Antigen dose-dependent inhibition of Thl cell proliferation generates ..... density-dependent inhibitory effect is represented ... 2C and D the equilibria defined.
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Dynamic interaction between CD4 + T cells and parasitic helminths: mathematical models of heterogeneity in outcome A. N. SCHWEITZER and R. M. ANDERSON Wellcome Research Centre for Parasitic Infections, Department of Biology, Imperial College, London SW7 2BB (Received 20 January 1992; revised 23 April 1992; accepted 1 May 1992) SUMMARY Potential mechanisms of immunoregulation have been investigated for the capacity to generate heterogeneity in the outcome of infection with helminth parasites. We have developed a mathematical model of the interaction between T cell and parasite populations, based on the assumption that activation of a T h l C D 4 + T cell response is required for host resistance. Antigen dose-dependent inhibition of T h l cell proliferation generates heterogeneity in the outcome of host response to infection, with relatively low levels of exposure inducing resistance, and high levels of exposure associated with host susceptibility. Heterogeneity is additionally predicted in the duration of infection before individuals of the resistant class clear infection, with infection becoming more prolonged as the level of exposure rises. Similar categories of response are predicted if an alternative regulatory mechanism, that of interferon y-regulated control of Thl cell differentiation, is substituted into the model. However, the relationship between level of exposure and duration of infection is reversed. Results are discussed in the context of how these simple models of parasite-immune system interactions might be used to make predictions concerning specific examples of parasitic infection. Key words: helminths, T cell, regulation, dynamics, resistance, susceptibility.
INTRODUCTION
Frequency distributions of numbers of helminth parasites per host are typically highly aggregated or heterogeneous in form (Crofton, 1971; Anderson, 1978), where a few hosts harbour many parasites and many hosts harbour a few parasites. The factors that generate such patterns have been the subject of much discussion in the ecological and epidemiological literatures (e.g. Anderson & Gordon, 1982). These factors range from genetic variability in the control of host immune responses directed against parasite invasion (either specific or non-specific), to environmental or behavioural factors that determine host exposure to infection. A variety of laboratory studies have been conducted to demonstrate the influence of such factors under controlled experimental conditions (Wakelin, 1984; Keymer & Anderson, 1979). More recently, however, improvements in the techniques available to researchers to measure and quantify immune responses to parasitic helminths in both human and laboratory animal models, have made it possible to study the relationship between heterogeneity in parasite burden and heterogeneity in the immune responses directed against the infectious agent. This area of investigation is sometimes referred to as immuno-epidemiology in the context of the study of infections of humans in their natural environment. Pioneering work of this kind on helminths was conducted by Butterworth and
colleagues with respect to schistosome infections in Kenya, and more recent work on heterogeneity in immunological responsiveness to parasitic invasion has focused on intestinal nematodes (HaswellElkins, Elkins & Anderson, 1987; Bundy et al. 199U) and filarial parasites (Bradley et al. 1991). Sometimes such studies have provided both qualitative and quantitative data concerning the relationship between host immunological status and the level of infection. A good example is provided by filarial infections, where a proportion of individuals appear to be immune to infection, while many are chronically infected and may suffer prolonged immunosuppression (e.g. Ward et al. 1988; Maizels & Lawrence, 1991). When the quantitative detail of specific immune responses is explored, marked heterogeneity is often observed within the vertebrate immune system. This may relate to exposure to a particular antigen(s), to different amounts of antigen, or to host genetic background. One example of this is the presence or absence of delayed-type hypersensitivity responses, where presence is inversely associated with high antibody titres (Borel, Fauconnet & Miescher, 1965; Loewi, Holborow & Temple, 1966; Parish & Liew, 1972). Recently, the basis of this dichotomy following exposure to certain parasitic infections has become better understood due to the observation that mouse C D 4 + T cell clones can be divided into distinct subsets based on their growth requirements and cytokine secretion profiles (Mosmann
Parasitology (1992), 105, 513-522 Copyright © 1992 Cambridge University Press
