MODELS FOR HOST-MACROPARASITE SYSTEMS: A COMPARISON BETWEEN SPATIALLY STRUCTURED AND UNSTRUCTURED MODELS

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1 MODELS FOR OST-MACROPARASITE SYSTEMS: A COMPARISON BETWEEN SPATIALLY STRUCTURED AND UNSTRUCTURED MODELS FABIO AUGUSTO MILNER Department of Mathematics, Purdue University 5 North University St., West Lafayette, IN , U.S.A phone: +) , fax: +) milner@purdue.edu and CURTIS A. PATTON Epic Systems Corporation 53 Tokay Blvd., Madison, WI 537-7, U.S.A phone: +) , fax: +) cpatton@epicsystems.com Abstract We consider classical host-parasite systems consisting of ordinary differential equations for the total numbers of hosts, parasites,and larvae. We also consider a newer, spatially structured model that consists of a partial differential equation for each of those populations. We finally compare the behavior of solutions for the spatially homogeneous model to that of the solutions for the spatially structured model through numerical simulations. Key words: host-macroparasite system, continuous deterministic models, numerical simulations. Supported in part by the National Science Foundation under Grant DMS Supported in part by the National Science Foundation under Grant INT

2 MILNER, PATTON Introduction Most host-parasite models in the literature consist of systems of ordinary differential equations that describe the evolution of the numbers of hosts and parasites, and they do not incorporate any other structure. Following ideas of Anderson [,,3,,5,6,7] models usually start from one ODE for each class of hosts n having exactly n parasites and then summing all these classes to arrive to an equation for the total number of hosts. In order to have a closed system, assumptions are made about the distribution of parasites among the hosts. An alternative conceptual approach based on the introduction of a continuous structure variable representing parasite load in the hosts was introduced in [5,6,7,3]. In the last two of these papers the model for the evolution of the hosts is an advection-diffusion equation, but the diffusion term is in the variable representing parasite load, not space. A host-parasite model that introduces spatial diffusion was considered in [8] where existence and uniqueness are considered, but not asymptotic behavior nor the relation with spatially-independent models. The aim of the present paper is to compare the dynamics of models that are spatially structured and those that are not. In the first part of this paper we introduce the models we shall compare: the spatially structured model of [8] and several unstructured models based on Anderson s. In the second part of this paper we compare the behavior of solutions of the two models by examining results from simulations. We should point out that the spatially structured system of three populations introduces some singularities, thus making showing existence, uniqueness and positivity very difficult. So we only present some preliminary results from [8] for the PDE model and leave a more complete analysis for future work. The models we consider contain three populations: hosts, adult) parasites, and their larvae. They include births and natural mortality in the host population. owever, in the simulations we ignore the latter, because we are thinking of situations in which it is not important, such as fish farms in which the length of time of each farming cycle is much shorter than the life span of the fish. Description of the spatially structured model We begin by describing the spatially structured model from [8]. Let h j) = h j) x, t), j IN, denote the spatial density of hosts having j parasites at a point x and time t. Then, j) = j) t) = h j) x, t) dx and = x, t) = h j) Ω represent, respectively, the total number of hosts in the region Ω with a burden of j parasites, and the spatial density of the total number of hosts. j=

3 ost-macroparasite systems 3 Define now the following parameters: b, x, t), the natural density-dependent fertility rate of hosts at a point x and time t. γj), a factor reducing the fertility of hosts in the class h j) ; γj). m h, x, t), the natural density-dependent mortality of hosts at a point x and time t. αj), the parasite-induced mortality of hosts from a burden of j parasites. d h h j), the spatial flux of the host population. The natural fertility and mortality rates here are allowed to depend on but not on the parasite distribution as assumed in [8], since any influence of the parasites in host fertility and mortality should be incorporated in γ and α. Next, let P = P x, t) denote the spatial density of parasites at a point x and time t,.) P = j= jh j). Parasite deaths are due to three mortality sources: their natural mortality at a rate µj) in hosts of class h j), the mortality due to the natural host mortality, and the mortality due to the parasite-induced mortality of hosts from a burden of j parasites, related to αj). These last two sourcess are due to the fact that if a host dies, the parasites it has must also die. Finally, let the spatial density of larvae at a point x and time t be given by L = Lx, t), and define the following parameters: νl, x, t), the natural death rate of larvae at a point x and time t; ˆλj), the laying rate per host in the class h j) of parasites at time t; R j h l) ) l, L, x, t ), the recruitment rate of larvae on hosts of class h j) ; d L L, the spatial flux of the larvae population. We describe next the dynamics of the hosts, starting with hosts having no parasites. We make the following assumption: [ All hosts are born free from parasites. That is, all newborn hosts belong to ) the class h ). Besides spatial migration, the rate of change of h ) consists of the following four terms: m h, x, t) h ) = m h h ), l, the total natural mortality rate of hosts having parasites at a point x and time t. R h l) ) l, x, t ) h ), l, the total rate of loss of hosts having parasites at a point x and time t that recruit larvae at a per capita rate R.

