MODELLING AND ANALYSIS OF THE SPREAD OF MALARIA: ENVIRONMENTAL AND ECOLOGICAL EFFECTS

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1 Journal of Biological Systems, Vol. 13, No. 1 (2005) 1 11 c World Scientific Publishing Company MODELLING AND ANALYSIS OF THE SPREAD OF MALARIA: ENVIRONMENTAL AND ECOLOGICAL EFFECTS SHIKHA SINGH,J.B.SHUKLA and PEEYUSH CHANDRA, Department of Mathematics, Indian Institute of Technology Kanpur (U.P.) India Centre for Modelling, Environment and Development MEADOWS Complex, 18-Navsheel Dham, Kanpur , India LNM IIT, Jaipur , India peeyush@iitk.ac.in Received 31 March 2004 Revised 14 September 2004 In this paper, general SIS and SIRS models with immigration of human population for the spread of malaria are proposed and analyzed. Effects of natural as well as human population density related environmental and ecological factors, which are conductive to the survival and growth of mosquito population, are considered. It is shown in both the cases that as the parameters governing environmental and ecological factors increase, the spread of malaria increases. It is also found that due to immigration, this infectious disease becomes more endemic. Keywords: Malaria; Mosquito Population; Immigration. 1. Introduction Vector borne diseases, such as malaria, are a great threat to health of populations in both developed and developing countries. Malaria is caused by parasites such as Plasmodium (P) falciparum, P. vivax, P. malariae and P. ovale. 1,2 It spreads by bites of female mosquito of Anopheline species (Anopheles (An) stephensi, An. dirus, An. gambiae, An. freeborni). It is during the course of blood feeding that the protozoan parasites of malaria is ingested by mosquitoes and later transmitted after reproductions to humans. The phases of the cycle of malaria parasite in mosquitoes (definitive hosts) and humans (intermediate hosts) represent a complex series of processes. 3,4 The production of malarial parasite in human host begins in liver cells and undergoes asexual multiplication in red blood cells by the process of schizogony which produces merozoites that eventually lyse the infected cells and invade other erythrocytes. During this process, certain merozoites develop into sexual forms, the male and female gametocytes. Mosquitoes become infected when they feed and ingest human blood containing mature gametocytes. In Corresponding author. 1

2 2 Singh et al. the mosquito midgut, the macrogametocytes (female) and microgametocyte (male) shed the red blood cell membranes that surround them and develop into gametes. These macrogametocytes mature into macrogametes, which accept microgametes to become zygotes. The zygote then elongates to become ookinete. The ookinete penetrates the midgut epithelium and oocyst development takes place in about 24 hours after the blood meal. The length of time required to go from ookinetes to mature sporozoites in oocyst depends upon the type of malaria parasite, mosquito species, the ambient temperature, the density of infection, etc. Later, sporozoites escape from oocysts to find their way to mosquito salivary glands. Only from there the sporozoites are transmitted to human host when the female mosquito next feeds. 4 6 It may be noted that to be a good vector the Anopheline mosquito must have appropriate environment (natural, manmade as well as physiological) to survive long enough for the parasite to develop in its gut and be ready to infect humans during the next blood meal. The prevailing environment in its habitat should also be conducive for fertilization and breeding to increase its population and thereby infecting more persons due to enhanced contacts. 4,7 9 It may be pointed out here that as mosquito feed on water, nectars, sugar solutions, blood, etc. and breed on store/stagnant water, vegetation and grasses in water, household wastes, etc. the following factors may be important for the spread of malaria: (i) Natural environmental and ecological factors conducive to the survival and growth of mosquito population: Examples are rain, temperature, humidity, vegetation and grasses in water, etc. 4,8 10 (ii) Human population density related factors (human made environment) conducive to the survival and growth of mosquito population. Examples are open drainage of sewage water, water ponds, water tanks, household wastes, hedges and damp parks, etc. 7,11 13 (iii) Demographic factors: Examples are growth of human population due to immigration, living conditions, etc. 14,15 Thus, a more systematic modelling study is required to predict the spread of malaria by considering the above-mentioned environmental and ecological factors. It may be noted that these factors grow naturally as well as due to human population density related factors. Hence in our study, we consider effects of these factors on the spread of malaria through human population density in the model. The modelling and analysis of the spread of malaria has been conducted by many researchers in the past. 1,7,11,16 18 It has been suggested that models for the spread of malaria are similar in mathematical structures to those of gonorrhea epidemic. 7,19,20 However, in these studies the effects of above-mentioned environmental and ecological factors on the spread of malaria have not been considered either directly or indirectly. In this paper, therefore, general SIS and SIRS models are proposed to study effects of above-mentioned natural and human population density related factors