A. N. Schweitzer and R. M. Anderson et al. 1986; Cherwinski et al. 1987). Delayed hypersensitivity-mediating T cells, specifically producing IL-2 (autocrine growth factor), interferon y (IFNy) and lymphotoxin/TNF, have been designated Thl cells, while cells producing IL-4 (autocrine growth factor also responsible for high levels of IgE production), IL-5 (inducing eosinophilia) and the more recently characterized IL-10 (Fiorentino, Bond & Mosmann, 1989; Moore et al. 1990) have been designated Th2 cells. Clones showing secretory profiles intermediate between these two extremes have also been isolated and are believed to represent short-term stimulated or transiently expressed phenotypes (Street et al. 1990; Mosmann & Moore, 1991 ; Gajewski et al. 1989) in contrast to the mature, long-term stimulated Thl or Th2 phenotype. Definitive evidence in man of a dichotomy in CD4+ T cell subset activation comparable to that observed in mice has recently been reported (Romagnani, 1991). Infection of mice with parasites such as Leishmania major (Locksley & Scott, 1991) and Schistosoma mansoni (Scott et al. 1989; Sher et al. 1990; Finkelman et al. 1991) clearly demonstrates an association of Thl-type responses with resistance and the failure to maintain a Thl-type response with prolonged or progressive susceptibility to infection. Immunity to other infections, in particular with intestinal nematodes, seems to be associated with Th2 rather than Thl responses (Finkelman et al. 1991; Else & Grensis, 1991; Carman et al. 1992). Regardless of effector subset, it therefore follows that heterogeneity in host responses to infection may be associated with the processes governing the bias in CD4 + T cell subset stimulation. We focus in the present investigation upon those examples of infection where immunity is Thl-associated, and where effective production of Thl cytokines tends to correlate with resistance in human populations. Defective proliferative responses, often associated with high IgE titre and eosinophilia, are characteristic of chronic infection, particularly with helminths (Sher & Colley, 1989). While a number of potential cross-regulatory activities have been characterized between the two subsets (Gajewski & Fitch, 1988; Fernandez-Botran et al. 1988; Peleman et al. 1989; Fiorentino et al. 1989), as yet little is understood about the underlying determinant(s) of response bias in vivo. In this paper we build on earlier studies (Schweitzer & Anderson, 1992; Schweitzer, Swinton & Anderson, 1992) to examine the dynamic interaction between Thl cells and helminth parasitic infection via the formulation and analysis of simple mathematical models of the principal variables involved. In particular, we have previously shown that specific assumptions concerning the regulatory process controlling T cell activation and the immune responses targetted at parasitic organisms can generate a variety of possible
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outcomes, such as host resistance or persistent susceptibility, depending on initial conditions and parameter value choice (Schweitzer & Anderson, 1991). Here, we examine in more detail the particular responses that are thought to be important in certain helminth infections of mammalian hosts. Biological background to model formulation (a) The parasite. Helminths are complex multicellular organisms which often develop through many antigenically distinct life-cycle stages within the vertebrate host. Of particular significance with respect to immunological defences in human hosts are the infective stages, which invade the host and ultimately develop into the reproductively mature adult worm, and the offspring produced by adult worms (for example, eggs or larvae), which often play a role in the induction of pathology and disease. In order to develop a simple mathematical model, we limit this investigation to a single life-cycle stage of the parasite. The mathematical description of parasite dynamics could equally well represent repeated exposure of individuals to infective-stage larvae, or the continuous production of secretory or excretory antigens, eggs or larvae by a relatively inert adult worm population. A realistic model of the latter would need to incorporate the added dimension of adult worm dynamics, which would in turn be influenced by the rate of exposure to infective stages and the extent of anti-infective stage immunity that had developed as a consequence of past infection. In this paper, however, we focus on the simplest of representations of the adult parasite population, or its offspring or secreted antigens, by describing the dynamics of the antigen as an immigration — death process (i.e. representing a constant rate of recruitment and a net rate of loss proportional to antigen concentration). (b) The immune system. The mechanism(s) underlying the differential control of T helper cell subset expression is unclear. A positive influence of IFNy and IL-4 on the differentiation of T h l and Th2-type cells respectively has been demonstrated (Gajewski et al. 1989; S. L. Swain, cited by Mosmann & Moore, 1991) but this effect is difficult to distinguish from the cross-regulatory properties of cytokines produced by the respective subsets, which may influence the selective outgrowth of one or other population. In particular, IFNy inhibits the proliferation of Th2-type but not Thl-type T cells (Gajewski & Fitch, 1988; Fernandez-Botran et al. 1988) although not affecting Th2-type cytokine secretion, while IL-10 decreases cytokine synthesis (in particular, IFNy) by Thl-type cells (Fiorentino et al. 1989). Although there is no evidence as yet that cytokine administration alone, at the time of challenge with live organisms, can significantly alter the