4 MILNER, PATTON µ)h ), the total rate of gain of hosts that come into the class h ) from class h ) because of natural mortality of their parasites. The fertility of hosts having j parasites is γj)b, x, t), and the total birth rate of hosts is γj)b, x, t) h j). j= Thus, the equation for hosts without any parasites is.) h ) t d h h ) = j= b, x, t) γj)h j) m h h ) R h ) + µ)h ). Next we consider parasitized hosts. The equation for the class h j), j, is similar to.) but, because of ), the birth term is not present and, since these hosts have parasites, there is an added, parasite-induced, mortality term, αj)h j). There is also an additional positive term on the right side of the equation, R j h j ), that describes hosts from class h j ) that recruit a parasite and enter the class h j). Finally, there is an additional negative term, µj)h j), that describes hosts from class h j) that have one parasite that dies moving them to the class h j ). The other terms are analogous to the ones in.). ence, the equation describing the dynamics of hosts with j parasites is.3) h j) t d h h j) = µj + )h j+) [ m h + αj) + R j + µj) ] h j) +R j h j ) = F j. We now add some hypotheses to make it easier to define the global dynamics for hosts, = j= h j). First, we assume that parasite recruitment is spatially homogeneous and does not depend on the distribution of parasites among the host classes, but rather on the total number of hosts, : 3) R j = R j h l) ) l, L, x, t ) = R, L). We consider two different functions for R:.) R, L) = βlx, t) where β is a positive constant, and.5) R, L) = ρl Ĉ +. The function.) was proposed by Anderson [] and is based on the assumption that recruitment of larvae is proportional to their number. The second

5 ost-macroparasite systems 5 function.5) is inspired in the ones proposed in [9] and [] to describe a reduced recruitment rate for large host densities. We also include the following hypotheses before we describe the global dynamics of hosts. [ The natural fertility rate of hosts is spatially and temporally homogeneous; it is just a density dependent rate: b, x, t) = b). ) [ The natural mortality rate of hosts is spatially and temporally homogeneous, it is just a density dependent rate: m h, x, t) = m h 3) ). ) d h. [ The natural mortality rate of larvae is spatially and temporally homogeneous, it is just a density dependent rate: ν L, x, t) = 5) νl). We are now ready to describe the global dynamics of hosts. Let X and Y denote random variables, X representing the number of parasites, and let EY ) denote the expected value of Y. Clearly, the probability that a host has j parasites is P X = j) = hj). We proceed as in [,,]: we sum equation.) and equations.3) over all classes to obtain.6) t d h = [b)eγx)) m h ) EαX))]. To obtain the global dynamics for parasites, we multiply equations.3) by j, sum over all j, and recall.). Then,.7) P t = d h j h j) + jf j, j= j= where F k was defined in.3). After some calculations using the fact that h j) = P X = j), we can model the dynamics of parasites by.8) P t d h P = [R, L) m h, x, t)ex) E µx)) E XαX))]. The larvae dynamics involves three phenomena: There is one source term that corresponds to the number of larvae that parasites lay. ˆλj)h j) is the number of eggs laid by parasites on hosts having j parasites, so in all hosts. j= There are two loss terms: ˆλj)h j) is the total number of eggs laid by parasites One is due to the recruitment of larvae by hosts at a point x and time t, R, L)x, t). The other is the rate at which larvae die from natural death at a point x and time t, νl)lx, t).

