COMPETITIVE DYNAMICS IN A MODEL FOR ONCHOCERCIASIS WITH CROSS-IMMUNITY

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1 CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 11, Number 4, Winter 2003 COMPETITIVE DYNAMICS IN A MODEL FOR ONCHOCERCIASIS WITH CROSS-IMMUNITY JIMMY P. MOPECHA AND HORST R. THIEME ABSTRACT. The interference of human and animal onchocerciasis which have different causative agents but the same vector is investigated by an ordinary differential equations model. The wasting of parasites on the wrong definitive hosts is found to lower the basic reproduction ratios of either disease while cross- immune reactions lead to classical competition phenomena. In particular, competitive exclusion of human onchocerciasis by animal onchocerciasis or visa versa may occur even if the basic reproduction ratios of both diseases exceed one. The zooprophylactic effects of varying the amount of cattle are discussed. 1 Introduction Onchocerciasis, commonly known as riverblindness, is a vector-borne parasitic disease. It is caused by filarial (threadlike) worms (Onchocerca volvulus in human hosts) and transmitted by the blackfly (Simulium damnosum). It occurs close to rivers because the egg, larva and pupa stages of the blackfly are aquatic [6]. The disease has been known for more than half a century as a constraint to social and economic development and causes considerable pain, suffering and blindness. Onchocerciasis is particularly prevalent in tropical Africa and parts of tropical America; in 1982, 20 million people were estimated to be affected [7], more recent estimates by WHO mention more than 17.7 million infected, visually impaired and another blind [30]. In West Africa, the fear of infection is one of the major causes of migration from fertile riverine areas into submarginal lands, which results in overcultivation and low productivity [47]. In spite of the success of the Onchocerciasis Control Program (OCP) and similar programs (2 million originally infected cured, 200,000 cases of blindness prevented, 25 The work of the first author was partially supported by NSF grants DMS and DMS The work of the second author was partially supported by NSF grants DMS and Copyright c Applied Mathematics Institute, University of Alberta. 339

2 340 JIMMY P. MOPECHA AND HORST R. THIEME million hectares of land available for resettlement [30]) onchocerciasis remains a health treat. After the end of OCP in 2002, control is almost entirely based on periodic mass treatment with ivermectin which lowers the microfilarial loads in affected individuals, but does not kill the adult worms [4], and will presumably not lead to eradication of the disease (at least not in West Africa) [25]. The risk of eventual ivermectin resistance calls for the development of complementary means like macrofilaricides or chemotherapy that sterilizes adult worms by targeting endosymbiotic Wolbachia bacteria [18, 21]. In this context, the interference of human and animal onchocerciasis is of interest. In North Cameroon, it has been found that 50%, as compared to 1% or less in other affected countries, of the infective filarial larvae found in the fly vectors of human onchocerciasis did not belong to the parasite of humans onchocerca volvulus [7], but to the parasite of cattle onchocerca ochengi [43, 45] or to the parasite of warthogs onchocerca ramachandrini [42]. In two study sites in North Cameroon, the annual proportion of flies infected with animal filariae was amazingly high (57 90% of all infections) [44], and high proportions of infection with animal filariae were also found in other study sites in North Cameroon [8, 26, 29, 43]. In the Guinea and Sudan savanna, O. ochengi is the prevailing filarial species in S. damnosum s.l., principally due to the abundance of cattle, mainly in the Adamawa highlands (about two times more cattle than humans), and to the flies preference of cattle as a blood-host [44]. The high degree of zoophily of S. damnosum s.l. in North Cameroon considerably lowers the vectorial capacity [11] of the vector populations for O. volvulus [27, 28]. This effect has been called zooprophylaxis. An additional effect became evident when the mean microfilarial loads in the above-mentioned two study villages, Galim and Karna, were related to the O. volvulus Annual Transmission Potential (ATP): even though the O. volvulus ATP (on man) was seven times higher in Galim than in Karna, the microfilarial load was a surprising 143 times lower in Galim than in Karna. Several other villages in the cattle-raising areas of the Adamawa highlands in Cameroon show a similar epidemiological situation as in Galim, i.e. high S. damnosum s.l. biting rates on man (and cattle), high infection rate of the flies with O. ochengi and low endemicity of human onchocerciasis (Wahl and Renz, unpublished observations). The disproportiately low endemicity of human onchocerciasis in Galim (and other cattle-farming villages in the Adamawa highlands), in conjunction with a very high O. volvulus ATP and an even higher O. ochengi ATP, led to the hypothesis [28, 44] that in these areas anthropoboophilic S. damnosum s.l. permanently inoculates O. ochengi larvae

3 ONCHOCERCIASIS MODEL WITH ZOOPROPHYLAXIA 341 into man, which cause a cross-reactive immunization against O. volvulus. This hypothesis of natural heterologous vaccination by O. ochengi was strengthened by immunological studies, which showed a high degree of homology between O. volvulus and O. ochengi in protein profile and serological recognition, and demonstrated significant differences in the serological reactivity between the patients from Galim and Karna [16, 17]. The low prevalence of palpable nodules in the population at Galim indicates that the putative cross-reactive immunity against O. volvulus also affects the pre-adult stages of the parasite, and not merely the microfilariae. That O. ochengi larvae are inoculated into humans during the flies bloodmeals became apparent from data collected in the dry season at Galim: from 188 parous S. damnosum s.l., which had accidentally taken a partial blood meal on the fly collectors, only 5 (2.7%) carried infective O. ochengi larvae, as compared to 132 (7%) of 1879 parous flies which did not have any traces of blood in their guts. In this paper we will present a mathematical model that, as simple as possible, describes the dynamics of both human and bovine onchocerciasis under the presence of cross-immunity reactions and will explore its consequences on the disease dynamics. We will assume that the inoculation of O. volvulus larvae into cattle also causes a cross-reactive immunization against O. ochengi. In Section 2, we will briefly explain the life cycle of the parasite. The model will be described in Section 3 and its well-posedness established in Section 4. In Section 5, we investigate the equilibrium points and the conditions under which they exist. Their local stability is studied in Section 6. Section 7 deals with extinction and persistence of the disease and uses persistence theory as it has been developed by G. Butler, H. I. Freedman, and P. Waltman in a series of seminal papers (see [46] for a survey). The results are summarized in Section 8. In Section 9, we establish stronger results concerning the global behavior under a monotonicity assumption which expresses that the immune reactions are moderate. Our approach uses dynamical systems which are strongly monotone with respect to a non-standard cone as they have been introduced in a general framework by S. B. Hsu, H. L. Smith, and P. Waltman [20]. In the Discussion, we highlight that cross-immunity makes the onchocerca system competitive between O. volvulus and O. ochengi. We also speculate how changes in the size of the cattle population may affect the disease prevalence.

