Modelling the spread of bacterial infectious disease with environmental effect in a logistically growing human population

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1 Nonlinear Analysis: Real World Applications ) Modelling the spread of bacterial infectious disease with environmental effect in a logistically growing human population Mini Ghosh a,,1, Peeyush Chandra a, Prawal Sinha a, J.B. Shukla b a Department of Mathematics, Indian Institute of Technology, anpur, anpur , India b Centre for Modelling, Environment and Development, MEADOW Complex, 18-Nav Sheel Dham, anpur , India Received 14 January 2005; accepted 25 March 2005 Abstract In this paper, an SIS model for bacterial infectious disease is proposed and analyzed where the growth of human population is logistic. It is assumed that disease is transmitted by direct contact of susceptibles with infectives as well as by bacteria. Further it is assumed that bacteria population too is growing logistically in the environment and it s growth is enhanced due to the environmental discharges caused by human sources. The stability of the equilibria are studied by using the theory of differential equation and computer simulation. It is concluded from the analysis that the spread of the infectious disease increases when the growth of bacteria caused by conducive environmental discharge due to human sources increases Elsevier Ltd. All rights reserved. eywords: Epidemic model; Bacteria; Simulation 1. Introduction Bacteria are group of micro-organisms. Some bacteria can cause diseases in human as well as in animal and also in plants. Disease in human caused by bacterial infection includes Corresponding author. Tel.: x 41109; fax: addresses: m.ghosh@massey.ac.nz M. Ghosh), peeyush@iitk.ac.in P. Chandra), prawal@iitk.ac.in P. Sinha). 1 Present address: Institute of Information and Mathematical Sciences, Massey University, Auckland, Albany, Private Bag North Shore Mail Centre, New Zealand /$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi: /j.nonrwa

2 342 M. Ghosh et al. / Nonlinear Analysis: Real World Applications ) diphtheria, gonorrhoea, tuberculosis, typhoid fever, etc. There are many infectious diseases in which infection transmission is caused by direct contact of susceptibles and infectives, while there are some diseases such as tuberculosis, typhoid, etc.) which are also transmitted indirectly by the flow of bacteria from infectives into the environment. There are several papers on mathematical modelling of infectious diseases in general [1 4,9 13]. In particularly, Gonzalez-Guzman [8] analyzed an SIS model for the spread of typhoid by considering the direct as well as indirect transmission with the flow of bacteria from infectives into the environment without considering the explicit equation for bacteria. Feng et al. [6] analyzed the two strain tuberculosis model and described the effects of variable period of latency on the disease dynamics. But now we know that bacteria can grow and survive in large numbers in almost every environment including polluted water in ponds, lakes, rivers, acidic and alkaline waters and also in air and soil [5]. For example the typhie bacteria multiply in milk products and other food wastes. Also cholera bacteria can survive by sheltering beneath the mucus outer coat of various algae and zooplankton and can grow due to the organic pollutants in water. Thus, bacteria can grow in the presence of household discharges into the environment caused by human source. Hence the increase in the population of bacteria in the environment enhances the spread of bacterial infectious diseases in the human population suggesting the need to consider bacterial growth equation explicitly in the formulation of the bacterial disease models. Therefore, here an SIS models for the spread of bacterial infectious diseases are proposed and analyzed by considering the growth of bacterial population caused by environmental discharges. It is assumed that both the human population and the bacterial population are growing logistically in the environment. 2. Mathematical model We consider an SIS model in which logistic growth of human population is assumed Fig. 1). The disease is assumed to spread by infectives directly as well as by flow of bacteria in the environment by infectives. The total population density Nt) is divided into susceptible class Xt) and infective class Yt). It is assumed that all susceptibles living in the habitat are affected by the bacteria population, whose density Bt) in the environment grows logistically with a given intrinsic growth rate and environmental carrying capacity. The growth of the bacteria density is further assumed to increase as the cumulative density of discharges in the environment, by human population, increases. Here both the birth and death rates are density dependent. The birth rate decreases and the death rate increases as the population size increases towards its carrying capacity [7]. The mathematical model is given by the following set of equations, Ẋ = b ar N } N d + 1 a) rn } X βy + λb)x + νy, Ẏ = βy + λb)x ν + α + d + 1 a) rn } Y, 0 a 1,

