ABSTRACT

1. Dynamic regulation of single- and mixed-species malaria infections: specific and non-specific mechanisms of control David E. Gurarie1, Peter A. Zimmerman2, and Charles H. King2 1Math. Department; 2Center for Global Health and Diseases, Case University, Cleveland, OH Fig.5: Randomly varying SS-clearing rate. Random drops represent either antigenic variation of the surface protein of Pf-species, or random inoculation with a heterologous strain, whereby cs ‘relaxes’ to its low to high value, via ‘affinity maturation’ (development of specific antibodies). Adaptive immunity brings about new dynamic features – transition from stable equilibrium to limit cycles (recrudescent parasitemia), or more complicated chaotic dynamics (Hopf bifurcation). We demonstrate it for a single spp model, by looking at the effect of time-lag L (Fig.6), and for mixed spp by varying SS-efficiency (Fig.7) m m J K a2 a1 F V Fig.6: Time series (parasitemia - solid black, SS-effector - solid gray, NS-effector - dashed) show transition from stable equilibrium to limit cycle (Hopf) past L=12. cn I sn cn F V V V a1<a a2<a< a1 << a<a2 y2 III y1 y1 y1 y2 m III y2 III I IV II II IV II IV x1 x2 F x1 x2 F x2 x1 F ss cs ss cs ABSTRACT DYNAMIC PATTERNS FOR ADAPTIVE SS-IMMUNITY ANALYSIS OF EQUILIBRIA • Intra-host immune regulation of parasite species involves a complex web of different cell types, cytokines and pathways, and exhibits complex dynamics (Ref. 1-2). Here we develop simplified mathematical models of single and mixed-species infections based on the immune regulatory mechanisms (stimulation - clearing). Our approach exploit differential-delay equations and dynamical systems (cf. Ref. 3-4). It explains species interaction and its outcomes in terms of ‘predation/ competition’. In many cases it allows a detailed analysis of the long-term behavior (equilibria, limit cycles, stability, bifurcations). Our analysis has several applications: • it explains ‘sequential patterns’ of parasitemia observed in many clinical and field studies • It exhibits possible clinical outcomes (species ‘coexistence’ or ‘domination’) • The latter has potential implications for chemotherapy control (drug resistance), and community transmission. Our models highlight the role of the adoptive species-specific immunity as the primary regulating mechanism of parasite growth. • DE system (1) is a version of ‘2-prey + 3-predator’ model. It has 7 essential parameters: • NS and SS efficiencies (“clearing”x”stimulation”/”decay”): eN, eF,eV • relative growth rate: a = a2/a1; • cross-reactivities: 0<x,h<1; (iv) time-lag L. • System (1) will typically equilibrate in a damped-oscillating pattern (Fig. 6-7), and its equilibrium (endemic) states are approximately described by the classical Volterra-Lotka competition We incorporate 2 elements of adaptive SS-immunity in our system (1): (i) time-lag L (several days) in stimulating SS-effectors; (ii) variable SS-clearing rate cs(t) (Fig. 5). Fig. 3: Phase-plane portraits of reduced system (1), stable equilibrium marked in black. Depending on parameters we get either species domination (F right, or V - left), or coexistence (middle). Remark: Changing relative growth a in Fig. 3 is relevant to analysis of drug treatment. Indeed, having ‘susceptible’ and ‘resistant’ strains, their growth rates are modified by persistent drug, that creates an effective ‘selection /domination’. Next plot (Fig.4) shows regions of coexistence and Fast-domination in in the full (5D) model for a range of cross-reactivities: 0<x<.8, and NS/SS efficiencies: 0< en;es <2.5. Conclusion: A hypothetical host population distributed over parameter region (0< en,s <2.5) has predominant coexistence equilibrium for low x (different species) , but it changes to a ‘dominant fast species’ for strongly cross-reactive (homologous) strains, high x. MODELING PRINCIPLES • We stress basic patterns and processes rather than detailed structure and components on the ‘immune-parasite’ system. Those are • Parasite replication • Stimulation of immune effectors (by parasite Ag) • Parasite