Modelling the spread of carrier-dependent infectious diseases with environmental effect

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1 Applied Mathematics and Computation 152 (2004) Modelling the spead of caie-dependent infectious diseases with envionmental effect Mini Ghosh a, *, Peeyush Chanda a, Pawal Sinha a, J.B. Shukla b a Depatment of Mathematics, Indian Institute of Technology, anpu , India b Cente fo Modelling, Envionment and Development, MEADOW Complex, 18-Nav Sheel Dham, anpu , India Abstact Many infectious diseases spead by caies such as flies, ticks, mites, snails, etc. In this pape an SIS model fo caie-dependent infectious diseases, like cholea, diahea, etc. caused by diect contact of susceptibles with infectives as well as by caies is poposed and analyzed assuming the gowth of both the human and the caie populations logistic. It is assumed futhe that the density of caie population inceases with the incease in the cumulative density of dischages by the human population into the envionment. The mathematical model is analyzed fo the following two cases: (i) the ate of cumulative envionmental dischages Q is a constant, and (ii) the ate of cumulative envionmental dischages Q is a function of the population density. This model is analyzed using usual theoy of diffeential equations and compute simulation. By compute simulation it is concluded that if the gowth of caie population caused by conducive household dischages inceases, the spead of the infectious disease inceases. Ó 2003 Elsevie Inc. All ights eseved. eywods: Epidemic model; Caie; Simulation * Coesponding autho. Pesent addess: Depatment of Mathematics, Univesity of Tento, Via Sommaive 14, Tento, Italy. addesses: ghosh@science.unitn.it (M. Ghosh), peeyush@iitk.ac.in (P. Chanda), pawal@iitk.ac.in (P. Sinha) /$ - see font matte Ó 2003 Elsevie Inc. All ights eseved. doi: /s (03)

2 386 M. Ghosh et al. / Appl. Math. Comput. 152 (2004) Intoduction Many infectious diseases spead by caies such as flies, ticks, mites and snails, which ae pesent in the envionment [10 12,21]. Fo example, ai-bone caies o bacteia spead diseases such as tubeculosis and measles; while wate-bone caies o bacteia ae esponsible fo the spead of dysentey, gastoenteitis, diahea, etc. [4,19]. Vaious kinds of household and othe wastes, dischaged into the envionment in esidential aeas of population, povide a vey conducive envionment fo the population gowth of some of these caies [15,18]. This enhances the chance of caying moe bacteia fom infectives to the susceptibles in the population leading to fast spead of caiedependent infectious diseases. Thus unhygienic envionmental conditions in the habitat caused by humans population become esponsible fo the fast spead of an infectious disease. In ecent decades, thee have been seveal investigations of infectious diseases using deteministic mathematical models with o without demogaphic change [2,3,5 9,13,14,16,20]. In paticula Geenhalgh [7] has studied an infectious disease model with population-dependent death ate using compute simulation. Gao and Hethcote [5] analyzed an infectious disease model with logistic population gowth. Zhou and Hethcote [20] have studied a few models fo infectious diseases using vaious kinds of demogaphics. Hethcote [13] has discussed an epidemic model in which the caie population is assumed to be constant. But in geneal the size of the caie population vaies and depends on the natual conditions of the envionment as well as on vaious dischages into it by the human population. Thus in this pape, the effect of vaiable caie population caused by envionmental dischages on the spead of an infectious disease is studied. 2. The model In this pape an SIS model with logistic gowth of human population is consideed so that both the bith as well as the death ates ae density dependent in such a manne that the bith ate deceases and death ate inceases as the population density inceases towads its caying capacity [5]. Hee the population density N ðtþ is divided into two classes: susceptibles X ðtþ and infectives Y ðtþ. It is assumed that all susceptibles living in the habitat ae affected by a caie population of density CðtÞ, which gows logistically with given intinsic gowth ate and caying capacity. The gowth ate of its density is futhe assumed to incease with the incease in the cumulative density of dischages by the human population into the envionment. eeping the above in mind and by consideing simple mass action inteaction, a mathematical model is poposed as follows:

3 M. Ghosh et al. / Appl. Math. Comput. 152 (2004) _X ¼ b a N N d þð1 aþ N X bxy kxc þ my ; _Y ¼ bxy þ kxc m þ a þ d þð1 aþ N Y ; _N ¼ 1 N N ay ; ð1þ _C ¼ sc 1 C dc þ s 1 EC; L _E ¼ QðNÞ E; X þ Y ¼ N; s > d; 0 6 a 6 1; with initial conditions: X ð0þ > 0; Y ð0þ P 0; Nð0Þ > 0; Cð0Þ P 0 and Eð0Þ > 0: Hee EðtÞ is the cumulative density of envionmental dischages conducive to the gowth of caie population; b and d ae the natual bith and death ates; ¼ b d > 0 is the gowth ate constant; is the caying capacity of the human population density in the natual envionment; b and k ae the tansmission coefficients due to the infectives and the caie population espectively; a is the disease elated death ate constant and m is the ecovey ate constant i.e. the ate at which individual ecoves and moves to the susceptible class again fom the infective class. The constant L is the caying capacity of the caie population in the natual envionment; s is its intinsic gowth ate; d is the death ate of caies due to contol measues, whee s > d; s 1 is the pe capita gowth ate coefficient of the caie population due to the cumulative envionmental dischages ate QðN Þ which is human population density dependent (an inceasing function of N) and is the depletion ate coefficient of the envionmental dischages. In witing the model (1), we use the tem tansmission coefficient in the sense as used by Andeson and May [1], which means that new cases of disease occu at the ates bxy and kxc due to inteaction of susceptibles with infectives and caies espectively. Fo 0 < a < 1, the bith ate deceases and the death ate inceases as N inceases to its caying capacity. When a ¼ 1, the model could be called simply a logistic bith model as all of the esticted gowth is due to a deceasing bith ate and the death ate is constant. Similaly, when a ¼ 0, it could be called a logistic death model as all of the esticted gowth is due to an inceasing death ate and the bith ate is constant. It is easy to note that the above model is well-posed in the egion of attaction T 1 given by

4 388 M. Ghosh et al. / Appl. Math. Comput. 152 (2004) T 1 ¼ ðy ; N; C; EÞ : 0 6 Y 6 N 6 ; 0 6 C 6 L QðÞ s d þ s 1 s 0 6 E 6 QðÞ : The model (1) is analyzed fo the following two cases: i(i) the ate of cumulative envionmental dischages Q is a constant, and (ii) the ate of cumulative envionmental dischages Q is a function of the population density. The function QðN Þ is such that it satisfies following conditions: Qð0Þ ¼Q 0 > 0; Q 0 ðnþ P 0; i.e. Q is an inceasing function of N. We conside the fom of QðN Þ as QðNÞ ¼Q 0 þ ln, whee l > 0 is a constant. ; 2.1. Case I: Q is a constant Q a In this case we note fom the last two equations of system (1) that lim sup EðtÞ ¼Q a and lim sup CðtÞ ¼ L Q a s d þ s 1 ¼ C m > 0: t!1 t!1 s Thus to see the global behavio of the system it is easonable to conside the following subsystem of system (1): _Y ¼ bðn Y ÞY þ kðn Y ÞC m m þ a þ d þð1 aþ N _N ¼ 1 N N ay ; Y ; whee C m inceases as the household dischage ate Q a inceases. The esult of an equilibium analysis is stated in the following theoem. Theoem 1. Thee exist the following two equilibia, namely (i) E 1 ð0; 0Þ and (ii) E 2 ðby ; bn Þ, which exists if m þ a þ d > ½ða Þ=ŠkC m. Poof. Existence of E 1 is obvious. The existence of second equilibium point is shown as follows (see Fig. 1): Setting the ight-hand side of (2) to zeo, we get Y ¼ a 1 N N; ð3þ ð2þ

5 M. Ghosh et al. / Appl. Math. Comput. 152 (2004) Fig. 1. Existence of equilibium point. hn by 2 b ð1 aþ o i N ðmþaþd þ kc m Þ Y kc m N ¼ 0: ð4þ It may be pointed out that in the N Y plane (3) gives a paabola with vetex ð=2; =4aÞ and passing though the points ð0; 0Þ and ð; 0Þ, while (4) gives a hypebola with a banch in the fist and fouth quadants and passing though ð0; 0Þ. The two cuves will intesect at a point ðby ; bn Þ povided the slope of paabola at ð0; 0Þ is moe than that of the hypebola banch in fist quadant at ð0; 0Þ, i.e. slope of (4) at ð0; 0Þ is less than that of slope of (3) at ð0; 0Þ, which gives m þ a þ d > a kc m : ð5þ Thus the condition fo the existence of second equilibium point ðby ; bn Þ is poved. Remak 1. Fom (4), dy kc m ¼ > 0; dn m þ a þ d þ kc m ð0;0þ ð6þ which inceases as C m inceases. Remak 2. It is also noted fom (4) that dby =dc m > 0 fo bn P =2, which implies that equilibium infective density inceases as C m inceases. Remak 3. It is noted that if b ¼ð1 aþ=, then (4) gives a paabola and in this case also thee exists a unique positive oot bn in ð0; Þ.