3 M. Ghosh et al. / Nonlinear Analysis: Real World Applications ) Contribution Interaction Total human population N) Susceptible X) Infective Y) Bacteria B Environmental discharges E) Fig. 1. Schematic illustration of the model. Ṅ = r Ḃ = sb 1 N ) N αy, 1 B L ) + s 1 Y s 0 B + δbe, Ė = QN) E = Q 0 + ln E. 1) Here Et) is the density of cumulative environmental discharges conducive to the growth of bacteria population; b and d are the natural birth and death rates; r = b d>0 is the growth rate constant; is the carrying capacity of the environment corresponding to the human population; β and λ are the transmission coefficients due to the infectives and bacteria population, respectively; α is the disease related death rate constant; ν is the recovery rate constant; s is the intrinsic growth rate of bacteria population; L is the environmental carrying capacity of bacteria population; s 0 is the death rate of bacteria due to control measures; s 1 is the rate of release of bacteria from the infective population in the environment; δ is the rate of growth of bacteria population due to the environmental discharges; Q is the cumulative rate of environmental discharges which may be population density dependent and is its depletion rate coefficient. We assume that s>s 0. In writing the model 1), we assume that new cases of disease occur at rate βxy and λxb due to the interaction of susceptibles with infectives and bacteria, respectively. For 0 <a<1, the birth rate decreases and the death rate increases as N increases to its carrying capacity. When a = 1, the model could be called simply a logistic birth model as all of the restricted growth is due to a decreasing birth rate and the death rate is constant. Similarly, when a = 0, it could be called a logistic death model as all of the restricted growth is due to an increasing death rate and the birth rate is constant. As X + Y = N, we ignore the variable X and find the region of attraction

4 344 M. Ghosh et al. / Nonlinear Analysis: Real World Applications ) T corresponding to 1) as follows: T = Y,N,B,E): 0 Y < r 4α, 0 <N,0 B B max, 0 E Q) } and the model is well posed in the region T, where B max is given by B max = L s s 0 + δ Q) } + s s0 + δ Q) } 2 + 4ss 1. 2s L We analyze the model 1) in the following two cases: i) the cumulative rate of environmental discharges Q is a constant, and ii) the cumulative rate of environmental discharges Q is a function of total population density, which we consider as Q = Q 0 + ln, where Q 0 and l are constants Case I: Q is a constant Q a Since X + Y = N, it is sufficient to consider the following equivalent system of 1), ν + α + d + 1 a) rn } Y, Ẏ = βy + λb)n Y) Ṅ = r 1 N ) N αy, Ḃ = sb 1 B L ) + s 1 Y s 0 B + δbe, Ė = Q a E. 2) To study the stability of the system 2) it is reasonable to reformulate the above system using the asymptotic value of E i.e. E m = Q a / in the above system as follows: ν + α + d + 1 a) rn } Y, Ẏ = βy + λb)n Y) Ṅ = r 1 N ) N αy, Ḃ = sb 1 B L ) + s 1 Y s 0 B + δb Q a. 3) The result of equilibrium analysis is stated in the following theorem. Theorem 2.1. There exist the following four equilibria, namely i) P 1 0, 0, 0) i.e. total extinction, ii) P 2 0, 0,B ), where B =L/s)s s 0 +δq a / )} > 0, i.e. bacteria only, iii) P 3 0,,0) i.e. only human and no bacteria and no infection and iv) P 4 Ŷ, ˆN, ), which is interior with both bacteria and human surviving. For α >r, this

5 M. Ghosh et al. / Nonlinear Analysis: Real World Applications ) exists if α r) ν + α + d> λb. r If α <r then there exists a unique nontrivial equilibrium P 4 under following sufficient conditions α < r 2, k 1 βr ν + α + d) 2 4α < 0. Proof. The existence of the first three equilibria is obvious. We will show the existence of the fourth equilibrium point P 4 by the isocline method. Setting the right-hand sides of 3) to zero, we get the following for N = 0 and. Y = r α N 1 N s s 0 + δ Q a s L B2 [ k1 β N rα N 1 N where k 1 = β 1 a)r. Also 6) can be written as or ), 4) ) B s 1r α 1 N ) N = 0, 5) ) + λb + ν + α + d }] r 1 N ) + λb = 0, 6) α B = βr2 /α 2 2 )N )N 2 +α/βr)k 1 βr/α))n α/βr)ν+α+d)} λr/α)n 1 α/r))} B = βr λα N )N N 1 )N N 2 ) N N, 7) ) where N 1 is the negative and N 2 is the positive root corresponding to the quadratic in the numerator of the above equation and N = 1 α/r)). Also when N = 0, ν + α + d) B = λ1 α/r)) = B 1 say). Clearly in the N B plane, 5) is an ellipse passing through 0, 0) and, 0) with major and minor axes parallel to the coordinate axes and origin at 2, L s s 0 + δ Q )) a. 2s We consider the following two cases: Case I: α >r, i.e. N is negative.