3 Modelling and Analysis of the Spread of Malaria 3 on the spread of malaria. It is assumed here that the mosquito population density is governed by a generalized logistic model, where the per capita growth rate and the modified carrying capacity are functions of human population density and they increase as human population density increases. Further, since immigration of human population increases the population density, this aspect has also been considered in the modelling process. 2. SIS Model of Malaria with Constant Immigration We consider a human habitat which is affected by natural as well as human population density related environmental and ecological factors. We assume that the human population density N 1 (t) is divided into the susceptible class X 1 (t) anhe infective class Y 1 (t). The mosquito (anopheline species) population density N 2 (t) is also divided into two classes, the susceptible class X 2 (t) and the infective class Y 2 (t). Keeping in view the above discussion and by considering the criss-cross interactions between mosquito and human populations, an SIS model with immigration is proposed as follows. 7,20 dx 1 dy 1 dn 1 dx 2 dy 2 dn 2 = A β 1 X 1 Y 2 + νy 1 d 1 X 1 = β 1 X 1 Y 2 (ν + α 1 + d 1 )Y 1 = A d 1 N 1 α 1 Y 1 = s 2 (N 1 )N 2 s 20N 2 2 L 2 (N 1 ) β 2X 2 Y 1 (d 2 + α 2 )X 2 (2.1) = β 2 X 2 Y 1 (d 2 + α 2 )Y 2 = s 2 (N 1 )N 2 s 20N 2 2 L 2 (N 1 ) (d 2 + α 2 )N 2 X 1 + Y 1 = N 1, X 2 + Y 2 = N 2, X 1 (0) > 0, Y 1 (0) 0, X 2 (0) 0, Y 2 (0) 0. In model (2.1), A is the constant immigration rate of human population, d 1 is the natural death rate of human population, β 1 is the disease transmission coefficient involving susceptible humans and infective mosquito population, β 2 is the disease transmissioncoefficientinvolving susceptible mosquito population and infective humans, ν is the recovery rate coefficient of infective class, α 1 is the disease related death rate constant. The constant d 2 is the natural death rate of mosquito population and α 2 is the death rate of mosquito population caused by control measures. The sixth equation in model (2.1) governs the density of mosquito population where s 2 (N 1 ) is the growth rate per capita of its population density such that s 2 (N 1 ) (d 2 + α 2 ) is the intrinsic growth rate as compared to the usual logistic

4 4 Singh et al. model. It is assumed here that this growth rate increases with human population density related factors, so we have, ds 2 (N 1 ) 0, (2.2) dn 1 and s 20 is the growth rate of mosquito population caused by conducive natural factors only. Thus in case s 2 (N 1 ) is independent of human population density then s 2 (N 1 ) s 20. It may be pointed out here that in the case of more favorable natural environmental and ecological factors for the survival and growth of mosquito population, the magnitude of s 20 would be correspondingly large. Similarly, L 2 (N 1 ) is the modified carrying capacity of the mosquito population and its value as compared to the usual logistic model is: [ ] s2 (N 1 ) (d 2 + α 2 ) L 2 (N 1 ). We assume here that this modified carrying capacity also increases with human population density, thus we have, dl 2 (N 1 ) 0, (2.3) dn 1 and L 20 is the modified carrying capacity without any interference with human population. It is again noted here that L 20 increases due to more favorable natural environmental and ecological factors. It is further noted from (2.2) and (2.3) that even in absence of human population related factors, the ( mosquito population density increases in its natural habitat and it tends to L 20 1 d 2+α 2 ) s.thus,whens20 20 increases, this limiting modified carrying capacity increases, causing an increase in mosquito population. Since X 1 + Y 1 = N 1 and X 2 + Y 2 = N 2, the system (2.1) can be reduced to the corresponding system involving the variables Y 1,N 1,Y 2 and N 2 only. It can be seen that the set Ω = [ (Y 1,N 1,Y 2,N 2 ):0 Y 1 N 1 A/d 1, 0 Y 2 N 2 N ] 2 attracts all solutions (initiating in the positive octant) of the reduced system, where N 2 =[{s 2 (A/d 1 ) (d 2 + α 2 )} L 2 (A/d 1 )]/s 20. (2.4) s Equilibrium Analysis In the following, we give results of equilibrium analysis, the details of which are given in Appendix A. Theorem 3.1. The reduced model has three equilibria, which are given below, (i) E 0 =(0,A/d 1, 0, 0) (ii) E 1 =(0,A/d 1, 0, N 2 ) which exists provided s 2 (A/d 1 ) >d 2 + α 2 where N 2 is given by (2.4). (iii) E 2 =(Y1,N1,Y2,N2 β ) exists provided G 0 = 1β 2A N 2 d > 1. 1(d 2+α 2)(ν+α 1+d 1)