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Helminths and CD4+ T cells eventual outcome of infection (Mosmann & Moore, 1991; Locksley & Scott, 1991), differentiation of Thl cells does appear to be selectively promoted following immunization with IFNy or TNFa in the presence of non-living antigen (Scott, 1991; Liew et al. 1991). A potentially important difference between the two subtypes is the increased sensitivity of IL-2 production (by Thl cells) over IL-4 production (by Th2 cells) to factors affecting T cell receptor-linked intracellular signalling mechanisms (Mueller, Jenkins & Schwartz, 1989; Gajewski, Schell & Fitch, 1990; Munoz et al. 1990). Interestingly, a number of investigators have observed a ligand densitydependent inhibition of proliferative (Thl-type) T cell responses at supra-optimal antigen concentration (Matis et al. 1982, 1983; Lamb et al. 1983; Suzuki et al. 1988) suggesting a possible manifestation of signalling differences between the two subsets which may be relevant in the context of mathematical model formulation. A variety of additional suggestions have been put forward to explain the preferential induction of one or other subset (Swain et al. 1988; Powers, Abbas & Miller, 1988; Chang et al. 1990). We limit our investigation to a comparison of the implications of two particular areas of interest: the ligand densitydependent inhibition of Thl-type IL-2-dependent T cell proliferation, and cytokine environmentregulated differentiation. Underlying both hypothesized mechanisms are the assumptions that T h 2 type responses are not necessary to regulate Thl cell growth and that such responses are expressed only when IFNy production is sub-inhibitory. The flow chart shown in Fig. 1 outlines the assumptions about the parasite and the T cell populations upon which the mathematical model, described below, is based. The mathematical model We consider the simple situation of two interacting populations: an activated T cell population (X) under autocrine control, and a continuously produced parasite stimulus (A) susceptible to immunological killing, or to clearance in the case of secretory antigens. We nevertheless take into account the density of two additional variables, the precursor T cell population (M), and cytokine (I) (= IL-2) produced by, and driving the proliferation of, activated cells. T cell precursors are produced at a constant rate A per unit time, have a mean natural life-span of \/u and contact antigen at a constant per capita rate g per antigen concentration per unit time. Proliferation occurs maximally at a rate r. However, we assume that two factors act to limit the magnitude of this rate. In the absence of antigen, autocrine proliferation of T cells saturates when high levels of IL-2 are present (e.g. Toribio et al. 1989). This density-dependent inhibitory effect is represented
by a saturation function with parameter d. Antigenspecific activation is frequently associated with a decrease in the proliferative response at supraoptimal doses of antigen (Matis et al. 1982, 1983; Lamb et al. 1983; Suzuki et al. 1988). This effect is represented by a further saturation function in association with the parameter p. The net rate of T cell proliferation is therefore assumed to adopt the form rXI/[(l + dX\){\ +pXA)]. The mean activated life-span of a T cell is assumed finite, and equal to \/w. Cytokine is produced at a net rate a per activated cell, and removed or inactivated at a net rate b. Thus, — = A-mM-gMA, dt dX
It
= gMA + -
rXI
(1) -wX,
^ = aX-bl. dt
(2) (3)
This set of equations describing T cell dynamics has been investigated elsewhere in terms of activation by non-replicating antigen (Schweitzer & Anderson, 1992) and a replicating parasitic organism such as Leishmania spp. (Schweitzer et al. 1992). In this paper we examine model behaviour in the context of a helminth parasite population that grows in size by an immigration—death process. The parasite population, or population of excreted or secreted antigen, is assumed to grow at a constant rate h, to have a mean natural life-span I / a (death rate a per unit time) and to be killed or cleared at a rate s per T cell per parasite (or unit concentration of antigen), where: dA
=
h-aA-sAX.