6 346 M. Ghosh et al. / Nonlinear Analysis: Real World Applications ) B N* < N 1 ) B 1 a) b) N* N 1 0, 0) N 2 N 2 N B N* >N 1 ) B 1 a) b) N 1 N* N 0, 0) 2 N 2 N Fig. 2. Existence of equilibrium point when α >r. In this case, a) when k 1 > ν + α + d>0 0 <N 2 < and b) when k 1 is negative or 0 <k 1 < ν + α + d <N 2 <. The graphs of 5) and 7) for both conditions a) and b) are shown in Fig. 2. We note that for the existence of positive intersecting point with N less than carrying capacity, we must have B 1 >B 2, where B 2 is given by 5) when N = 0. We see that B 2 = B. Thus in this case, the condition for existence of ˆN, ) is α r) ν + α + d> λb. r Case II: 0 < α <r, i.e. N is positive. In this case, a) when k 1 > ν + α + d>0 0 <N 2 < and b) when k 1 is negative or 0 <k 1 < ν + α + d <N 2 <. The graphs of 5) and 7) for both conditions a) and b) are shown in Fig. 3. It is noted that the equilibrium value Nˆ 1 >/2 surely, if i) N >/2 and ii) N 2 >/2, which give

7 M. Ghosh et al. / Nonlinear Analysis: Real World Applications ) B N 2 > N* ) a) b) N 1 B1 0, 0) N* N N 2 N 2 B N 2 < N*) N 1 0, 0) N 2 N* N B1 Fig. 3. Existence of equilibrium point when α <r. rise to the following sufficient conditions on the parameters, 0 < α < r 2, k 1 2 ν + α + d) βr 4α < 0. 8) Stability analysis Now we present the stability analysis of these equilibria. The local stability results of these equilibria are stated in the following theorem.

8 348 M. Ghosh et al. / Nonlinear Analysis: Real World Applications ) Theorem 2.2. The equilibrium P 1 is unstable, the equilibrium P 2 is locally asymptotically stable provided α r) r otherwise if α r) r λb > ν + α + d λb < ν + α + d it is unstable. The equilibrium P 3 is unstable and the equilibrium point P 4 is locally asymptotically stable if the condition a 3 > 0 and a 1 a 2 a 3 > 0 are satisfied, where a 1,a 2 and a 3 are given explicitly in the proof of the theorem. It is noted here that existence condition for P 4 is the opposite of the stability condition of P 2 i.e. it exists only when P 2 is unstable. Proof. See Appendix II. Remark. It is easily seen that the local stability conditions are satisfied if ˆN >/2. Hence for numerical purposes, the set of parameters are chosen in such a way that ˆN >/ Nonlinear analysis and simulation The system 3) is integrated by the fourth-order Runge utta method considering the following set of parameters, which satisfy the local stability conditions. β = , λ = , ν = 0.012, α = , d = , a = 0.3, r = , = , s = 1, L= , s 0 = 0.65, s 1 = 10, δ = , Q 0 = 20, = The equilibrium values for this set of parameters are obtained as Ŷ = , ˆN = , = Simulation is performed for different initial positions 1, 2, 3 and 4 as shown in Fig. 4. From this figure, it is clear that this equilibrium may be globally stable provided that we start away from the other equilibria. Also in Figs. 5 9, the variation of infective population is shown for different s, s 1, δ, L and A, respectively. It is concluded that with the increase of these parameters, the infective population increases. It is observed that the parameter s the intrinsic growth rate of bacteria population, the carrying capacity of bacteria L and the growth rate of bacteria due to environmental discharges δ are the key parameters which we need to control to control the infectious disease.