5 Modelling and Analysis of the Spread of Malaria 5 It may be noted here that both E 0 and E 1 are disease free equilibria (E 1 is in the absence of any infective population) and E 2 corresponds to persistence of malaria. In view of (2.2) and (2.3), it can be checked that dn2 dn 1 0. Hence mosquito population increases as human population density increases. It is also found that dy 1 dn 2 0. This implies that as the density of mosquito population increases, the number of infectives in human population increases. Thus, it is concluded that conducive natural as well as human population density related environmental and ecological factors enhance the spread of malaria. 4. Stability Analysis The local stability results are stated in the following theorem: Theorem 4.1. The equilibrium E 0 is unstable if E 1 exists. The equilibrium E 1 if exists, is stable if G 0 < 1 and it is unstable if G 0 > 1. TheequilibriumE 2 if exists, is locally asymptotically stable provided the following conditions are satisfied, β 1 β 2 2 (N 1 Y 1 )2 (N 2 Y 2 )2 < Y α 1 Y 2 1 L 2 2(N 1 )(ν + α 1 + d 1 ) 2 (ν + α 1 + d 1 )(d 2 + α 2 ) 2, (4.1) } 2 <β 1 d 1 Y2 s 2 20(N2 Y2 ) 2. { s 2(N 1 )+ s 20N 2 L 2(N 1 ) L 2 2 (N 1 ) (4.2) It is remarked here that the first part of the theorem regarding the equilibria E 0 and E 1 can be obtained by applying Routh Hurwitz criteria. Further, E 2 can be shown to be locally asymptotically stable under above-mentioned conditions by using the following positive definite function, corresponding to the linearized form of reduced system: V = α 1k 1 2β 1 Y (Y 1 Y1 ) 2 + k 1 2 (N 1 N1 ) (Y 2 Y2 ) 2 + k 3 2 (N 2 N2 ) 2, (4.3) 2 where k 1 and k 3 can be chosen in view of the conditions (4.1) and (4.2) (for detailed analysis ). We note here that when the growth rate per capita and modified carrying capacity of mosquito population are independent of human population density related factors (i.e. s 2 (N 1)=0,L 2 (N 1) = 0), the inequality (4.2) is satisfied automatically. This implies that human population density related factors have destabilizing effects on the system. Now, we state the result for nonlinear stability of E 2. Theorem 4.2. In addition to assumptions (2.2) and (2.3), let s 2 (N 1 ) and L 2 (N 1 ) satisfy 0 s 2(N 1 ) p and 0 L 2(N 1 ) q (for some positive constants p and q)

6 6 Singh et al. in Ω, then E 2 is nonlinearly asymptotically stable provided the following inequalities are satisfied: A 2 β 1 β2 2 (N 2 Y 2 )2 < d2 1 Y 2 (ν + α 1 + d 1 )(d 2 + α 2 ) 2 (4.4) { α 1 A 2 L 2 2(N1 )(ν + α 1 + d 1 ) p + s 20 N } 2 2 q L 2 <β 1 s 2 20d 3 1Y2 (N2 Y2 ) 2. (4.5) 20 This can be proved by using the following positive definite functions, corresponding to the reduced system of (2.1), V = α 1k 1 2β 1 Y2 (Y 1 Y1 )2 + k 1 2 (N 1 N1 )2 + 1 ( 2 (Y 2 Y2 ) 2 + k 3 N 2 N2 N2 ln N ) 2 N2 (4.6) where k 1 and k 3 can be chosen in view of the conditions (4.4) and (4.5). It is noted in this case also that when p = q =0, the inequality (4.5) is automatically satisfied. This again implies that human population density related factors have destabilizing effect on this system. The above theorems imply that under certain conditions, if the density of mosquito population increases due to natural as well as human population density related environmental and ecological factors, then the number of infectives in the human population increases leading to fast spread of malaria. They also suggest that such an infectious disease becomes more endemic due to immigration. 5. SIRS Model with Constant Immigration In this section, we propose and analyze a SIRS model governing the spread of malaria with constant immigration. In this case, human population density is divided into three classes: the susceptibles X 1 (t), the infectives Y 1 (t) and the removed class Z 1 (t). The mosquito population density is divided into two classes only as in the case of SIS model, i.e. the susceptibles X 2 (t) and the infective class Y 2 (t). The mosquito population density N 2 (t) is assumed to be governed by the same equation as in the case of SIS model discussed earlier. In view of the above and the considerations in Sec. 2, a mathematical model is proposed as follows, dx 1 = A β 1 X 1 Y 2 + δ 1 Z 1 d 1 X 1 dy 1 = β 1 X 1 Y 2 (ν 1 + α 1 + d 1 )Y 1 dz 1 = ν 1 Y 1 (d 1 + δ 1 )Z 1 dn 1 = A d 1 N 1 α 1 Y 1 (5.1)

