

The second one crosses the P-axis at points A(-(k/@,O), B((krl/(ak and when P + foe lim N = (Ks(& ak))/r& N-axis at E(0, KI), The P nullclines are P=O and N=z(k+kP).

* - bp)),O),

(10) and the

(11)

The second one crosses the P-axis at A(-(k/i),O) (same point as in the previous case (10)) and the N-axis at D(0, @r/a)). When KI > p~/a, there are two nontrivial steady states in the positive orthant E(0, KI) and F(P*, N*) (see Appendix B). The study of the Jacobian shows that E is unstable and the only

Parasitism and Host Patch Selection

77

stable steady state is F(P*,N*) one (see Appendix B). In this case, we see that the system host-parasite gets stabilized. When K1 < /.~/a and r& > ale, there are three nontrivial equilibrium points in the positive orthant (see Appendix B and Figure 3). A stability analysis shows that E and H are stable while G is unstable. There is a separatrix passing through G dividing the phase plane in two parts. According to the initial condition, either the trajectory goes to E (the parasite gets extinct) or it goes to H and host and parasites coexist. The comparison between the density independent case and this case shows an interesting result. When K1 < pi/a, in the density independent case, the parasite always gets extinct; on the contrary, when K1 < pi/a and T& > ak, the density dependence of the host migration, i.e., the adaptation of the hosts (leaving Patch 1 when many parasites are here, repulsive effect) allows us to stabilize the host-parasite system.

DISCUSSION The difficulty of manipulating complex models could be the main reason explaining why behaviour and spatial patterns are rarely taken into account in epidemiological modelling. The aggregation methods allow us to deal with such complexity. Here, we have proposed a simple model of habitat’s choice linked to the presence of a parasite. This very simple model leads to an aggregated model with complex dynamics. Sizes of patches, infection rate, and parasite induced mortality drastically influence the stability of the system. The first observation to note is that a very simple density dependence of the migration of the hosts leads to a new aggregated model quite different from the aggregated model of the density independent case. Furthermore, the qualitative properties of both models (density dependent and independent) are not the same. In particular, new equilibria can appear. This is the case of the density dependent model of Figure 3 with three nontrivial equilibria in the positive orthant (instead of two only in the density independent case). The new equilibrium H can be stable in some cases leading to a new situation in which trajectories can go either to point E or to H according to the initial conditions. In this case, we see that the density dependence plays a stabilizing role because the trajectory does not tend necessarily to E (extinction of the parasite) but, can also tend to G where host and parasite coexist. This first application of the aggregation methods to epidemiology is very promising. These methods allow us to deal with more real assumptions about the behavioural interplay between hosts and parasites. For example, to our knowledge, the only examples of modelling concern the manipulation hypothesis, i.e., the way by which parasites increase their transmission efficiencies by debilitating their hosts [10,19]. However, these models show their limitation when one tries to incorporate more realistic assumptions. The aggregation methods permit to reduce the number of variables if different times scales are associated to the dynamics of variables, which seems the case in host-parasite system. Host patch selection is a fast decision, while parasite infection is a slow process. In the future, we would like to explore more complex behaviours involving, for example, mate choice [20], host decision-making [21], and other patterns which permit to avoid infected individuals.

APPENDIX Derivation

of the Aggregated

A

Model

We start from equations (1) which are recalled in (A.l): drill

c-g-=kn2-bl+e

(A-1)

78

S. MORAND et al.

dnz &-+nl-knz+&rzn2 (A.l)(cont.) 4-a = E(mpl % We add the two first equations

- MPl).

to get the time derivative

of the total

population

N:

$!=rlnl(l-~)+r2n2(l-~)-anlpl,

(A.21

dpl

YiF-= e(mn

- ~lpd.

As N is a constant of motion for the migration, the fast part does not occur in (A.2). The next step is to replace ni and ns in terms of the fast equilibrium, i.e., respectively, by v:N and v,*N and pi by P into (A.2): dN=rN dt

K > -

dP = P(-p1 dt

+ crN).

Equations (A.3) are identical to equations Where the parameters are as follows:

T =

I_!! (

cxNP, (A.3)

(4) and constitute

the aggregated

model.

rlv; + 7-221; and (A-4)

a =

au;.

In the case of density independent migration, the patch frequencies are constant and the model is of the form (4). In the density dependent case, the frequencies are functions of P given by (8b) leading to the aggregated model (9) obtained from (A.3), where

T = QV;(I’)

+7-24(P)

and

f

=

+ g(v;(PJ2)

($(v;(P))’

;

1

,

(A.5)

a = au;(P).

APPENDIX Analysis of the Aggregated

B

Model in the Case of Density Dependent Migration

CASE 1. Kl > pi/a. Figure 2 shows representation of nullclines in the PN-plane in the case depends of the sign of x7-s - ak, where KI > PI/a. If P + foe, the position of the asymptote but whatever case conclusions about steady states are the same. There are six steady states, two of them (nontrivial) in the positive orthant. One obtains for Jacobian:

J(O,O) =

(0,O) is always unstable

(saddle

-“,I)?

point).

J(0, K1) =

E(0, KI) is a saddle

(;

-” ( 0

T

(i(n + ~2) aK1 -

point when KI > PI/a (case of Figure

~1

2).

ak)

,

Parasitism

and Host Patch Selection

79

Steady states

??

P

Figure 2. Nullclines of the aggregated

model (9), case KI > pi/a.

When Ki > /.~i/a (case of Figure 2), there are nontrivial equilibria in the positive orthant E(0, Ki) which is a saddle point and F(P*, N*) which is stable (trace negative, determinant positive). In this case, F is globally asymptotically stable. CASE 2. K1 < pi/a AND rziE > ak. (0,O) is still a saddle point. There equilibria. When Ki < pi/a, E(0, Ki) is a stable node. The equilibria are found when the two following nullclines intersect:

N=KlKz This equation as follows:

(krl + (L-2 -

has the root A(-(k/k), prrzKik2 uk

We limit our study

ak) P) (k + LP)

(rlKzk2 + r2Klk2P2)

P

2

and

0). A simple calculation

- KlKz (krz - uk) P - KlKzkrl

N = 5

are three

nontrivial

(k + kP) .

leads to a second degree equation +

cLlnK2k

= 0.

U

to the case of a positive

discriminant,

4pirir2K2 K2

>

uK1 (kr2 - uk)

This condition can be obtained by an adequate it is easy to check that, in this case, two positive (see Figure 3).

i.e., when

2 (b-K+

choice of the parameters. real roots corresponding

Then, when rzk > uk, to point G and H exist

N

??

Steady states

P Figure 3. Nullclines of the aggregated

model (9), case K1 < pi/a and T& > ak.

80

S. MOUND

et al.

A stability analysis shows that E and H are stable while G is unstable. There is a separatrix. According to the initial condition, either the trajectory goes to E (the parasite becomes extinct) or it goes to H and host and parasites coexist.

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