9 M. Ghosh et al. / Nonlinear Analysis: Real World Applications ) E.P. Infective population Susceptible population Fig. 4. Variation of infective population with susceptible population Case II: Q is a variable In this case let us consider following system of equations Ẏ = βy + λb)n Y) ν + α + d + 1 a) rn Ṅ = r 1 N ) N αy, Ḃ = sb 1 B ) + s 1 Y s 0 B + δbe, L } Y, Ė = Q 0 + ln E. 9) The result of equilibrium analysis is stated in the following theorem. Theorem 2.3. There exist the following four equilibria, namely i) E 1 0, 0, 0,Q 0 / ) i.e. total extinction ii) E 2 0, 0,B,Q 0 / ) i.e. bacteria only, where B =L/s)s s 0 +δq 0 / )}, iii) E 3 0,,0,Q 0 +l)/ ) i.e. bacteria and infection free with only human and environmental discharge, and final one iv) E 4 Ŷ, ˆN,,Ê), the interior equilibrium, this exists if ) 4ss 1 r δl 2 αl >. For α >r, we should have an additional condition as ν + α + d>α r)/r)λb. Proof. The existence of each of the first three equilibria is obvious. We present in the following, the proof of the existence of the fourth equilibrium E 4. Setting the right-hand

10 350 M. Ghosh et al. / Nonlinear Analysis: Real World Applications ) Infective population s = 1.5 s = Time in years Fig. 5. Variation of infective population with time for different intrinsic growth rate of bacteria population. sides of 9) to zero, we get the following for N = 0 and, Y = r α N 1 N ), 10) s L B2 s s 0 + δ Q ) 0 + ln B s 1r 1 N ) N = 0, α 11) B = βr2 /α 2 2 )N )N 2 + α/βr)k 1 /) βr/α))n α/βr)ν + α + d)}, 12) λr/α)n 1 α/r))} where k 1 = β 1 a)r. Clearly 11) is an ellipse if ) 4ss 1 r δl 2 αl >, 13) which passes through 0, 0),, 0), 0,B ) and, L/s)s s 0 + δ/ )Q 0 + l)}). Also 12) is the same as 7). Hence as before, plotting 11) and 12) in the N B plane, we will get a positive intersecting point ˆN, ) Figs. 10 and 11) and corresponding to it we get Ŷ and thus we have the fourth equilibrium point E Stability analysis Now we present the stability analysis of these equilibria. The local stability results of these equilibria are stated in the following theorem.

11 M. Ghosh et al. / Nonlinear Analysis: Real World Applications ) Infective popualtion s 1 = 15 s 1 = Time in years Fig. 6. Variation of infective population with time for different growth rate of bacteria population due to infected human population Infective population δ = δ = Time in years Fig. 7. Variation of infective population with time for different growth rate of bacteria due to environmental discharges. Theorem 2.4. The equilibrium E 1 is unstable, the equilibrium E 2 is stable if λb + ν + α + d>r and α r) r λb > ν + α + d

12 352 M. Ghosh et al. / Nonlinear Analysis: Real World Applications ) Infective population L = L = Time in years Fig. 8. Variation of infective population with time for different carrying capacity of bacteria population Infective population r = r = Time in years Fig. 9. Variation of infective population with time for different growth rate of human population. otherwise if λb α r) + ν + α + d<r or λb < ν + α + d r it is unstable. The equilibrium E 3 is unstable and E 4 is locally asymptotically stable provided a 0 > 0, a 3 a 2 a 1 > 0 and a 1 a 3 a 2 a 1 ) a 0 a3 2 > 0, where a 0,a 1,a 2, and a 3 are given

13 M. Ghosh et al. / Nonlinear Analysis: Real World Applications ) B N* < N 1 ) B 1 a) b) N* N 1 0, 0) N 2 N 2 N B N* >N 1 ) B 1 a) b) N 1 N* N 2 0, 0) N 2 N Fig. 10. Existence of equilibrium point for α >r. explicitly in the proof of this theorem. Here too it is noted that E 4 exists only when E 2 is unstable. Proof. See Appendix I. Remark. We note that the second condition is satisfied for ˆN >/2. So for simulation we choose our set of parameters such that ˆN >/ Nonlinear analysis and simulation The system 9) is integrated by the fourth-order Runge utta method using same set of parameters as given in Case I, with Q 0 = Q a and an additional parameter l = , which satisfies the local stability conditions.