6 6 MILNER, PATTON.9) In view of 5), we model the larvae dynamics as follows: L t d L L = EˆλX)) R, L) νl)l. Putting equations.6),.8), and.9) together, the dynamics of the hostparasite system is described by t d h = b)eγx)) [m h ) + E αx))],.) P t d h P = [R, L) m h )EX)] [E µx)) + E XαX))], L t d L L = EˆλX)) R, L) νl)l. For biological reasons, all parameters are nonnegative. Thus, we assume { mh ), b), 6) ˆλk), νl), αk), µk), R, L). We also assume that the natural death rate of hosts m h ) is given by {, for a Malthusian growth with b m, 7) m h ) = m + r, r = b m, for a logistic growth with b > m, K where K represents the carrying capacity of the environment. We define the functions that depend on k following May and Anderson: { µk) = µk l and αk) = αk l for l =,, 8) γk) = γ k, < γ, ˆλk) = ˆλk, where l = means that natural parasite death rates or induced mortality of hosts are density dependent on parasites, whereas l = means that those rates do not depend on how many parasites are present on hosts. In order to have a closed model in terms of, P and L, one usually makes some assumptions on the distribution of parasites in hosts. Some classical assumptions are that parasites are distributed in the host population according to a Poisson law with mean z = P/ or a negative binomial with the same mean and a clumping parameter [,5]. The second distribution is characterized by the fact that a few hosts carry a high parasite burden, while the majority of hosts have few parasites [,,3,,5,8,9,]. This type of distribution is said to be overdispersed. Four types of models denoted I to IV are obtained for each of these distributions by assuming the induced mortality of hosts to be density dependent or independent, and the natural death rate of parasites to be density dependent or not. For each model, using a Poisson or a binomial distribution, one can calculate the corresponding moments in the model.), and both these

7 ost-macroparasite systems 7 distributions fit the the generic model t d h = [η, P ) m + r) M, P )], P.) t d h P = R, L) P [m + r + µ + α) + Q, P )], L t d L L = ˆλP R, L) νl)l, where M, P ) = EαX)), η, P ) = b) EγX)), and Q, P ) satisfies P [µ + α + Q, P )] = [EµX)) + EXαX))] see the second equations of.) and.)). Using the Poisson or negative binomial distributions, one obtains the forms for η, P ), M, P ) and Q, P ) explicitly, listed in Tables and below. M, P ) Q, P ) Model Ia α P α P Model IIa α P α P + P ) 3 + P ) Mz) αz αz + z) Qz) αz M, P ) Q, P ) Model IIIa α P µ + α) P Model IVa α P + P ) P µ + 3α + α P ) Mz) αz αz + z) Qz) µz µ + α)z Table. Functions M, M, Q and Q for a Poisson distribution.

8 8 MILNER, PATTON M, P ) Q, P ) Model Ib α P ) + P α α P Model IIb + + ) [ + P α 3 + ) ) P + ) ] P Mz) αz αz + Qz) α z + [ ) + 3 αz + ) ) z + ) ] z M, P ) Q, P ) Model IIIb α P µ + α) P + α P Model IVb [ + + ) ) [ + P µ + 3α + α ) ] P + ) ] P [ Mz) αz αz + Qz) [ α z + + ) ] [ µ µ + ) ) ] z ) α + α + ) ] z z Table. Functions M, M, Q and Q for a negative binomial distribution. η, P ) = be P γ ) b + γ)p for a Poisson distribution ) for a negative binomial distribution In model I the parasite-induced mortality of hosts is density independent, and so is the natural death rate of parasites. This means αk) = αk and µk) = µk.

9 ost-macroparasite systems 9 In model II the parasite-induced mortality of hosts is density dependent, and the natural death rate of parasites is density independent. This means αk) = αk and µk) = µk. In model III the parasite-induced mortality of hosts is density independent, and the natural death rate of parasites is density dependent. This means αk) = αk and µk) = µk. In model IV the parasite-induced mortality of hosts is density dependent, and so is the natural death rate of parasites. This means αk) = αk and µk) = µk. 3 Comparison between the spatially homogeneous model and the spatially structured model For our analysis we neglect host demography, which is appropriate when considering, for example, farm fish of the same age living in tanks with average life-span much larger than the time until harvesting. Thus, we set b) = m) =. 5 3 : number of hosts 6 8 days a) P: number of parasites L: number of larvae days Fig.. Dynamics of hosts a), parasites and larvae b) populations Poisson distribution). b) 5 : number of hosts P: number of parasites L: number of larvae days a) days Fig.. Dynamics of hosts a), parasites and larvae b) populations negative binomial distribution). b)