7 Modelling and Analysis of the Spread of Malaria 7 dx 2 dy 2 dn 2 = s 2 (N 1 )N 2 s 20N 2 2 L 2 (N 1 ) β 2X 2 Y 1 (d 2 + α 2 )X 2 = β 2 X 2 Y 1 (d 2 + α 2 )Y 2 = s 2 (N 1 )N 2 s 20N 2 2 L 2 (N 1 ) (d 2 + α 2 )N 2 X 1 + Y 1 + Z 1 = N 1 and X 2 + Y 2 = N 2 X 1 (0) > 0, Y 1 (0) 0, Z 1 (0) > 0, X 2 (0) 0, Y 2 (0) 0. In (5.1), ν 1 is the transfer rate coefficient from infective class Y 1 to removed class Z 1 and δ 1 is the recovery rate coefficient of Z 1 joining the susceptible class X 1.The other variables and parameters in (5.1) are the same as defined in the SIS model. Since X 1 + Y 1 + Z 1 = N 1 and X 2 + Y 2 = N 2, the system (5.1) can be simplified to the corresponding reduced system involving the variables Y 1,Z 1,N 1,Y 2 and N 2 only. It can be seen that for this reduced, the set W = {(Y 1,Z 1,N 1,Y 2,N 2 ): 0 Y 1 N 1 A/d 1, 0 Z 1 N 1 A/d 1, 0 Y 2 N 2 N 2 }, attracts all solutions initiating in the positive orthant, where N 2 is given by (2.4). In the following, we give the result of equilibrium analysis. Theorem 5.1. There exist following three equilibria of the reduced system of (5.1): (i) P 0 =(0, 0,A/d 1, 0, 0) (ii) P 1 =(0, 0,A/d 1, 0, N 2 ), which exists provided s 2 (A/d 1 ) >d 2 + α 2, where N 2 is given by (2.4). (iii) P 2 =(Ŷ1, Ẑ1, ˆN 1, Ŷ2, ˆN 2 ) exists provided G 00 > 1 where G 00 = β 1 β 2 A N 2 d 1 (d 2 + α 2 )(ν 1 + α 1 + d 1 ) (Ŷ1, Ẑ1, ˆN 1, Ŷ2, and ˆN 2 ) are given by the set of relations which are obtained by putting the right hand sides of the corresponding reduced system of (5.1) equal to zero). Here also we note that while both P 0 and P 1 are disease free equilibria, P 1 is in the absence of any infective population. Also the equilibrium P 2 corresponds to the endemic nature of malaria. In this case also, dy1 dn 2 0, i.e. the infective human population density increases as the mosquito population density increases. This again shows that conducive natural as well as human population density related environmental and ecological factors enhance the spread of malaria.