(4)
RESULTS
(1) Multiple steady states or outcomes To facilitate understanding of the dynamics of immunoregulation in the context of infection we refer to a phase plane of parasite abundance (A) versus Thl cell abundance (T). In order to reduce the dimensionality of the interaction we assume that both precursor cell production and cytokine secretion occur on much faster time-scales than the parasite—T cell interaction. Assuming that M and I rapidly attain equilibrium values M* = A/(tt+#P) and I* = aX/b, obtained by setting dM/dt = 0 and dl/dt = 0 in (1) and (3) respectively, the system of equations may be reduced in size to: dX gAA r'X2 -wX, dt u+gA (\+d'Xl)(l+pXA) dA -— = h-aA-sAX, dt where r' = ra/b and d! = da/b.
(5) (6)
A. N. Schweitzer and R. M. Anderson
516
Parasite
IL4;
Fig. 1. Flow chart showing the assumptions upon which the mathematical model is based. ( ) Populations and positive interactions incorporated into the basic model; ( ) negative interactions; ( ) modification (incorporating IFNy-mediated differentiation) subsequently considered; (—) way in which the model fits into the broader context of current understanding of CD4+ T cell subset activation and cross-regulation. Phase plane analysis (Fig. 2) for particular parameter values reveals three possible classes of outcome depending on the level of exposure to infection, namely: (a) under-exposed non-immune, (b) immune and (c) ' immunosuppressed' nonimmune. In Fig. 2 A, exposure is insufficiently high to stimulate an immune response of sufficient magnitude to clear the parasite burden, and a low but steady state of infection is retained (labelled U). In Fig. 2B the host will progress to stable immunity with only a negligible parasite burden persisting (labelled R). In Fig. 2C and D the equilibria defined on the phase plane suggest two possible outcomes: stable immunity (labelled R) and stable tolerance of the parasite burden (labelled S). In reality, only the tolerant state will be attained following exposure of a naive individual (A(0) = 0 and X(0) = 0) to high levels of infection. This is due to a boundary, or stable manifold, passing through the unstable equilibrium separating the two end states. In Fig. 2C this boundary intersects the T cell axis and initial conditions in the unimmunized, uninfected host will always lie on the 'susceptible' side. In Fig. 2D the parameter values assigned to the parasite population are dramatically reduced relative to those used in Fig. 2C, and the boundary intersects the parasite axis. Under these conditions, the 'resistant' state only is accessible to the naive host under steady exposure conditions.
the immune class (Fig. 2B), namely in the duration of infection prior to the development of immunity (Fig. 3 A). The curve in Fig. 4 (solid line) suggests that the duration of infection increases as the level of exposure approaches the ' immunosuppressed' threshold (i.e. moving from the situation in Fig. 2B to that in Fig. 2C). In fact, the duration of infection is also prolonged at levels of exposure just above the 'underexposed' threshold. Indeed, in Fig. 3D, where ligand density-dependent inhibition is ignored (/> = 0), a slight delay in parasite clearance is nevertheless evident at very low levels of exposure to infection. The observed delay occurs when the parasite and T cell variables pass through a region of the phase space where the two null isoclines dX./dt = 0 and dA/dt = 0 are in close proximity but are not intersecting. The growth or decline of both populations will concurrently approach zero, but neither will actually cease growth or decay altogether and hence the interacting system will eventually move out of this region of phase space to approach a defined, stable equilibrium state, in this case representing immunity. (3) The effect of cytokine-regulated differentiation-adaptation of the model
(2) Progression of infection over time
As mentioned earlier, another area of interest in understanding the determination of T h l versus Th2-type responses is the effect of the cytokine environment on the subsequent phenotypic differentiation of cells. An example of this is the promotion of Thl-type T cell differentiation in the presence of IFNy (Gajewski et al. 1989), and of Th2-type differentiation in the absence of IFNy and/or the presence of IL-4 (Gajewski et al. 1989; S. L. Swain, cited by Mosmann & Moore, 1991). The frequent cloning of T cells showing cytokine secretion profiles intermediate between the two subsets, particularly under non-immunized conditions (Street et al. 1990), supports the hypothesis that these cells represent short-term stimulated . phenotypes in the differentiation pathway of T cells. According to the hypothesized lineage, precursor stimulation will introduce IFNy into the environment early on, regardless of the secretory profiles of the cells produced eventually. We modified our model to incorporate the assumption that following contact with antigen, precursor cells produce a range of cytokines, including IFNy, before becoming committed to one or other phenotype. The latter phenotype would be influenced by the amount of IFNy produced at this early stage, and the effector function of this early IFNy rise would also contribute to parasite control.