10 MILNER, PATTON For the data we used to run the simulations presented above, we chose the natural death rate of parasites as density independent µk) = µk), and the induced death rate of hosts as density dependent αk) = αk ) see models IIa and IIb). We also chose ˆλk) = ˆλk, and we assumed ˆλ > µ+α. The values for the parameters are the following: ρ =, ν = 5, C =, λ = 7, α =., µ =.3, γ =.6, =. We took here the following initial conditions: = 5, P = 5, L =. We can see that before going to extinction, parasites and larvae see their numbers increase very quickly, whereas hosts always decrease in number. owever, hosts converge faster than parasites and larvae to their equilibrium point. For these ODE models there is always extinction of the parasite population, and hosts tend to a limit, >, whatever the chosen distribution. For the spatially structured model we take the same diffusion coefficient for hosts and parasites, d =.8, and we choose a larger diffusion coefficient for larvae, d L =, because larvae are supposed to move faster than hosts. 5 osts 8 6 Parasites space a) time space b) 6 time 8 Larvae space c) 6 time 8 Fig. 3. Dynamics of hosts a), parasites b) and larvae c) when the parasites distribution is given by a Poisson distribution

11 ost-macroparasite systems The initial conditions for larvae were chosen to be constant in space: L x). We took the spatial domain to be x, and intialized hosts and parasites with the functions x) = 5 63 x x ) and P x) = 5 63 x x ), so the total numbers of hosts, parasites, and larvae initially are the same as for the ODE model since x x ) dx = ) osts 8 6 Parasites space a) time space b) 6 time 8 8 Larvae space c) 6 time 8 Fig.. Dynamics of hosts a), parasites b) and larvae c) when the parasites distribution is given by a negative binomial distribution The spatial structure does not seem to visibly change the system dynamics. With all coefficients being space-independent and the boundary conditions being homogeneous Neumann, the effect of the diffusion term is as one should expect to equidistribute the populations throughout the spatial domain. The host population decreases more rapidly and approaches a smaller value, possibly even zero. The parasite and larvae populations die out quickly, as was the case with the ODE model.

12 MILNER, PATTON Conclusions We introduced a general, continuous, spatially structured model for host-parasite systems as in [8]. In order to obtain a finite system of equations, we assumed that parasites are distributed in the host population according to a Poisson or a negative binomial distribution. The choice of distribution along with the choices we have for the factor reducing the fertility of host due to parasitism and for the mortality rate of parasites, 8), result in eight different possibilities for right hand sides of this generic model.) see the Tables and for the different forms the functions η, M, and Q can take). We then eliminated demography in the host population in order to analyze the model for a system where the natural birth and death rates of the host population were not important. We did not prove that the long term behavior of the solution of the continuous spatially structured model is space-independent, but the numerical simulations we performed seemed to indicate that that is the case see Figures 3 and ). Existence of a solution was demonstrated in [8]. Other issues such as uniqueness, positivity and asymptotic behavior of the solution of the spatially structured system will be discussed elsewhere. Some useful extensions of this work include a similar analysis as done for this model to a host-parasite system where demography in the host population is important or to a system where birth and death rates may not be spatially homogeneous that would probably lead to long term stable profiles that are not spatially homogeneous. 5 Bibliography. R.M Anderson, Mathematical models of host-helminth parasite interactions. Ecological stability. Usher M.B & Williamson M.. Eds., Chapman and all, London: ).. R.M Anderson, Dynamics aspects of parasite population ecology. Ecological aspects of Parasitology Kennedy C.R. Ed., North-olland Publishing Company, Amsterdam: ). 3. R.M Anderson, The regulation of host population growth by parasitic species, Parasitology, Vol 76: ).. R.M Anderson, Regulation and stability of host-parasite population interactions. I. Regulatory processes. Journal of Animal Ecology, Vol. 7: ). 5. R.M Anderson, Regulation and stability of host-parasite population interactions. II. Destabilizing processes. Journal of Animal Ecology, Vol 7: ). 6. R.M Anderson, The influence of parasitic infection on the dynamics of host population growth. Population dynamics, Anderson R.M., Turner

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