14 354 M. Ghosh et al. / Nonlinear Analysis: Real World Applications ) B N 2 > N*) a) b) N 1 B1 0, 0) N* N N 2 N 2 B N 2 < N*) N 1 0, 0) N 2 N* N B 1 Fig. 11. Existence of equilibrium point for α <r. The equilibrium values for this set of parameters are determined as Ŷ = , ˆN = , = , Ê = Simulation is performed for different initial positions 1, 2, 3 and 4 shown in Fig. 12. From this figure, it is clear that this equilibrium is globally stable provided that we start away from

15 M. Ghosh et al. / Nonlinear Analysis: Real World Applications ) Infective population E.P Susceptible population Fig. 12. Variation of infective population with susceptible population Infective population s = 1.5 s = Time in years Fig. 13. Variation of infective population with time for different intrinsic growth rate of bacteria population. the other equilibria. Also effects of various parameters such as s, s 1, δ,l,l and r, onthe infective population are shown in Figs As in previous model, here too the parameter s the intrinsic growth rate of bacteria population, the carrying capacity of bacteria L and the growth rate of bacteria due to environmental discharges δ are important and increase in any of these parameters is causing the increase in the endemic level of infected human population. Also increase in the growth rate of human population causes the increase in the endemic level of infected population means disease spreads fast if population is dense and endemic level of infected human remains high.

16 356 M. Ghosh et al. / Nonlinear Analysis: Real World Applications ) s 1 = 15 s 1 = 10 Infective population Time in years Fig. 14. Variation of infective population with time for different growth rate of bacteria population due to infected human population Infective popualtion δ = δ = Time in years Fig. 15. Variation of infective population with time for different growth rate of bacteria population due to environmental discharges. 3. Conclusions In this paper an SIS model for bacterial infectious diseases caused by direct contact of susceptible with infectives as well as by bacteria is proposed and analyzed. It is assumed

17 M. Ghosh et al. / Nonlinear Analysis: Real World Applications ) Infective population L = L = Time in years Fig. 16. Variation of infective population with time for different carrying capacity of bacteria population l = l = Infective population Time in years Fig. 17. Variation of infective population with time for different l. that both the human and the bacteria population are growing logistically in the environment. It is further assumed that the growth of bacteria population is enhanced due to the presence of conducive environmental discharges caused by human source. This model is analyzed for the following two cases, i) the cumulative rate of environmental discharges Q is a constant, and ii) the cumulative rate of environmental discharges Q is a function of the total human population density. Equilibrium analysis is presented and it is found that in each case the

18 358 M. Ghosh et al. / Nonlinear Analysis: Real World Applications ) Infective population r = r = Time in years Fig. 18. Variation of infective population with time for different growth rate of human population. nontrivial equilibrium is locally symptotically stable under certain conditions. By simulation it is shown that nontrivial equilibrium is globally stable under local stability conditions for that set of parameters. It is concluded from the analysis that if the intrinsic growth rate or the carrying capacity of bacteria population or the growth rate of bacteria due to environmental discharges increases, the endemic level of infected human population increases. So we need to control the growth of bacteria and hence the environmental discharges conducive to growth of bacteria to control the spread of infectious disease. It is noted that if the growth rate of human population increases, the spread of the bacterial disease further increases and it becomes more endemic. Appendix I Proof of Theorem 2.4. The variational matrix M at Y, N, B, E) is m 11 m 12 λn Y) 0 α r 2r M = N 0 0 s 1 0 s s 0 + δe 2s L B δb, 0 l 0 where m 11 = βn 2βY λb ν + α + d + 1 a)r/)n}, m 12 = βy + λb 1 a)r/)y.

19 M. Ghosh et al. / Nonlinear Analysis: Real World Applications ) The variational matrix M 1 at equilibrium point E 1 is given by ν + α + d) α r 0 0 M 1 = s 1 0 s s 0 + δ Q l 0 Here two characteristic roots of the above matrix are positive implying that E 1 is unstable. The variational matrix M 2 at equilibrium point E 2 is given by λb + ν + α + d) λb 0 0 α r 0 0 M 2 = s 1 0 s s 0 + δ Q ) 0 δb. 0 l 0 Here two characteristic roots are and s s 1 + δq 0 / )), and the other roots are given by the following quadratic ψ 2 +λb + ν + α + d r}ψ λb + ν + α + d)r + λb α = 0. Hence using the Routh Hurwitz criteria, the equilibrium E 2 is stable if λb α r) + ν + α + d>r and λb >ν + α + d) r and unstable if λb α r) + ν + α + d<r or λb <ν + α + d). r The variational matrix M 3 at equilibrium point E 3 is given by β ν + α + d + 1 a)r} 0 λ 0 α r 0 0 M 3 = s 1 0 s s 0 + δ Q ) 0 + l 0. 0 l 0 Two characteristic roots of the above matrix are and r, the other roots are given by the following quadratic equation, )} ψ 2 Q0 + l β ν + α + d + 1 ar) + s s 0 + δ ψ δ )} 0 Q0 + l + s s 0 + δ β ν + α + d + 1 ar)} λs 1 = 0. By the Routh Hurwitz criteria the equilibrium E 3 is locally stable if ) Q0 + l β 1 a)r ν + α + d) + s s 0 + δ < 0 δ )} 0 Q0 + l and s s 0 + δ β ν + α + d + 1 ar)} > λs 1