4 342 JIMMY P. MOPECHA AND HORST R. THIEME 2 Life cycle of Onchocerca volvulus Onchocerciasis or riverblindness is caused in humans by onchocerca volvulus, a filarial (threadlike) worm. The infection is transmitted from person to person by blackflies belonging to the genus Simulium. Man is the definitive host of O. volvulus: fertilized female worms which live in subcutaneous human tissue produce microfilariae which can then be picked up from the skin by the vector flies during their blood-meals. Most of the microfilariae thus ingested die or are digested together with the blood-meal. The few that are successful in penetrating the wall of the fly s stomach settle in the thoracic muscles. After passing through three larval stages, they finally become free larvae capable of infecting the human host. When a female Simulium fly bites man, infective-stage larvae of onchocerca volvulus enter the skin through the wound caused by the bite of the fly. From there they migrate to subcutaneous tissue and mature [7]. The worms mature slowly and may require as long as a year to reach full size, but occasionally onchocerca nodules have been detected in infants 3 to 10 months old [10, p. 305]. Adult female worms are thought to live up to 15 years [23, p. 103], [10, p. 308], 12 years on average according to [31, p. 5] and 8 to 10 years according to [1, Table 15.2(a)]. Fertilized females can produce millions of embryos (microfilariae). Microfiliariae may live in the skin for as long as 30 months according to [10, p. 308] citing Duke (1972) and around two years according to [31, p. 5]. The life cycle of onchocerca ochengi is similar to that of O. volvulus except that cattle rather than man is the definitive host. The blackfly lives for up to four weeks [31, p. 6]. The life expectation of O. volvulus larvae in blackflies has been estimated to be 14 to 28 days [1, Table 15.2(b)]. It was estimated that, during a blood meal in the Cameroon forest, 80 % of infective larvae escape from the fly and 40% of infective flies become non-infective [7]. Wahl et al. [44] calculated that flies shed 48% of their load of infective O. ochengi larvae during partial blood meals on men (the wrong definitive host). It is fair to assume that infective O. volvulus larvae are wasted on cattle (from the parasite s point of view) in a similar way. 3 Model description In order to explore the effects of immunity and cross-immunity on the disease dynamics we keep the model as simple as possible and neglect many important features incorporated in other models (see [7, 25, 13, 2, 3] and the references therein). As [2, 3], we divide each of the two parasite populations into three stages, microfilariae (in the definitive host), larvae (in the fly), filariae (in the definitive

5 ONCHOCERCIASIS MODEL WITH ZOOPROPHYLAXIA 343 host). This leads to six dependent variables which all depend on one independent variable, time t. Dependent Variables x v1 x v2 x v3 x o1 x o2 x o3 number of volvulus microfilariae in humans number of volvulus larvae in flies number of infective volvulus filariae (adult worms) in humans number of ochengi microfilariae in cattle number of ochengi larvae in flies number of infective ochengi filariae (adult worms) in cattle Model equations The numbers of human and bovine hosts and of flies is assumed to remain constant. The rates of change for the number of parasites in their three different stages and for their two host species are described by a system of six ordinary differential equations. To save some space we only list the three differential equations for the parasites of one host species, x uj, where u = v, o, and introduce the following notation, ˆv = o and ô = v: x u1 = β ux u3 ( ) µ u1 + ( ν u + b u p u1 xu1, x u2 = b up u1 p u2 x u1 µ u2 + ν F + b uh u F q uu + b ) ûhû F q uû x u2, x u3 = b ( ) uh u F q b u uux u2 f u η uu F q b u uux u2 + η uû F q ûuxû2 (µ u3 + ν u )x u3. Parameters In the model system above the following parameters and parameter functions have been used. They are listed in the order of occurrence. β u per capita birth rate of u-parasites µ uj per capita mortality rate of u-parasite in stage j ν u per capita mortality rate of definitive host of u- parasite b u average rate at which a typical u-definitive host is bitten by flies p u1 probability that a u-microfilaria leaves the u-definitive host and enters the fly when the fly takes a blood meal. p u2 probability that u-microfilariae entering the fly develop into larvae ν F per capita mortality rate of flies H u number of definitive hosts for u-parasite F number of flies

6 344 JIMMY P. MOPECHA AND HORST R. THIEME q uw probability that a u-larva leaves the fly when the fly bites a w-definitive host, u, w {v, o} f u ( ) probability that a u-larva that leaves a fly and enters its definitive host develops into an adult worm η uw measure of strength and length of the immune reaction per w- larvae in the u-definitive host Model Explanation (1) The number of microfilariae in u-definitive hosts, x u1, increases by birth from adult worms at per parasite rate β u. It decreases as a result of their natural death, at per parasite rate µ u1, of the death of the u-definitive host, at per host rate ν u, or of being picked up by a fly during a blood meal on the u-definitive host, at per parasite rate b u p u1. The last rate is the product of the average rate at which a typical definitive host for u-parasites is bitten by flies, b u, and the probability that a u-microfilaria leaves the u-definitive host and enters the fly when the fly takes a blood meal, p u1. The rate b u = b u (F, H v, H o ) depends on the number of flies and the number of right and wrong definitive hosts. But since these are assumed not to change, we suppress this dependence in the notation. (2) The number of u-larvae in flies, x u2, increases as a result of the fly picking microfilariae up during a blood-meal on the u-definitive host. The respective rate is multiplied by the probability that a microfilaria develops into a larvae, p u2. The same number decreases as a result of their natural death, at per larva rate µ u2, of the death of the fly, at per fly rate ν F, or of inoculation into humans and cattle during a blood-meal. The rate at which one typical u-larva is transmitted from the flies to the right definitive hosts, b u (H u /F u )q uu, is compounded from the per host rate of being bitten by flies, b u, multiplied by the number of definitive hosts, H u, divided by the number of flies, F, and multiplied by the probability that a u-larva leaves the fly when the fly bites a definitive host, q uu. An analogous consideration applies to the transmission to wrong definitive hosts. (3) The number of filariae (adult worm) increases through the maturation of larvae that have been transmitted by flies. The transmission rate is as described in the previous paragraph. The probability that a transmitted larvae develops into a mature worm is affected by an immune reaction of the host described by the function f u. This immune reaction is triggered both by the larvae that enter the host as right definitive host or as wrong definitive host. The strength of the reaction is assumed to be proportional to the rate at which the larvae enter the host.