Numerical simulations of the course of infection in an individual over time demonstrate that an additional level of heterogeneity can be observed within
We assume that the proportion of antigencontacted cells entering the Thl-type differentiation pathway increases from 0 to 1 as the amount of IFNy
517
Helminths and CD4+ T cells
50
dA/dt = 0 \ /
40
\ \
30
dX/dt = 0
20
\
10
dX/dt = 1 ^
0 -1
o
-1
T cells (log scale - arbitrary units) Fig. 2. Phase plane showing the possible outcomes (stable equilibria) following different levels of host exposure to parasite antigens. ( ) Null isoclines (dX/dt = 0 and dA/dt = 0 as indicated - note that the position of dX/dt = 0 is common to all four situations). In (A) the level of exposure is very low (h = 0 5 ; a = 0-2 and s = CM throughout) and an unstable equilibrium produces a boundary, or stable manifold ( , also in (C) and (D)), between the two stable equilibria which represent underexposed (U) and resistant (R) individuals. Note that if the very first exposure with parasite is high, but further exposure is very low as shown, the interaction between T cells and parasites may lie on the ' resistant' side of the boundary, where immunity/resistance would result. In (B) exposure is relatively low (h = 10) and the 'susceptible' equilibrium state is not defined, therefore individuals are always resistant (R). In (C), exposure is high (h = 40) and two stable equilibria representing resistance (R) and susceptibility (S) are defined, separated again by an unstable saddle point equilibrium through which a boundary (or stable manifold) passes. T h e naive state (no parasites or T cells) lies within the 'susceptible' sector of the phase plane and susceptibility will always result. In Fig. 2 D the numerical coordinates of the equilibria are identical to those in Fig. 2 C but the dynamics of the parasite population are reduced overall, such that h = 0 4 , a. = 00002, s = 00001. T h e boundary passing through the saddle point now intersects the parasite axis, the naive state lies within the 'resistant' sector of the phase plane, and immunity will always result. This figure holds for one combination of host parameters {g = 0001, A = 1, u = 0 1 , r = 0 1 , d = 00001, w = 0 1 , p = 005). Genetic differences within a heterogeneous host population would alter the relative parameter values such that a single level of exposure might produce comparably heterogeneous outcomes between individuals adding a further dimension to the level of heterogeneity within a population (see Anderson, 1991).
(F) increases, according to the arbitrary saturation function / (with parameter q)
f=
(7)
1+gF'
where the net production of IFNy per unit time is determined by dt
(8)
u+gA
The parameter q determines the relative effect on differentiation for each unit of IFNy, the latter being produced at a net rate of nl and n2 per newly stimulated precursor and activated cell respectively. Equation 2 is consequently modified to r'X2
dX lit
u+gA'J
(\+d'X2)(\+pXA)
-wX.
