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.the disease has direct transmission, and that bothKeeping in mind the potential need to incorp- healthy and infected hosts can move between hab-orate these complex spatial processes when con- itats at constant per capita rates (though possibly atsidering some specific host parasite systems, simple different rates in the two habitats).A model for thismodelling approaches can nevertheless capture the scenario which describes the initial stage of infectionmain properties of the dynamics of host parasite is given by duplicating the above SI model, with twointeractions in a spatial context.pools of infected individuals (recall, for the momentwe are assuming that the host population is initiallyall healthy hosts, with fixed densities).The modeldescribing the initial stages of infection is5.4 Host parasite interactions incoupled, heterogeneous patchesdI11S1I1 (d12 1)I1 m12I1 m21I2(A2/A1),One straightforward way to incorporate space intodthost pathogen systems is to imagine that the envir-dI22S2I2 (d22 2)I2 m21I2 m12I1(A1/A2).onment is comprised a number of distinct habitats.dtIn each of these, there is a well-mixed population ofhosts and pathogens, described by standard epi- All the parameters in the above earlier SI modeldemiological models.The habitats are then coupled have now been made habitat-specific.In addition, THE SPATI AL DI MENSI ON 75we have assumed that infected individuals can growth rate reflects a nonlinear averaging of themove between habitats, at rates that are also growth rates of the two habitats.potentially habitat-specific.(Healthy hosts may 2 If we let both movement rates be equal, and thenalso be moving, but if so, they are assumed to do so take the limit as they get very large in the abovein a manner that does not alter the pattern of abun- expression, the growth rate simply becomes thedances between habitats).Because the variables are arithmetic average over the two habitats, (r1 r2)/2.cast in terms of density, the fluxes between habitats In this limit, the landscape is actually just one hab-have to be adjusted to account for the fact that the itat with internal heterogeneity.Given rapid move-number of individuals per unit time moving from ment of infected individuals, landscapes with thehabitat i to habitat j equals the product of the den- same average growth rate (averaged among hab-sity in habitat i, the area of habitat i, and the migra- itats) but different degrees of internal heterogeneity,tion rate; whereas, the impact this influx of nonetheless should have the same growth rate forindividuals has on density in habitat j has to be the infection.scaled against the area of habitat j.3 If both intrinsic growth rates are positive, so is theThe above model is a pair of coupled linear differ- overall growth rate; conversely, if both growth ratesential equations, so it can be fully analysed.In par- are negative, the overall growth is also negative.ticular, the dominant eigenvalue of the characteristic Parasite invasion requires that there be at least onematrix defines the growth rate of the infection over habitat in the landscape which is intrinsicallyboth habitats, after an initial transient phase.For favourable, and could potentially sustain an invasionsimplicity, we combine infection, death, and recov- on its own (were infected individuals not to move).ery into a habitat-specific intrinsic growth rate for 4 If we let m21 approach zero, while keepingthe infection, when rare: the other movement rate fixed, all movementswill be from habitat 1 into habitat 2.In this case,ri iSi (di2 i).the growth rate overall converges on the growth rateA habitat may foster a high initial growth rate for theof the better habitat (which we have assumed to beparasite simply because there is a high density ofhabitat 1) minus losses to emigration.Thus, an infec-hosts there, or instead because of individual impactstious disease may grow in habitats where it has anof the infection upon hosts (e.g.locally low deathintrinsic growth rate less than zero, provided it isrates of infected individuals).We assume that habitatmaintained in habitats where it has a positive growth1 has the higher growth rate.With this notation, therate.All else being equal, such spillover modes ofgrowth rate of the infection can be shown to beinvasion by an infection should be most noticeable in1[ m m21 r1 r2habitats with lower than average host densities (this122result emerges from inspecting the eigenvector(m12 m21 r1 r2)2 4( m21r1 m12r2 r1r2)].describing the distribution of individuals betweenthe two habitats, when the invasion has settled into(see Holt 1985 for an analogous treatment of popu-its equilibrial rate of change).lation increase in a two-habitat environment, albeitwith symmetrical movement).The above bit of theory provides insight into howThis expression can be manipulated to makespatial heterogeneity can influence parasite estab-some general statements about how habitat hetero-lishment.We would be the first to admit that thisgeneity influences the establishment of a disease.provides just a first pass through this problem.1 Note that the relative habitat areas drop out.The A full analysis of this issue would require one toslightly counterintuitive result is that a combina- analyse more complex landscapes, alternativetion of intrinsic habitat qualities influences invasion transmission dynamics, additional classes (e.g.rates, but relative habitat areas do not.The reason hosts with acquired immunity) and so on.for this is basically that with reciprocal movement, Moreover, we have not paid attention to feedbacksthe descendents of any given individual cycle via depression of healthy host numbers, or tothrough both habitats.Overall, the asymptotic transient dynamics. 76 PARASI TI SM AND ECOSYSTEMSSome insights into the consequences of habitat In a heterogeneous landscape, flows among habi-heterogeneity for disease dynamics can be also tats can permit greater parasite loads to be main-gleaned from parallel studies of the classical tained in some habitats than would be expected justLotka Volterra predator prey model, in which prey from local dynamics [ Pobierz całość w formacie PDF ] |
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