7 ONCHOCERCIASIS MODEL WITH ZOOPROPHYLAXIA 345 b u q uu x u2 (t)(1/f ) is the rate of u-larvae entering one typical u-definitive host (the right definitive host). b u qûu xû2 (1/F ) is the rate of û-larvae entering a typical u-definitive host (which is the wrong host for them). η uu and η uû are measures of the strength and lengths of the respective immune reactions per larva. Since the maturation probability is a decreasing function of the immune reactions, f u is a monotone decreasing function. We neglect the time delay it takes for a larvae to develop into a mature worm (at least one year [1, Table15.3(a)]). The number of filariae is reduced either by their natural death, at per parasite rate µ u3, or by death of the host, at per host rate ν u. Non-dimensionalization of the model In a first step towards formulating the model in dimensionless variables and parameters, we introduce the following variables, z uj = x uj H u, j = 1, 3, u = v, o the number of parasites of type u in the micro-filarial and filarial stages per definitive host. We also introduce y u2 (t) = η uu b u q uu x u2 (t) 1 F, which is a dimensionless measure of the strength of the immune reaction caused by u-larvae in the u-definitive host. ηuu 1y u2(t) is the rate at which u-larvae are inoculated into the u-definitive hosts. Further we introduce the immunity ratios (3.1) r u = η uûb u qûu ηûû bûqûû. Then r u yû2 is a dimensionless measure of the immune reaction caused in an average u-definitive host by the larvae of the other host type. Recall that ˆv = o and ô = v. With this scaling, we obtain the following system, z u1 = β u z u3 (µ u1 + ν u + b u p u1 )z u1, y b u H u u2 = η uu p u1 p u2 b u q uu F z u1 ( µ u2 + ν F + b uh u F q uu + b ) ûhû F q uû y u2, z u3 = y u2 f u (y u2 + r u yû2 ) (µ u3 + ν u )z u3. η uu

8 346 JIMMY P. MOPECHA AND HORST R. THIEME Since 1/ν F is the life expectation of a fly [41, p. 10], we realize that (3.2) m u = b u H u F is the average number of blood meals a typical fly takes from the definitive hosts of parasite type u during it s life time. 1 (3.3) D u1 = µ u1 + ν u + b u p u1 is the average sojourn time of u-microfilaria in its definitive host, (3.4) D u2 = = 1 µ u2 + ν F + buhu F 1 ν F q uu + b ûhû F q uû 1 µ u2 + ν F (1 + m u q uu + mûq uû ) is the average sojourn time of a u-larva in a fly, and 1 (3.5) D u3 = µ u3 + ν u is the expectation of adult life of a u-filaria in its definitive host [41, pp. 10,11]. (3.6) π u2 = m u q uu ν F D u2 is the probability of a u-larva to survive the stage in the fly and get into the definitive host [41, p. 11]. With these new parameters we have the system z u1 = 1 (β u D u1 z u3 z u1 ), D u1 y u2 = 1 (η uu b u p u1 π u2 p u2 z u1 y u2 ), D u2 z u3 = 1 ( ) Du3 y u2 f u (y u2 + r u yû2 ) z u3. η uu D u3 We make z u1 and z u2 dimensionless, (3.7) y u1 = η uu b u p u1 π u2 p u2 z u1, y u3 = η uu β u D u1 b u p u1 π u2 p u2 z u3 = η uu π u1 π u2 p u2 β u z u3,

9 ONCHOCERCIASIS MODEL WITH ZOOPROPHYLAXIA 347 where (3.8) π u1 = b u p u1 D u1 is the probability of surviving the u-microfilarial stage and being picked up by a fly [41, p. 11]. We rewrite the system in these dimensionless dependent variables, with y u1 = 1 (y u3 y u1 ), D u1 y u2 = 1 (y u1 y u2 ), D u2 y u3 = 1 ( ) yu2 R u (y u2 + r u yû2 ) y u3, D u3 (3.9) R u (y) = D u3 β u π u1 π u2 p u2 f u (y). We realize that (3.10) ρ u = D u3 β u is the average number of microfilariae which a single average u-filaria can produce during its life-time. We recall that π u1 is the probability of making it through the microfilarial stage in the u-definitive host and being picked up by the flies, p u2 is the probability of developing into a larva within the fly, π u2 is the probability of a larva to make it into the definitive host, and f u is the probability of a larvae in the definitive host to mature. In short, R u (y u2 + r u yû2 ) is the reproduction ratio of the u-parasite (i.e. the number of microfilariae born to one average filarial worm that make it to the filarial stage), when the strengths of the immune reactions caused by u-larvae and û-larvae are y u2 and yû2 respectively. R u = R u (0) is the basic reproduction ratio, i.e., the reproduction ratio if there is no immune reaction by the definitive u-host. After a normalization at 0, g u (y) = R u (y)/r u (0), the system becomes y u1 = 1 (y u3 y u1 ), D u1 y u2 = 1 (y u1 y u2 ), D u2 y u3 = 1 ( ) Ru y u2 g u (y u2 + r u yû2 ) y u3, D u3

10 348 JIMMY P. MOPECHA AND HORST R. THIEME with (3.11) R u = ρ u π u1 π u2 p u2 g u (0) and strictly monotone decreasing functions g u : [0, ) (0, 1], g u (0) = 1. Dietz [7], using a different model for human onchocerciasis, estimates R v, the basic reproduction ratio for O. volvulus, to be 3.1, 3.5, and 50.3 at three different locations in Cameroon, 9.0, 33.7, and 74.0 at three different locations in Burkina Faso, and at one location in Ivory Coast. These values were basically confirmed and added to by Basáñez and Boussinesq [2]. The life expectations are about three weeks for flies, 14 years for adult filarial worms in humans, 55 years for humans, and 12 years for cattle in North Cameroon. The development within the fly to an infective larvae takes a minimum of 6 days [10, 31]. [2] suggests yearly per capita mortality rates of µ v1 = 0.8, µ v2 = 52, and µ v3 = 0.1, ν F = 26, ν v = The values for cattle may be similar except ν o = /D u1 may be considerable smaller though than 1/µ u1 depending on the biting rate b v. Still, D u2 will be much smaller than D u1 and D u3 ; so, in cases where R u is not too large, it is justified to take the quasi steady state in the second equation, y u1 = y u2, and reduce the system to four equations, y u1 = 1 (y u3 y u1 ) D u1, u = v, o. y u3 = 1 ( ) yu1 R u g u (y u1 + r u yû1 ) y u3 D u3 In the following, we will mainly analyze this reduced four-dimensional system. The large-time behavior of model solutions extends to the full system if D u2 is small enough compared with D u1 and D u3 and R u not too large [19] or if the monotonicity hypothesis (M) in Section 9 is assumed. 4 Well-posedness of the model We introduce the dependent variables x 1 = y v1, x 2 = y v3, x 3 = y o1, x 4 = y o3, and the parameters a 1 = 1/D v1, a 2 = 1/D v3, a 3 = 1/D o1, a 4 = 1/D o3, and obtain the

11 ONCHOCERCIASIS MODEL WITH ZOOPROPHYLAXIA 349 system (4.1) x 1 = a 1(x 2 x 1 ), x ( ) (4.2) 2 = a 2 x1 R v g v (x 1 + r v x 3 ) x 2, (4.3) x 3 = a 3(x 4 x 3 ), (4.4) x 4 = a 4 ( x3 R o g o (r o x 1 + x 3 ) x 4 ). This is a particular case of a system x = F (x), where x is the vector with coordinates x 1,..., x 4 and F : R 4 + R 4 is the vector field associated with (4.1) (4.4). We show that unique non-negative solutions exist for the above general system as suggested by the epidemiological interpretation. We make the following assumptions throughout this paper: All constants in the model equations are positive. The functions g u : [0, ) (0, 1], u = v, o, are continuously differentiable and g u(ξ) < 0 for all ξ 0. g u (ξ) 0 as ξ, u = v, o. This implies that F : R 4 + R4 is continuously differentiable and so locally Lipschitz continuous. It is not difficult to check that F j (x) 0 if x R 4 + and x j = 0. By [41, Theorem A.4], for any initial data in R 4 +, there exists a unique solution of x = F (x) in R 4 + defined on some interval [0, b) such that b = or lim sup t b 4 x j (t) =. j=1 In the next step, we show that any non-negative solution to our system is bounded which implies that b =. Since g u (ξ) 0 as ξ by assumption, we can find d u > 0 and δ u (0, 1) such that x 1 R v g v (x 1 + r v x 3 ) d v + δ v x 1 (4.5), x 1, x 3 0. x 3 R v g v (r o x 1 + x 3 ) d o + δ o x 3 Let V (t) = αx 1 (t) + x 2 (t) with α > 0 to be determined later. Then V = αa 1 (x 2 x 1 ) + a 2 (d v + δ v x 1 x 2 ) = a 2 d v + (a 2 δ v αa 1 )x 1 + (αa 1 a 2 )x 2.