(9)
Note that if the newly stimulated cells are not assumed to produce IFNy, then a Thl-type T cell response would never be activated regardless of the 35
magnitude of the variable A, since when X = 0, dX/dt = 0. In order to avoid the complication of IFNy produced by an initial bias of Thl-type cells, we assume production by newly activated cells only («, = 0). Fig. 3B shows the predicted time-course for IFNy-regulated differentiation of Thl-type T cells in the absence of any significant density-dependent inhibition (p = 0). This simplified situation demonstrates that when q is small, as the level of exposure moves above the threshold for induction of an immune response, cytokine regulated differentiation prolongs the duration of infection over a very much wider range of supra-threshold exposure levels than occurs in the absence of controlled differentiation, or when parameter q associated with differentiation is large, and the function/ approaches unity (Fig. 3 D). This reflects the enhanced degree of nonlinearity resulting from the inclusion of cytokineregulated differentiation. When density-dependent inhibition is also assumed, the duration of infection for a given level of exposure approaching the PAR 105
A. N. Schweitzer and R. M. Anderson
518 Immunosuppressed
0
20 40 60 80 100 120140 160 180 200 20 40 60 . 80 Exposure (egg/larva production) (arbitrary units)
100
Fig. 4. Relationship between the duration of infection (time in arbitrary units until T cell abundance reaches 100) and the level of exposure (h). ( ) Relationship observed for numerical solution of the basic model; ( ) solutions when cytokine-regulated differentiation is assumed (q = 0-1). Other parameter values are as for Fig. 2.
-O
0
20 40 60 80 100 120 140 160 180200
' immunosuppressed' threshold is similarly prolonged (Fig. 3 C). A comparison of the relationship between duration of infection and level of exposure with, and without, cytokine-regulated differentiation is shown for the full model (p == | 0) in Fig. 4. The net effect is to increase the length of time for which an infected individual remains infected over that pertaining in the absence of this complication.
0
0
20 40 60 80 100 120 140 160 180 200
20 40 60 80 100 120 140 160 180 200 Time (arbitrary units)
Fig. 3. Numerical solution over time for the progression of infection for different levels of host exposure. In each example, the parameter h is varied through 0-5; 10; 15; 20; 30; 40 (low to high as indicated in each graph). The assumptions upon which the Thl cell population is modelled are varied: (A) the basic model; (B) IFNy regulated differentiation with p = 0, q = 0-1; (C) IFNy regulated differentiation with p = 0-05, q = 0-1; (D) basic model (no regulation of differentiation) with p = 0. Other parameter values as for Fig. 2.
DISCUSSION
At present, the mechanisms governing the differential activation of C D 4 + T cell subsets, particularly during infection, represent an area of key interest to immunologists. We have begun to approach this problem by investigating the population biology of the immune system and its interaction with invading parasitic organisms (Schweitzer & Anderson, 1991, 1992; Schweitzer et al. 1992). In this paper we have considered the interaction between helminth parasites and C D 4 + T cells with the aim of determining the potential role of some of the experimentally characterized regulatory features of T cells and cytokines in generating observed patterns of infection. The antigenic complexity of the organism with which we are dealing means that our simple model which mirrors responses directed at one life-cycle stage only should be viewed as a starting point for the addition of complexity. However, the qualitative patterns generated by the model suggest that simple assumptions can generate complicated patterns of dynamical behaviour. When interpreting observed trends it is clearly important to be aware of this property.
Helminths and CD4+ T cells Dose-response curves generated in vitro suggest that ligand density-dependent ; nhibition of IL-2 driven C D 4 + T cell proliferation is an intrinsic property of Thl-type T cells (Matis et al. 1982, 1983; Lamb et al. 1983; Suzuki et al. 1988). The basic model of T cell dynamics, which incorporates this assumptions generates patterns of infection comparable to those observed, when considered in association with a helminth-type stimulus. Two possible outcomes are predicted following adequate exposure to the parasite, namely resistance reflected by control of the parasite population following relatively low levels of exposure, and tolerance of the parasite due to a failure to raise an effective immune response following relatively high levels of exposure. This is consistent with circumstantial evidence that many individuals with persisting patent infection in endemic areas have continuously been more heavily exposed (Ward et al. 1988). There is also a growing body of epidemiological evidence that points to predisposition to heavy infection with helminth parasites, due perhaps to behavioural and environmental factors (Schad & Anderson, 1985; HaswellElkins et al. 1987). In addition, controlled infection of laboratory animals demonstrates high dose suppression of parasite clearance by otherwise resistant mice (Wassom et al. 1984). The mathematical structure of the model is nevertheless compatible with the alternative suggestion that inhibition is specifically caused by factors produced by the parasite (Pritchard, Ali & Behnke, 1984; Pearce et al. 1991). Simple models have recently been published that mimic this assumption (Anderson, 1991) and