clearing by immune effectors • Mathematical models can be either • Deterministic (continuous -‘Differential Equations’ or • discrete -‘finite-difference’), or stochastic, or combine both. Fig.7: Two-spp model (1) undergoes 2 bifurcation as SS-efficiency changes from a low value .2 to high 11.2 ‘Parasite + immunity’ regulatory circuit • Two species: F,V • growth (solid arrows) at rates a1,2 • Clearing (dashed arrows) by (i) fever F; (ii) non-specific (NS) immune effector I; (iii) specific (SS) immune effectors J,K • Immune effectors: I,J,K degrade (deactivate at certain rates m) • Species F,V stimulate production of immune effectors and trigger ‘fever switch’ Fig. 4 Fig.8: Randomly varying SS-clearing creates ‘chaotic’ sequential patterns, and could bring about the demise of a ‘slow spp’ (dashed). Upper and lower panels show ‘linear’ and ‘log’ plots of the process, left and right columns compare ‘fixed SS’ vs. ‘random SS’ (Fig.5). SUMMARY AND CONCLUSIONS Fig.1: Two-species ‘parasite-effector’ circuit • Intra-host immune regulation of Plasmodium species (single or mixed) is modeled by systems of differential, or differential-delay equations, that resemble some familiar patterns of population Biology (‘predation competition' ). • We combine analytic and numeric tools to explore possible outcomes: ‘coexistence’, ‘domination’, ‘cycles’, sequential patterns, in terms of the basic parameters: NS and SS efficiencies, cross-reactivities, fitness (relative grwoth rates), time lags (in SS induction). • Further applications (in progress) include: (i) Individual-based models of community transmission; (ii) Development of drug resistance; • (iii) Prediction of disease outcome and control strategies (treatment/vaccination) on individual and community level • References: • Boyd, M.F., Kitchen, S.F. (1937) Am J Trop Med 17, 855-861; 18, 505-514. • Bruce, M.C., Day, K.P. (2002), Curr Opin Microbiol 5, 431-7; 2003, Trends Parasitol. 19,271-7. • Anderson, R.M., May, R.M., Gupta, S., (1989), Parasitology, 99 Suppl, S59-79. • Mason, D.P., McKenzie, F.E., (1999), Am J Trop Med Hyg 61, 367-74; J Theor Biol 198, 549-66. • Gurarie,D., P.A. Zimmerman, C.H. King, (2005), J Theor Biol (to appear) 5 variables: F,V-spp,I,J,K- effectors, obey a coupled differential (Ref.4), or differential-delay system (Ref.5) (1) Fig.2 : Fever controller F(F+V), turns on and off past ‘threshold’ levels • We studied potential impact of area-wide control strategies on spatially-distributed endemic schistosome infection, using a distributed Macdonald-type model. An optimal age-targeted strategy was determined for a range of values r. Focal control fails to achieve the desired impact, due to linked environment. • Future work (in progress) will extend the above models to include immunity, chronic morbidity, and seasonally varying snail ecology. • Macdonald, G. (1965). Trans R Soc Trop Med Hyg59, 489-506. • D. Gurarie, C. King, Parasitology, 2005 (to appear) • We studied potential impact of area-wide control strategies on spatially-distributed endemic schistosome infection, using a distributed Macdonald-type model. An optimal age-targeted strategy was determined for a range of values r. Focal control fails to achieve the desired impact, due to linked environment. • Future work (in progress) will extend the above models to include immunity, chronic morbidity, and seasonally varying snail ecology. • Macdonald, G. (1965). Trans R Soc Trop Med Hyg59, 489-506. • D. Gurarie, C. King, Parasitology, 2005 (to appear) • We studied potential impact of area-wide control strategies on spatially-distributed endemic schistosome infection, using a distributed Macdonald-type model. An optimal age-targeted strategy was determined for a range of values r. Focal control fails to achieve the desired impact, due to linked environment. • Future work (in progress) will extend the above models to include immunity, chronic morbidity, and seasonally varying snail ecology. • Macdonald, G. (1965). Trans R Soc Trop Med Hyg59, 489-506. • D. Gurarie, C. King, Parasitology, 2005 (to appear)