12 350 JIMMY P. MOPECHA AND HORST R. THIEME Choose δ (δ v, 1) and α such that αa 1 = a 2 δ. Then there exists some ɛ > 0 such that V a 2 d v ɛv. Hence V is bounded on [0, b) and so are x 1 and x 2. Similarly, we show that x 3 and x 4 are bounded on [0, b). This implies b =. Now V is bounded on [0, ), actually lim sup t V (t) a 1 d v /ɛ. x 1 and x 2 are bounded as well and lim sup x 1 (t) a 1d v t ɛα, lim sup t x 2 (t) a 1d v. ɛ An analogous result holds for x 3 and x 4. Theorem 4.1. For any x 0 R 4 +, there exists a unique solution x of (4.1) (4.4) defined on [0, ) with values in R 4 + and initial values x 0. The solution is bounded on [0, ). More precisely there exists some c (0, ) which is independent of x 0 such that x(t) [0, c) 4 for all sufficiently large t. For later use we state that the solutions of the system (4.1) (4.4) induce a continuous semiflow [41, Cor. A.30]. Corollary 4.2. The definition T (t, x 0 ) = x(t) where x 0 R 4 + and x is the unique solution of (4.1) (4.4), defined on [0, ) with values in R 4 + and initial values x 0, defines a continuous semiflow on R 4 +, i.e., the map T : [0, ) R 4 + R 4 + is continuous, and T t T s = T t+s, t, s 0, where T t represents the map T (t, ) : R 4 + R 4 +. Moreover T (0, x 0 ) = x 0 for all x 0 R Equilibrium points At equilibrium, time derivatives are 0 and we obtain the following relations from (4.1) (4.4), (5.1) x 2 = x 1 =: x v, 0 = x v ( Rv g v (x v + r v x o ) 1 ), x 4 = x 3 =: x o, 0 = x o ( Ro g o (r o x v + x o ) 1 ). Obviously, the origin, E 0 = (0, 0, 0, 0) is a solution to the above system.

13 ONCHOCERCIASIS MODEL WITH ZOOPROPHYLAXIA Boundary equilibria For the boundary equilibrium, where O. volvulus is present, but O. ochengi is extinct, suppose x o = 0, and x v > 0. Then R v g v (x v ) = 1. If R v 1, this equation has no solution x v > 0, because g v (0) = 1 and g is strictly decreasing. If R v > 1, there is a unique solution x v > 0 by the intermediate value theorem, because we have assumed that g v (ξ) 0 as ξ, x v = gv 1 (1/R v ). This gives rise to the unique boundary equilibrium where O. volvulus is present (called the volvulus boundary equilibrium for short), (5.2) E 1 = (ˆx v, ˆx v, 0, 0), ˆx v = g 1 v (1/R v ). A similar consideration applies to an ochengi boundary equilibrium with x v = 0 and x o > 0. It exists if and only if R o > 1 and is given by (5.3) E 2 = (0, 0, ˆx o, ˆx o ), ˆx o = g 1 o (1/R o ). 5.2 The interior alias coexistence equilibrium For the interior equilibrium, we have x v, x o > 0, and (5.4) R v g v (x v + r v x o ) = 1, R o g o (r o x v + x o ) = 1. System (5.4) only has a positive solution if R v > 1 and R o > 1 because g u (ξ) < 1 for ξ > 0. Necessarily, x v + r v x o = g 1 v (1/R v ), r o x v + x o = g 1 o (1/R o ). This system has a solution (which is unique) if and only if the determinant of the matrix of the system, = 1 r v r o, is differentfrom zero. Assuming that 0 and the solutions are positive, we have the unique interior equilibrium (5.5) E = ( x v, x v, x o, x o ), x v = g 1 v ( 1 R v ) r v go 1( 1 R o ) x o = g 1 o ( 1 R o ) r o gv 1 ( 1 = 1 r v r o. If > 0, x v > 0 and x o > 0 if and only if (5.6) r v < g 1 v (1/R v ) go 1 (1/R o ) < 1. r o R v ),

14 352 JIMMY P. MOPECHA AND HORST R. THIEME If < 0, the inequalities above will be reversed, and we will have (5.7) 1 < g 1 v (1/R v ) r o go 1 (1/R o ) < r v. If = 0, there is no solution unless r v = g 1 v (1/R v ) go 1 (1/R o ) = 1, r o in which case we have a continuum of solutions. 6 Local stability of equilibria The Jacobian matrix, J(x), of system (4.1) (4.4) at x = (x 1, x 2, x 3, x 4 ) is given by a 1 a (6.1) J(x) = a 2 [R v g v (η) + ζ] a 2 a 2 r v ζ a 3 a 3, a 4 r o ω 0 a 4 [R o g o (θ) + ω] a 4 with η = x 1 + r v x 3, θ = r o x 1 + x 3, ζ = x 1 R v g v(η), ω = x 3 R o g o(θ). 6.1 Local stability of the origin At the origin, ζ = ω = 0 and (6.1) comprises two blocks of 2 2 matrices. The trace of each block is negative, and the origin will be stable if the determinant of each block is greater than zero. Since g u (0) = 1, the origin, E 0, is stable if R o < 1 and R v < 1. If R o > 1 or R v > 1, the origin is a saddle, in particular unstable. 6.2 Local stability of the boundary equilibria For the volvulus boundary equilibrium E 1 = (ˆx v, ˆx v, 0, 0), (6.1), ω = 0 and J(E 1 ) has block structure. The traces of both block matrices are negative. This boundary equilibrium is stable if the determinant of each block is positive, i.e., since R v g v (η) = 1, if ˆx v R v g v(ˆx v ) < 0 and 1 R o g o (r oˆx v ) > 0. The first condition is satisfied because g v (ξ) < 0 for all ξ 0 by assumption. ( We substitute the value for ˆx v into the second condition, 1 R o g o ro gv 1 (1/R v ) ) > 0. Since g o and so go 1 are strictly monotone decreasing, the second condition is equivalent to (6.2) g 1 v (1/R v ) g 1 o (1/R o ) > 1 r o.