these generate similar broad patterns of behaviour (i.e. resistant and susceptible states) to those discussed in the present paper. Evidence has also been presented of an 'underexposed ' population in the field (Kurniawan et al. 1990) as predicted by the model. An earlier model developed to describe T cell activation by helminth parasites generated a similar dichotomy of resistant and susceptible individuals (Schweitzer & Anderson, 1991) but failed to predict an 'underexposed' group. The principle divergence between the assumptions of the earlier and the present model relates to the manner in which T cell proliferation occurs. In the earlier model, T cells simply proliferate following antigen-driven activation, whereas in the present case the activated T cell is stimulated to produce cytokine which secondarily drives proliferation. In addition, saturation terms are included within the function describing the net rate of proliferation. Numerical solutions of the set of equations demonstrates considerable heterogeneity not only in outcome, but in the duration of infection before a relatively sudden development of immunity within the resistant group. This suggests that within a population of hosts, some of those classified as belonging to the 'tolerant' class at one point in time
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might naturally progress to resistance at a later timepoint. Indeed, field studies suggest that this is the case (Bundy, Grenfell & Rajagopalan, 1991a), highlighting the importance of longitudinal as well as cross-sectional field studies. Controlled experimental infection of cats with Brugia pahangi shows clearly that while some cats remain indefinitely microfilaraemic, others suddenly clear microfilaraemia after an apparently unpredictable period (Fletcher et al. 1986). Adaptation of the model to take into account a possible role of IFNy in determining T h l cell differentiation broadens the range of exposure across which an inverse relationship between intensity of exposure and duration of infection is predicted, provided that parameter q is relatively small (in other words, only a proportion of antigen-contacted cells will differentiate at the level of I F N y produced during relatively high levels of exposure). On the other hand, even if q were small, contributions to the local level of IFNy at the time of precursor differentiation by extrinsic sources would lead to parasites being expelled more rapidly than predicted here. This might arise through direct, sustained administration of IFNy, or through priming of additional IFNy producing cell subsets such as C D 8 + cells and natural killer cells, which have not been incorporated into the present model. Differentiation as a potential point of regulation is pertinent to an envisaged cross-regulatory role for Th2-type cytokines, in particular, IL-10 mediated inhibition of IFNy production (Fiorentino et al. 1989). In the model, the effect of this Th2-type activity would be to reduce the magnitude of the parameter q, effectively prolonging the delay imposed by IFNy-regulated differentiation. Under these circumstances, the relationship between level of exposure and duration of infection in the field would depend critically on the range over which the level of exposure varied. This could be an important consideration as regards therapeutic intervention. The basic antigen density-dependent model predicts that decreasing the level of exposure could potentially promote the development of immunity and the further clearance of parasites. If a significant influence on differentiation is assumed, increasing the level of exposure might on the other hand promote the development of immunity, depending on the average level of exposure to infection in a defined community. For infections such as schistosomiasis, there is evidence that exposure to infective stages does, in fact, decrease with age in endemic areas, with immunity developing only slowly, and only in some individuals (Butterworth & Hagan, 1987). However, evidence to date has associated the development of this immunity to reinfection with IgE responses (Hagan et al. 1990) which are T h 2 mediated, emphasizing the importance of determining the type of response induced, and the stage upon
A. N. Schweitzer and R. M. Anderson which it exerts its effects, when applying these models to specific examples of infection. The relationship between exposure level and duration of infection clearly has important implications for the net transmission intensity in a defined area of endemic infection. Future work will blend these models of the course of infection in individual hosts into a framework describing transmission dynamics within a population of hosts. The model described in this paper must be interpreted with caution in view of the generalizations and simplifications upon which it is based. Nevertheless, the structure of the model provides a framework on which to build, and within which to consider more specific examples of interactions between parasites and the immune system. We have also highlighted here the value of comparing the predictions of models based upon different biological assumptions concerning principal immunoregulatory mechanisms, and of considering heterogeneity in the time-course of infection as well as heterogeneity in eventual outcome. A.N.S. thanks the Medical Research Council for Fellowship support and R.M.A. thanks the Wellcome Trust for research grant support. Discussions with Bryan Grenfell, Don Bundy and Rick Maizels have been greatly appreciated.
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