20 360 M. Ghosh et al. / Nonlinear Analysis: Real World Applications ) are satisfied and it is unstable if Q0 + l β 1 a)r ν + α + d) + s s 0 + δ δ 0 Q0 + l or s s 0 + δ ) > 0 )} β ν + α + d + 1 ar)} < λs 1 and E 4 exists. Clearly both conditions are not satisfied simultaneously, hence E 3 is unstable. The variational matrix M 4 at equilibrium point E 4 is given by ) βŷ + λ ˆN βŷ + λ 1 a) r Ŷ Ŷ λ ˆN Ŷ) 0 α r 2r M 4 = ˆN 0 0 ). s s 1 0 L + s 1Ŷ δ 0 l 0 The characteristic polynomial in this case is given by where ψ 4 + a 3 ψ 3 + a 2 ψ 2 + a 1 ψ + a 0 = 0, [ ] a 3 = βŷ + λ [ ˆN + r + 2r ] [ ] Ŷ ˆN s + L + s 1Ŷ + > 0, } a 2 = βŷ + λ ˆN + 2r ˆN Ŷ r + s L + s 1Ŷ ) ) + βŷ + λ ˆN 2r ˆN Ŷ r + s L + s 1Ŷ ) ) 2r ˆN + r s L + s 1Ŷ + α βŷ + λ 1 a) r } Ŷ s 1 λ ˆN Ŷ) )] )] )] = [ βŷ + λ ˆN 2r ˆN + [ Ŷ r s + [ L + s 1Ŷ [ ) )] [ ) )] + βŷ + λ ˆN 2r ˆN Ŷ r 2r ˆN + r s L + s 1Ŷ ) [ + α βŷ + λ 1 a) r Ŷ }] + βŷ + λ ˆN Ŷ s L + βŷ s 1Ŷ + s 1λŶ,

21 M. Ghosh et al. / Nonlinear Analysis: Real World Applications ) ) ) ) ) ) a 1 = βŷ + λ ˆN 2r ˆN Ŷ r + βŷ + λ ˆN 2r ˆN Ŷ r s L + s 1Ŷ ) ) ) ) 2r ˆN + r s L + s 1Ŷ s + L + s 1Ŷ βŷ + λ ˆN Ŷ + α βŷ + λ 1 a) r } ) Ŷ + s L + s 1Ŷ } s 1 λ ˆN Ŷ) + 2r ˆN r, ) a 0 = βŷ + λ ˆN 2r ˆN Ŷ r ) ) s L + s 1Ŷ + α βŷ + λ 1 a) r } Ŷ s s 1 λ ˆN Ŷ) L + s 1Ŷ ) ) 2r ˆN r + αλlδ ˆN Ŷ). By Murata [14], conditions for local stability of the system are a 3 > 0, a a 3 a 1 a 1 a 2 > 0, 3 a a a 2 a 0 > 0, 1 a 2 a 0 0 > 0. 0 a 0 a 3 a 1 3 a a 2 a 0 First inequality is obviously true, so if other inequalities are satisfied, then this equilibrium is locally asymptotically stable. Appendix II Proof of Theorem 2.2. The variational matrix M at Y,N,B)corresponding to the system 3), is given by βn 2Y) λb ν + α + d + 1 a) rn } βy + λb λn Y) M = α r 2r N ) 0. s 1 0 s s 0 + δ Q a 2 s L B The variational matrix M 1 at equilibrium point P 1 0, 0, 0) is ν + α + d) 0 0 M 1 = α r 0 s 1 0 s s 0 + δ Q. a. Clearly one eigenvalue of the above matrix is positive, this equilibrium is unstable.