15 ONCHOCERCIASIS MODEL WITH ZOOPROPHYLAXIA 353 Let us summarize: The volvulus boundary equilibrium is locally asymptotically stable if (6.2) holds, and an (unstable) saddle if the opposite strict inequality holds. Similarly, the ochengi boundary equilibrium E 2 = (0, 0, ˆx o, ˆx o ) is locally asymptotically stable, if (6.3) g 1 v (1/R v) go 1 (1/R o ) < r v. and an (unstable) saddle if the opposite strict inequality holds. 6.3 Local stability of the coexistence equilibrium For the coexistence equilibrium, E, recall that R v g v (η ) = R o g o (θ ) = 1. The Jacobian, evaluated at E, has the same sign as = 1 r v r o. Since the determinant is the product of the eigenvalues, the coexistence equilibrium is unstable if < 0. The dimensions of the stable and unstable manifolds are one and three, or three and one. A lengthy but straightforward calculation and application of the Routh-Hurwitz criterion [9, 41, 37] shows that the coexistence equilibrium, whenever it exists, is locally asymptotically stable if > 0 [22]. 7 Extinction and persistence of onchocerciasis The basic reproduction ratios R v and R 1 play a crucial role in the large-time behavior of the onchocerciasis system. Theorem 7.1. (a) Let R v 1. Then O. volvulus dies out, i.e., x 1 (t) 0 and x 2 (t) 0 as t for all solutions of (4.1) (4.4). (b) Let R o 1. Then O. ochengi dies out, i.e., x 3 (t) 0 and x 4 (t) 0 as t. for all solutions of (4.1) (4.4), (c) If both R v 1 and R o 1, then both O. volvulus and O. ochengi die out, i.e., all solutions of (4.1) (4.4) converge towards the origin as t. (d) If R v > 1 or R o > 1, then onchocerciasis persists uniformly strongly as a whole, in the following sense that there exists some ɛ > 0 such that lim inf t (x 1(t) + x 3 (t)) ɛ, lim inf t (x 2(t) + x 4 (t)) ɛ, for all solutions of (4.1) (4.4) with non-negative initial conditions satisfying x 1 (0) + x 3 (0) > 0 or x 2 (0) + x 4 (0) > 0.

16 354 JIMMY P. MOPECHA AND HORST R. THIEME In proving these results we will use the following notation for a function ψ : [0, ) R: ψ = lim inf t ψ(t), ψ = lim inf t ψ(t). Lemma 7.2. The following relations hold: (a) x 1 x 2, x 2 R v g v x 1, x 3 x 4, x 4 R o g o x 3, where g v = g v (x 1 + r v x 3 ) and g o is defined similarly. (b) x 1 x 2, x 2 R v g v x 1, x 3 x 4, x 4 R o g o x 3, where g v = g v (x 1 + r vx 3 ) and g o is defined similarly. Proof. Let s show the first estimate in (a). We apply the method of fluctuations [15]. By [41, Prop. A.22], there exists a sequence t n as n such that x 1 (t n ) x 1, x 1 (t n) 0, as n. By (4.1), 0 x 1 (t n) = a 1 (x 2 (t n ) x 1 (t n )). Hence x 1 = lim t x 2 (t n ) x 2. Thus x 1 x 2. The rest of the estimates in (a) and (b) are shown in exactly the same way. Recall that g u is monotone decreasing. Proof of Theorem 7.1. (a) Consider the functional V 1 (t) = a 2 x 1 (t) + a 1 x 2 (t). Since x i 0 by Theorem??, V 1 0. But (7.1) V 1 = a 1a 2 x 1 (R v g v (x 1 + r v x 3 ) 1). By assumption, 0 g v 1. So R v 1 implies that V 1 0. Thus V (t) is monotone non-increasing in t 0. Since V (t) is non-negative, it converges to some limit V 0 as t. Since x 1, x 2, V are bounded, so are x 1, x 2 and V as solutions of (4.1) (4.4) and (7.1). Hence x 1, x 2 and V are uniformly continuous on [0, ), and so are x 1, x 2 and V. By Barbalat s Lemma [41, Cor. A.18], 0 V (t) a 1 a 2 x 1 (t) ( g v (x 1 (t) + r v x 3 (t)) 1 ), t. We choose a sequence t n as n such that x 1 (t n ) x 1, as n. If x 1 > 0, we would have ( 0 lim x ( 1(t n ) g v x1 (t n ) + r v x 3 (t n ) ) ) 1 n x ( 1 gv (x 1 ) 1 ) < 0, because g v : [0, ) 2 (0, 1] is strictly monotone decreasing. This contradiction shows that x 1 = 0. Hence x 1(t) 0 as t. We apply

17 ONCHOCERCIASIS MODEL WITH ZOOPROPHYLAXIA 355 Barbalat s Lemma to (4.1) and obtain x 2 (t) 0 as t. O. volvulus dies out. (b) The proof is analogous to (a). (c) The assertion follows from (a) and (b) above. (d) Let R v > 1. We first show that there is an ɛ > 0 such that (7.2) lim sup(x 1 (t) + x 3 (t)) ɛ, whenever x 1 (0) + x 3 (0) > 0, t for every solution x of the system (4.1) (4.4). Suppose not, then for every ɛ > 0 there exists a solution of (4.1) (4.4) with non-negative initial data such that x 1 (0) + x 3 (0) > 0 and x j < ɛ, j = 1, 3. Recall the functional V 1 (t) = a 2 x 1 (t)+a 1 x 2 (t), V 1 (t) 0, V bounded. Since g v is decreasing, by (7.1), V 1(t) a 1 a 2 x 1 (t) ( R v g v ( (1 + rv )ɛ ) 1 ), for large t > 0. Since R v > 1, we can choose ɛ small enough such that R v g v ((1 + r v )ɛ) = 1 + δ, for some δ > 0. Hence V 1 (t) δx 1(t) 0, implying that V 1 (t) is non-decreasing. But V 1 (t) is bounded, thus it has a limit as t. By Barbalat s Lemma, V 1(t) 0 as t. The expression for V 1 (t) above shows that x 1(t) 0 as t. By Lemma 7.2 (a), x 2 = 0. This implies that V 1 (t) 0 as t, a contradiction because V is strictly positive and non-decreasing. This proves (7.2). In order to complete the proof of (d), we use the persistence theory in [41, Sec. A.5]. Let X = R 4 + and ρ(x) = x 1 + x 3 for x X. If ρ(x(0)) = x 1 (0) + x 3 (0) > 0, then ρ(x(t)) > 0 for all t 0. As we have shown above, the semiflow induced by solutions of (4.1) (4.4) (recall Corollary 4.2) is uniformly weakly ρ persistent. The compact set B = [0, c] 4, with c > 0 as in Theorem 4.1, attracts all solutions of (4.1) (4.4) with non-negative initial data. Thus the compactness condition (C) of [41, Thm. A.32] is satisfied, and the semiflow induced by solutions of (4.1) (4.4) is uniformly strongly ρ persistent, lim inf t (x 1(t) + x 3 (t)) ɛ > 0. Using this result, one can now show there exists an ɛ 1 > 0 such that lim inf t (x 2(t) + x 4 (t)) ɛ 1.