8 8 Singh et al. 6. Stability Analysis The local stability results of equilibria are stated in the following theorem: Theorem 6.1. The equilibrium P 0 is unstable if P 1 exists. The equilibrium P 1 if exists, is stable, if G 00 < 1 and it is unstable if G 00 > 1. The equilibrium P 2 (Ŷ1, Ẑ1, ˆN 1, Ŷ2, ˆN 2 ) if exists, is locally asymptotically stable provided the following conditions are satisfied, β 1 β2 2 ( ˆN 1 Ŷ1 Ẑ1) 2 ( ˆN 2 Ŷ2) 2 < 16 45Ŷ2(ν 1 + α 1 + d 1 )(d 2 + α 2 ) 2, (6.1) { α 1 Ŷ 1 L 2 2 ( ˆN 1 )(ν 1 + α 1 + d 1 ) s 2 ( ˆN 1 )+ s 20 ˆN 2 L 2 ( ˆN 1 ) L 2 2 ( ˆN 1 ) } 2 <β 1 d 1 Ŷ 2 s 2 20 ( ˆN 2 Ŷ2) 2. (6.2) The first part of the theorem regarding P 0 and P 1 follows from the Routh Hurwitz criteria. The conditions for local stability of P 2 can be obtained by using the following positive definite function corresponding to the linearized form of (5.1): V = α 1k 1 (Y 1 Ŷ1) 2 + α 1 (Z 1 Ẑ1) 2 + k 1 2β 1 Ŷ 2 ν 1 2 (N 1 ˆN 1 ) (Y 2 Ŷ2) 2 + k 3 2 (N 2 ˆN 2 ) 2, where k 1 and k 3 can be chosen in view of the conditions (6.1) and (6.2). It is pointed out here that condition (6.2) is automatically satisfied if s 2 (N 1 )andl 2 (N 1 ) are independent of N 1. In the following, we give the nonlinear stability result of non-trivial equilibrium P 2 (Ŷ1, Ẑ1, ˆN 1, Ŷ2, ˆN 2 ). Theorem 6.2. In addition to assumptions (2.2) and (2.3), let s 2 (N 1 ) and L 2 (N 1 ) satisfy 0 s 2(N 1 ) p and 0 L 2(N 1 ) q (for some positive constants p and q) in W, then P 2, if exists, is nonlinearly asymptotically stable under the following conditions: A 2 β 1 β2( 2 ˆN 2 Ŷ2) 2 < d2 1Ŷ2(ν 1 + α 1 + d 1 )(d 2 + α 2 ) 2, (6.3) α 1 A 2 L 2 2 ( ˆN 1 )(ν 1 + α 1 + d 1 ) { p + s 20 N 2 q L 2 20 } 2 <β 1 s 2 20 d3 1Ŷ2( ˆN 2 Ŷ2) 2. (6.4) This theorem can be proved by using the following positive definite function corresponding to the system (5.1), V = α 1k 1 ( ) 2 α 1 Y1 2β 1 Ŷ Ŷ1 + (Z 1 2 2ν Ẑ1) 2 + k 1 ( N1 1 2 ˆN ) ) 2 Y2 2( Ŷ2 + k3 (N 2 ˆN 2 ˆN 2 ln N ) 2 ˆN 2 where, k 1 and k 3 can be chosen under conditions (6.3) and (6.4).

9 Modelling and Analysis of the Spread of Malaria 9 In this case also, the above two theorems imply that the spread of malaria increases as the density of the mosquito population (caused by conducive natural as well as human population density related environmental and ecological factors) increases and the disease becomes more endemic due to immigration. It also shows that human population density related factors have a destabilizing effect on the system. 7. Conclusions It is noted here that the conducive environmental and ecological factors are important in the spread of malaria as they are responsible for the following aspects: (i) survival of infected anopheline mosquitoes for a period long enough for the malaria parasite to develop and to infect humans, (ii) growth of anopheline mosquito population caused by accelerated fertilization and breeding. In this paper, therefore, we have proposed and analyzed SIS and SIRS models with immigration for the spread of malaria by considering effects of natural as well as human population density related environmental and ecological factors. In both these cases, it has been shown that as factors conducive to the survival and growth of mosquito population increase, the number of infective human population increases, leading to an increased spread of malaria. It has also been shown that due to immigration the disease becomes more endemic. Acknowledgments The authors are thankful to the reviewers for their constructive suggestions. References 1. Dietz K, Molineaux L, Development and validation of new mathematical models of Plasmodium falciparum malaria, AMVTN Newsl 9:2 3, Eichner M, Diebner HH, Nolineaux L, Collins WE, Jeffery GM, Dietz K, Genesis, sequestration and survival of Plasmodium falciparum gametocytes. Parameter estimates from fitting a model to malaria the reply data, Trans Roy Soc Trop Med Hyg 95: , Das PK, Amalraj DD, Biological control of malaria vectors, Indian J Med Res 106: , Lopez-Antunano, Schmunis (eds.), Diagnosis of Malaria, Prentice-Hall of India, New Delhi, Harlow J, Votava P, Running S, Monitoring and Prediction of Malaria Outbreaks, Numerical Terradynamic Simulation Group, University of Nontana, Nissoula, Manoharan A, Jumblingam A, Das PK, Utility of catalytic models in the estimation of incidence and prevalence of malaria in a hyperendemic situation, South-East Asian JTropMedPublicHealth27: , 1996.