22 362 M. Ghosh et al. / Nonlinear Analysis: Real World Applications ) The variational matrix M 2 at equilibrium point P 2 is λb + ν + α + d} λb 0 M 2 = α r 0 s 1 0 s s 0 + δ Q ). a Clearly one eigenvalue is negative and the other eigenvalues are given by the following quadratic ψ 2 +λb + ν + α + d r}ψ + αλb rλb + ν + α + d) = 0. So this equilibrium is stable provided α r)/r)λb > ν + α + d. Clearly for α <r, this equilibrium is unstable. The variational matrix M 3 at the equilibrium point P 3 is M 3 = β ν + α + d + 1 a)r} 0 λ α r 0 s 1 0 s s 0 + δ Q a. The characteristic polynomial corresponding to the above matrix is [ r + ψ) ψ 2 s s 0 + δ Q } a + β ν + α + d + 1 a)r) ψ δ 0 + s s 0 + δ Q ) ] a β ν + α + d + 1 a)r) λs 1 = 0. So one root is negative and the other roots have negative real part if β 1 a)r ν + α + d) + s s 0 + δ Q a < 0 14) and also s s 0 + δ Q ) a [β 1 a)r ν + α + d)] > λs 1. 15) Clearly both conditions are not satisfied simultaneously, so this equilibrium is unstable. Now the variational matrix M 4 at equilibrium point P 4 is given by ) βŷ + λ ˆN βŷ + λ 1 a) rŷ λ ˆN Ŷ) Ŷ M 4 = α r 2r ˆN 0. ) s s 1 0 L + s 1Ŷ The characteristic polynomial corresponding to the above matrix is ψ 3 + a 1 ψ 2 + a 2 ψ + a 3 = 0,

23 M. Ghosh et al. / Nonlinear Analysis: Real World Applications ) where a 1 = βŷ + λ ˆN + r Ŷ 2 ˆN ) + s L + s 1Ŷ, ) } } a 2 = βŷ + λ ˆN r Ŷ 2 ˆN ) + s L + s 1Ŷ + r 2 s ˆN ) L + s 1Ŷ } + α βŷ + λ 1 a) rŷ λs 1 ˆN Ŷ), ) ) a 3 = βŷ + λ ˆN s Ŷ L + s 1Ŷ r 2 ˆN ) } ) + α βŷ + λ 1 a) rŷ s L + s 1Ŷ r 2 ˆN )λs 1 ˆN Ŷ). Hence by the Routh Hurwitz criteria, the system is locally stable if a 1 > 0,a 3 > 0 and a 1 a 2 a 3 > 0. However since a 1 > 0, the system is stable if a 3 > 0 and a 1 a 2 a 3 > 0 and it is unstable if a 3 < 0ora 1 a 2 a 3 < 0. References [1] R.M. Anderson, R.M. May, Population biology of infectious diseases: Part I, Nature ) [2] N.T.J. Bailey, The Mathematical Theory of Epidemics, Griffin, London, [3] N.T.J. Bailey, The Mathematical Theory of Infectious Diseases and its Application, Griffin, London, [4] N.T.J. Bailey, Introduction to the modelling of venereal disease, J. Math. Biol ) [5] A.P. Dufour, Disease outbreak caused by drinking water, J. WPCF 54 6) 1982) [6] Z. Feng, M. Ianneli, F.A. Milner, A two strain tuberculosis model with age infection, SIAM J. Appl. Math. 62 5) 2002) [7] L.Q. Gao, H.W. Hethcote, Disease transmission models with density dependent demographics, J. Math. Biol ) [8] J. Gonzalez-Guzman, An epidemiological model for direct and indirect transmission of Typhoid fever, Math. Boisci ) [9] D. Greenhalgh, Some threshold and stability results for epidemic models with a density-dependent death rate, Theor. Pop. Biol ) [10] D. Greenhalgh, Some results for an SEIR epidemic model with density dependence in the death rate, IMA J. Math. Appl. Med. Biol ) [11] H.W. Hethcote, Qualitative analysis of communicable disease models, Math. Biosci ) [12] H.W. Hethcote, H.W. Stech, P. Van den Driessche, Periodicity and stability in epidemic models, in: S.N. Busenberg,. Cooke Eds.), A Survey in Differential Equations and Applications in Ecology, Epidemics and Pollution Problems, Academic Press, New York, [13] R.M. May, R.M. Anderson, Population biology of infectious diseases: Part II, Nature ) [14] Y. Murata, Mathematics for Stability and Optimization of Economic Systems, Academic Press, New York, 1977.