18 356 JIMMY P. MOPECHA AND HORST R. THIEME The details are left to the reader. Notice that x 2 (0) + x 4 (0) > 0 implies that x 1 (t) + x 3 (t) > 0 for t > 0. Hence we get the same persistence results under the first condition. If R o > 1 and x 1 (0) + x 3 (0) > 0 or x 2 (0) + x 4 (0) > 0, we proceed analogously. The next result gives us more precise information for the case that one and only one of the two basic reproduction ratios exceeds one. Then the respective onchocerca species persists and converges towards an equilibrium, while the other onchocerca species dies out. Theorem 7.3. (a) If R v 1 < R o, then every solution of the onchocerca system where O. ochengi is initially present converges to the ochengi boundary equilibrium. (b) If R o 1 < R v, then every solution of the onchocerca system where O. volvulus is initially present converges to the volvulus boundary equilibrium. Proof. (a) Since R v 1, by Theorem 7.1(a), x 1 (t) and x 2 (t) converge to zero as t, while x 3 and x 4 are bounded in forward time. Fix x 1 and x 2 and consider the solution (x 3, x 4 ) of the associated two-dimensional ochengi subsystem which is asymptotically autonomous. Since R o > 1, the two-dimensional limiting system has the equilibrium (0, 0) and the interior equilibrium, (ˆx o, ˆx o ), which is locally asymptotically stable. We take the divergence of the vector field, a 3 (x 4 x 3 ) + ( ) a 4 x3 R o g o (x 3 ) x 4 = a3 a 4 < 0. x 3 x 4 Hence the ochengi subsystem satisfies Bendixson s criterion in R 2 +, and R 2 + possesses the Dulac property, meaning that there are no periodic orbits or cyclic chains of equilibria in R 2 +. By [40, Theorem 4.9], every forward bounded solution of the asymptotically autonomous ochengi subsystem converges to an equilibrium of the ochengi subsystem. This means that every solution of the full onchocerca system converges to the origin or the the ochengi boundary equilibrium. By Theorem 7.1, no solution of (4.1) (4.4) with x 3 (0) + x 4 (0) > 0 converges to the origin. So all these solutions of converge to the ochengi boundary equilibrium. (b) The proof is symmetric to that of (a). If both basic reproduction ratios exceed one, we need stronger conditions to specify whether it is O. volvulus or/and O. ochengi that persist.

19 ONCHOCERCIASIS MODEL WITH ZOOPROPHYLAXIA 357 Theorem 7.4. If R v > 1 and R o > 1, the following additional conditions specifically guarantee the strong persistence of O. volvulus (or O. ochengi): (a) If g 1 v (1/R v) go 1 (1/R o ) > r v, then O. volvulus persists uniformly strongly, in the sense that there exists some ɛ > 0 such that x 1 ɛ and x 2 ɛ for all solutions with non-negative initial data and x 1 (0) + x 2 (0) > 0. (b) If 1 > g 1 v (1/R v ) r o go 1 (1/R o ), then O. ochengi persists uniformly strongly. (c) If both conditions are satisfied, both O. volvulus and O. ochengi persist uniformly strongly. Proof. (a) We apply the persistence theory from [39]. Recall that the solutions of the system (4.1) (4.4) induce a continuous semiflow on X = R 4 + (recall Corollary 4.2). We set and X 1 = {x R 4 +; x 1 + x 2 > 0} X 2 = {(0, 0, x 3, x 4 ) R 4 +; x 3, x 4 0}. Then X = X 1 X 2, X 1 X 2 =, X 1 is open in X and forward invariant under the continuous semiflow T, and the compactness assumption (C 4.1 ) of [39] holds for X 2, because all solutions of (4.1) (4.4) that start in the positive cone R 4 + are attracted to the compact set [0, c] 4 by Theorem 4.1. We first apply [39, Thm. 4.4.]. By the Poincaré-Bendixson trichotomy ([41, Thm. A.10], e.g.), all solutions x starting in X 2 converge towards an equilibrium, namely (0, 0, ˆx o, ˆx o ) or (0, 0, 0, 0). Both equilibria are saddles (Sections 6.1 and 6.2) under our assumptions, and M 1 = {(0, 0, 0, 0)} and M 2 = {(0, 0, ˆx o, ˆx o } are isolated compact invariant sets. Let Ω 2 = {(0, 0, ˆx o, ˆx o ), (0, 0, 0, 0)}. Notice that X 2 is forward invariant and can be identified with R 2 +. We already established in the proof of Theorem 7.3 that R 2 + has the Dulac property for the autonomous ochengi subsystem, i.e., there are no periodic orbits or cyclic chains of equilibria in R 2 +. In particular, M 1 = {(0, 0, 0, 0)} and M 2 = {(0, 0, ˆx o, ˆx o )} are neither chained to one another nor to themselves. This means that Ω 2 has the acyclic covering M 1 M 2 [39, p. 425].

20 358 JIMMY P. MOPECHA AND HORST R. THIEME Next we show that no solution of the system (4.1) (4.4) that starts in X 1, with initial conditions satisfying x 1 (0) + x 2 (0) > 0, converges to either the ochengi boundary equilibrium or the origin. Let x be a solution of the system (4.1) (4.4) with x 1 (0) + x 2 (0) > 0. Suppose that it converges to the ochengi boundary equilibrium. This means that x 1 (t) 0, x 2 (t) 0 and x 3 (t) ˆx o, x 4 (t) ˆx o as t. We can find an ɛ > 0 such that x j (t) < ɛ, j = 1, 2, x j (t) < ˆx o + ɛ, j = 3, 4, for sufficiently large t > 0. Consider the functional V (t) = x 1 (t) + a 1 x 2 (t). Then V (t) 0 and V (t) 0 as t. Since g v is decreasing, V (t) = x 1 (t) ( R v g v ( x1 (t) + r v x 3 (t) ) 1 ) for large t > 0. By assumption x 1 (t) ( R v g v ( (1 + rv )ɛ + r v ˆx o ) 1 ), g 1 v (1/R v) go 1 (1/R o ) > r v, which, by (5.3), is equivalent to R v g v (r v ˆx o ) 1 > 0. We can choose ɛ > 0 such that R v g v ( (1 + rv )ɛ + r v ˆx o ) 1 > 0. Thus V (t) 0 for large t, implying that V (t) is eventually non-decreasing. This is a contradiction since V (t) is strictly positive and converges to zero. Hence no solution of (4.1) (4.4) that starts in X 1 converges to the ochengi boundary equilibrium. Since by assumption x 1 (0) + x 2 (0) > 0 and R v > 1, it follows from Theorem 7.1 (d) that x(t) does not converge to zero as t. We conclude that no solution of (4.1) (4.4) that starts in X 1 converges to any point of Ω 2. That is, each M k is a weak repeller for X 1 [39, p. 408]. By [39, Thm. 4.4], X 2 is a uniform weak repeller for X 1 [39, p. 408]. By Theorem 4.1, the compactness condition of [39, Thm. 1.4] is satisfied, and X 2 is a uniform strong repeller for X 1 [39, p. 408], i.e., O. volvulus persists uniformly strongly. (b) This case is symmetric to (a) and is proved analogously.