10 10 Singh et al. 7. Bailey NTJ, The Biomathematics of Malaria, Griffin, London, Reiter P, Climate change and mosquito-borne disease, Environ Health Perspect 109(1): , Rogers DJ, Randolph SE, The global spread of malaria in a future warmer world, Science 289: , Ross R, The amount of malaria depend on the number of the carriers, John Nurray, London, Aron JL, May RM, The population dynamics of malaria, in Anderson RM (ed.), Population Dynamics of Infectious Disease, Chapman and Hall, London, pp , Ghosh M, Chandra P, Sinha P, Shukla JB, Modelling the carrier dependent infectious diseases with environmental effects, Appl Math Comput 157: , Lindsay SW, Martens WJM, Malaria in the African highlands: past, present and future, Bull Wld Hlth Org 76(1):33 45, Hethcote HW, The mathematics of infectious diseases, SIAM Rev 42(4): , Singh S, Chandra P, Shukla JB, Modelling and analysis of the spread of carrier dependent infectious diseases with environmental effects, JBiolSyst11(3): , Aron JL, Mathematical modelling of immunity to malaria, Math Biosci 90: , Brauer F, Vanden Driessche P, Models for transmission of disease with immigration of infectives, Math Biosci 171(2): , Molineaux L, Diebner HH, Eichner M, Collins WE, Jeffery GM, Dietz K, Plasmodium falciparum parasitaemia described by a new mathematical model, Parasitology 122: , Hethcote HW, Yorke JA, Gonorrhea transmission dynamics and control, Lecture Notes in Biomathematics, Vol. 56, Springer, Nallaswamy R, Shukla JB, Effects of dispersal on stability of gonorrhea endemic model, Math Biosci 61:63 72, Shukla JB, Agarwal AK, Dubey B, Sinha P, Existence and survival of two competing species in a polluted environment: a mathematical model, JBiolSyst9:1 15, Shukla JB, Agarwal AK, Dubey B, Sinha P, Modelling the effects of primary and secondary toxicants on renewable resources, Nat Resour Model 16:99 122, Shukla JB, Dubey B, Modelling the depletion and conservation of forestry resources: effects of population and pollution, JMathBiol36:71 94, Appendix A The existence of E 0 or E 1 is obvious. To show the existence of non-trivial equilibrium E 2 =(Y1,N 1,Y 2,N 2 ), we proceed as follows: The components of E 2 are given by the positive solution of the following system of equations, which is obtained by substituting X 1 = N 1 Y 1 and X 2 = N 2 Y 2 in (2.1) and putting the right hand sides of the resulting equations to zero: β 1 (N 1 Y 1 )Y 2 (ν + α 1 + d 1 ) Y 1 =0 (A.1) A d 1 N 1 α 1 Y 1 =0 (A.2)

11 Modelling and Analysis of the Spread of Malaria 11 β 2 (N 2 Y 2 )Y 1 (d 2 + α 2 )Y 2 =0 (A.3) s 2 (N 1 ) s 20N 2 L 2 (N 1 ) (d 2 + α 2 )=0 (A.4) Eliminating N 1 from (A.1) and (A.2) and N 2 from (A.3) and (A.4) and after a little simplification, we note that Y1 satisfies F (Y1 d 1 (ν + α 1 + d 1 ) F (Y 1 )= β 1 {A (α 1 + d 1 )Y 1 } β 2 {s 2 (N 1 ) (d 2 + α 2 )} L 2 (N 1 ). s 20 (β 2 Y 1 + α 2 + d 2 ) It is noted that F (0) < 0providedG 0 > 1andF ( ) A α 1+d 1. We also find that F (Y 1 ) > 0in0<Y 1 < A α 1+d 1. Hence, there exists a unique root Y1 1)=0 in 0 <Y 1 < A α 1+d 1. Once, the value of Y1 is known, the values of N1,Y2 and N2 can be calculated from (A.2), (A.3) and (A.4).