21 ONCHOCERCIASIS MODEL WITH ZOOPROPHYLAXIA Summary of results for general immune reactions We can now give the following almost complete case analysis: Case 1. R v 1 and R o 1. There are no boundary equilibria and no coexistence equilibrium. The disease-free equilibrium is globally asymptotically stable, both O. volvulus and O. ochengi die out. Case 2.1. R v > 1 and R o 1. There exists a unique volvulus boundary equilibrium, but no ochengi boundary equilibrium and no coexistence equilibrium. The disease-free equilibrium is a saddle (hence unstable) and the volvulus boundary equilibrium is locally asymptotically stable. O. ochengi dies out and O. volvulus tends to the volvulus boundary equilibrium. Case 2.2. R v 1 and R o > 1. (symmetric to Case 2.1) O. volvulus dies out and O. ochengi tends to the ochengi boundary equilibrium. Case 3. R v > 1 and R 0 > 1. Both boundary equilibria exist, and onchocerciasis as a whole persists uniformly strongly. 1 Case 3.1. < g 1 v (1/R v ) r o go 1 (1/R o ) < r v (Bi-stability). A unique coexistence equilibrium exists. The coexistence equilibrium is a saddle and so unstable. The dimensions of the stable and unstable manifolds are one and three, or three and one. The boundary equilibria are both locally asymptotically stable. Case 3.2. r v < g 1 v (1/R v) go 1 (1/R o ) < 1 (Locally Stable Coexistence). r o A unique coexistence equilibrium exists. The coexistence equilibrium is locally asymptotically stable. The boundary equilibria are saddles and so unstable. Both O. volvulus and O. ochengi persist uniformly strongly. Case 3.3. r v = g 1 v (1/R v) go 1 (1/R o ) = 1. r o The two boundary equilibria are connected by a continuum of coexistence equilibria. We cannot say anything definitive about the stability of any of these equilibria. Case 3.4. r v, 1 < g 1 v (1/R v ) r o go 1 (1/R o ).

22 360 JIMMY P. MOPECHA AND HORST R. THIEME There is no coexistence equilibrium. The volvulus boundary equilibrium is locally asymptotically stable and the ochengi boundary equilibrium is unstable. O. volvulus persists uniformly strongly. Case 3.5. g 1 v (1/R v ) go 1 (1/R o ) < r 1 v, (symmetric to Case 3.4). r o There is no coexistence equilibrium. The ochengi boundary equilibrium is locally asymptotically stable and the volvulus boundary equilibrium is unstable. O. ochengi persists uniformly strongly. Case 3.6. r v < g 1 v (1/R v) go 1 (1/R o ) = 1. r o There is no coexistence equilibrium. Both the ochengi and the volvulus boundary equilibria exist. The ochengi boundary equilibrium is unstable. Under the present assumptions (cf. Section 9), we cannot say anything definitive about the stability of the volvulus boundary equilibrium. Case 3.7. r v = g 1 v (1/R v) go 1 (1/R o ) < 1 (symmetric to Case 3.6). r o There is no coexistence equilibrium but both boundary equilibria exist. The volvulus boundary equilibrium is unstable, but nothing definite can be said about the stability of the ochengi boundary equilibrium. 1 Case 3.8. < g 1 v (1/R v) r o go 1 (1/R o ) = r v. There is no coexistence equilibrium. The volvulus boundary equilibrium is locally asymptotically stable and O. volvulus persists uniformly strongly. Under the present assumptions (cf. Section 9), we cannot say anything definitive about the stability of the ochengi boundary equilibrium. 1 Case 3.9. = g 1 v (1/R v ) r o go 1 (1/R o ) < r v (symmetric to Case 3.8). There is no coexistence equilibrium. The ochengi boundary equilibrium is locally asymptotically stable and O. ochengi persists uniformly strongly. Under the present assumptions (cf. Section 9), we cannot say anything definitive about the stability of the volvulus boundary equilibrium. 9 Global stability analysis for moderate immune reactions In order to derive global stability results in the case that both basic reproduction ratios exceed 1, we make the additional assumption that the immune reaction functions g v and g o have the following property:

23 ONCHOCERCIASIS MODEL WITH ZOOPROPHYLAXIA 361 (M) d dr [rg u(r)] > 0 for all r 0, u = v, o. This property can be interpreted as moderate immune responses by the hosts. Examples are given by g u (r) = 1/(1 + br γ ) ξ with constants b, γ, ξ > 0 and γ + ξ 1. We will apply the results of [36]. Though we concentrate on the reduced four-dimensional system, our approach equally works for the original six-dimensional system. Let X = R 4 and represent it as X = X 1 X 2 where X 1 = R 2 is the space for the volvulus subsystem and X 2 = R 2 the space for the ochengi subsystem. Since R 2 is a Euclidean space, it is an ordered Banach space with the standard ordering and the usual Euclidean norm. For i = 1, 2, X i + = R 2 + is the associated positive cone and its interior Int X + i equals (0, ) 2. We use the same symbol for the partial orders generated by the cones X + i. If x i and x i are elements of X i, then we write x i x i if x i x i X i +, x i < x i if x i x i and x i x i, and x i x i if x i x i Int X + i. We use the same notation for the norm in both X 1 and X 2, namely. Our state space is the cone R 4 +. For our purposes, the more important cone is K = R 2 + ( R 2 +) with interior Int K = Int R 2 + ( Int R 2 +) = (0, ) 2 (, 0) 2. It generates the partial order relations K, < K and K. In this case, x K x if and only if x 1 x 1 and x 2 x 2. A similar statement holds with K replacing K and replacing. In the terminology of [35], let C = X + = R 4 +. C is a closed convex subset of X +. Let C 0 = C \ (C 1 C 2 {E 0 }), C 1 = {(x 1, x 2, 0, 0) C : x 1 > 0 or x 2 > 0}, C 2 = {(0, 0, x 3, x 4 ) C : x 3 > 0 or x 4 > 0}. The system of ordinary differential equations (4.1) (4.4) obtained in Section 3 has been rewritten in the form x = F (x) in Section 4. The solutions x of x = F (x) induce a continuous semiflow on C = R 4 + (Corollary 4.2). Definition 9.1. A semiflow T is order preserving if T t (x) K T t (y) whenever x K y, and strictly order preserving if T t (x) < K T t (y) whenever x < K y. It is strongly order preserving on A, where A C is positively invariant, if it is order preserving and, whenever x, y A and x < K y, there exist open sets U and V in A, x U and y V, and t 0 0 such that T t0 (U) K T t0 (V ).

24 362 JIMMY P. MOPECHA AND HORST R. THIEME In order to show that T is order preserving and even strictly order preserving on C and strongly order preserving on C 0, we use the terminology and the results of [33, Chap. 3, Sec. 5]. Let m = (m 1, m 2, m 3, m 4 ) = (0, 0, 1, 1). Then K = {x R 4 : ( 1) mi x i 0; 1 i 4} =: K m. Obviously x K y if an only if x i y i if m i = 0 and x i y i if m i = 1. The assumptions of [33, Chap. 3, Prop. 5.1] are satisfied provided that x = F (x) is cooperative with respect to K, i.e., ( 1) mi+mj F i(x) x j 0, i j, x C. For illustration we check two of these conditions and leave the remaining ones to the reader: ( 1) m2+m1 F 2 x 1 = ( 1) 0 R v [g v (x 1 + r v x 3 ) + x 1 g v (x 1 + r v x 3 )] = R v [g v (x 1 + r v x 3 ) + (x 1 + r v x 3 )g v(x 1 + r v x 3 ) r v x 3 g v (x 1 + r v x 3 )] = R v (ηg v (η)) r v x 3 R v g v(η), with η = x 1 + r v x 3. By assumption (M) and the strict decrease of g v, this expression is positive. For the same reason ( 1) m2+m3 F 2 x 3 = ( 1) 1 r v x 1 R v g v(η) 0. So the system (4.1) (4.4) is cooperative with respect to K and T is strictly order preserving on C by [33, Prop. 5.1 in Chap. 3]. In order to show that the semiflow is strongly order preserving on on C 0, we consider the Jacobian matrix of the vector field. Definition 9.2. An n n matrix Q is irreducible if for every nonempty, proper subset I of N = {1,..., n} there exists i I and l L N \ I such that q il 0. Recall the Jacobian matrix of our system (6.1). This matrix is irreducible if ζ, ω > 0, i.e., if x 1 > 0 and x 3 > 0, in particular for x Int C. Consequently, the semiflow generated by the vector field

25 ONCHOCERCIASIS MODEL WITH ZOOPROPHYLAXIA 363 F is strongly order preserving on Int C [33, Thm. 1.1 in Chap. 4]. Notice that T t (C 0 ) Int C for t > 0 and T t (x) < K T t (x) if x < K y. This implies that T is strongly order preserving on C 0 {E 1, E 2 }. If x C, then O(x) = {T t (x) : t 0} is called the positive orbit of T. It s omega limit set is defined in the usual way. An equilibrium point is a point x such that O(x) = {x}, equivalently T t (x) = x for all t 0. We assume that R v > 1 and R o > 1. This implies that the boundary equilibria E 1 = (ˆx v, ˆx v, 0, 0) and E 2 = (0, 0, ˆx o, ˆx o ) exist. Most of the hypotheses (H0) (H5) in [36] easily follow from the considerations above or in Section 7. As for (H3) in [36], it is easily checked that T t (C j ) C j, j = 1, 2. The restriction of T to C 1 corresponds to the semiflow generated by the volvulus subsystem, x 1 = a 1(x 2 x 1 ), x 2 = a 2[R v g v (x 1 ) x 2 ]. The Jacobian matrix at an arbitrary point (x 1, x 2 ) is given by ( J 1 (x 1, x 2 ) = a 1 a 1 a 2 R v g v ( ) + a 2 x 1 R v g v ( ) a 2 Since g v (0) = 1 and R v > 1, the equilibrium point (0, 0) is unstable. The only other equilibrium point is (ˆx v, ˆx v ) where R v g v (ˆx v ) = 1. This equilibrium point is locally asymptotically stable. It follows from (M) that the Jacobian matrix is irreducible and the volvulus system is cooperative. By [33, Thm. 1.1 in Chap. 4], the induced semiflow strongly order preserving on R 2 +. All solutions except those starting at (0, 0) converge to (ˆx v, ˆx v ) [34, Cor. 3.12]. This implies that T t (x) E 1 for all x C 1. The statement concerning C 2 is proved analogously. (H4) in [36] may or may not hold depending on the assumptions we make. But whenever the interior equilibrium of T in C 0 exists, it is unique. (H5) holds, since (ˆx v, ˆx v ) Int X 1 + [36]. Results The order interval [E 2, E 1 ] K plays a distinguished role. As E 1 = (ˆx v, ˆx v, 0, 0), and E 2 = (0, 0, ˆx o, ˆx o ), [E 2, E 1 ] K = [0, ˆx v ] 2 [0, ˆx o ] 2. This order interval is positively invariant and attracts the orbits of all points since ω(x) [E 2, E 1 ] K for all x C 0 [36, p. 199]. In particular, all steady states belong to [E 2, E 1 ] K which implies the coordinates of the the coexistence equilibrium and the boundary equilibria are related as x v ˆx v and x o ˆx o. Let B i = {x : ω(x) = E i } denote the basin ).

26 364 JIMMY P. MOPECHA AND HORST R. THIEME of attraction of E i, i = 1, 2, and B = {x : ω(x) = E} the basin of attraction of E if E exists. We have the following cases (cf. Section 8): 1 Case 3.1. < g 1 v (1/R v) r o go 1 (1/R o ) < r v. Bi-stable competitive exclusion There exists a unique coexistence equilibrium. Both the volvulus and the ochengi boundary equilibrium are locally asymptotically stable. Almost all solutions converge towards one of the boundary equilibria by Theorem 3.4 of [36] and the subsequent remarks. We have bistable competitive exclusion: C = B 1 B 2 S where the pairwise disjoint sets B 1, B 2, and S have the following properties : B 1 and B 2 are open sets and their union is dense in R 4 +; every solution starting in B 1 converges to the volvulus boundary equilibrium, while every solution starting in B 2 converges to the ochengi boundary equilibrium. In other words: O. volvulus outcompetes O. ochengi for initial data in B 1, while O. ochengi outcompetes O. volvulus for initial data in B 2. The basins of attraction B 1 and B 2 contain the following sets, (9.1) [0, x v ] 2 [ x o, ) 2 B 2, [ x v, ) 2 [0, x o ] 2 B 1, x v = g 1 v ( 1 R v ) r v go 1 ( 1 R o ), x o = g 1 o ( 1 R o ) r o gv 1 ( 1 R v ), where x v and x o are the respective volvulus and ochengi coordinates of the coexistence equilibrium E = ( x v, x v, x o, x o ). S is an unordered, positively invariant set containing the disease-free equilibrium, the coexistence equilibrium and its basin of attraction B, and possibly other points. S has Lebesgue measure 0 and is a C 1 manifold [36, Remark 3]. Case 3.2. r v < g 1 v (1/R v) go 1 (1/R o ) < 1. Stable coexistence r o There exists a unique coexistence equilibrium. Both boundary equilibria are unstable. This corresponds to Case 1 in Section 5 of [36]. All solutions starting in C 0 (i.e., both O. volvulus and O. ochengi are present initially) converge towards the coexistence equilibrium. Case 3.3. r v = g 1 v (1/R v ) g 1 o (1/R o ) = 1 r o. Neutral case