6 390 M. Ghosh et al. / Appl. Math. Comput. 152 (2004) Stability analysis Now we pesent the stability analysis of these equilibia. The local stability esults ae stated in the following theoem. Theoem 2. The equilibium E 1 ð0; 0Þ is unstable and the equilibium E 2 ðby ; bn Þ is locally asymptotically stable povided bby þ k bnc m by! n ð2 bn Þþa b ð1 aþ o Y b þ kc m a > 0: Poof. The vaiational matix M 1 at E 1 ð0; 0Þ coesponding to the system of equations (2) is given by M 1 ¼ kc m ðmþaþdþ kc m : a Since one eigenvalue of M 1 is positive, E 1 is unstable. The vaiational matix bm at E 2 ðby ; bn Þ coesponding to the system of equations (2) is given by 0! bm ¼ b by þ k bnc m Y bby þ kc m ð1 aþ b 1 B by A : a ð 2 bn Þ The chaacteistic polynomial is given by ( ) w 2 þ bby þ k bnc m þ by ð2 bn Þ w þ ( ) þ a bby þ kc m ð1 aþ by ¼ 0: bby þ k bnc m by! ð2 bn Þ In ode that the above quadatic has oots which have negative eal pats, it is necessay that ð7þ bby þ k bnc m by þ ð2 bn Þ > 0 ð8þ and! bby þ k bnc m n by ð2 bn Þþa b ð1 aþ o Y b þ akc m > 0: ð9þ

7 M. Ghosh et al. / Appl. Math. Comput. 152 (2004) The fist inequality is obviously tue in view of the equilibium conditions i.e. (3) and (4). Thus only the second inequality gives the condition fo linea stability of E 2 ðby ; bn Þ. Hence the theoem. Remak. It may be noted that the inequality (9) is satisfied fo bn P =2 and b > ð1 aþ=. The condition bn P =2 is also compatible with the condition of dby =dc m > 0. Hence fom now onwad we assume the above mentioned condition Nonlinea analysis and simulation. Let us take the following Liapunov function: V ¼ Y by by ln Y þ 1 by 2 k 1ðN bn Þ 2 ; ð10þ whee k 1 is to be suitably chosen. Now using the system (2), we get _V ¼ b þ knc m ðy by Þ 2 k 1 Y by ðn þ bn ÞðN bn Þ 2 k 1 a kc m by b 1 a ðy by ÞðN bn Þ: Afte choosing k 1 ¼ 1 kc m þ b ð1 aþ ; a by we note that _V is negative definite in the egion =2 < N 6 such that =2 < bn 6, whee it is assumed that b > ð1 aþ=. Hence E 2 ðby ; bn Þ is globally asymptotically stable in a subegion of T 1. To see the global behavio of the nontivial equilibium and to see the effects of vaious paametes on the spead of the disease, the system (2) is integated by the fouth ode Runge utta method using following set of paametes in the simulation, which satisfies the local stability condition (9) of equilibium E 2. b ¼ 0: ; k ¼ 0: ; m ¼ 0:012; a ¼ 0:0005; a ¼ 0:3; d ¼ 0:0004; ¼ 0:0003; ¼ 50000; C m ¼ : The equilibium values of by and bn ae obtained as by ¼ 7299:645 and bn ¼ 29087:309. Simulation is pefomed fo diffeent initial positions 1, 2, 3, 4 as shown in Fig. 2. In this figue, the infected population is plotted against the susceptible population. Fom the solution cuves, we obseve that the system is globally stable fo this set of paametes, povided that we stat away fom othe equilibia. In Fig. 3, the infective population is plotted against time fo diffeent

8 392 M. Ghosh et al. / Appl. Math. Comput. 152 (2004) Fig. 2. Vaiation of infective population with susceptible population. Fig. 3. Vaiation of infective population with time fo diffeent C m. C m and fom this we obseve that the infective population inceases as C m inceases Case II: Q is a vaiable In this case we conside the following equivalent system of system (1) (using X þ Y ¼ N):

9 M. Ghosh et al. / Appl. Math. Comput. 152 (2004) _Y ¼ bðn Y ÞY þ kðn Y ÞC m þ a þ d þð1 aþ N Y ; _N ¼ 1 N N ay ; ð11þ _C ¼ sc 1 C dc þ s 1 EC; L _E ¼ QðNÞ E ¼ Q 0 þ ln E: The esult of equilibium analysis is stated in the following theoem. Theoem 3. Thee exist the following five equilibia, namely ii(i) E 1 ð0; 0; 0; Q 0 = Þ, i(ii) E 2 ð0; ; 0; Q 0 = Þ, (iii) E 3 ðy ; N ; 0; E Þ, which exists if b > ð1 aþ þ m þ a þ d, whee N ¼ b 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a ð1 aþ þ b 1 a ð1 aþ 2 þ 4 b ðm þ a þ dþ a ; Y ¼ a 1 N N > 0; 2 b a E ¼ QðN Þ ; (iv) E 4 ð0; 0; C; EÞ, whee C ¼ L Q 0 s d þ s 1 ; E ¼ Q 0 ; s i(v) E 5 ðby ; bn ; bc; beþ, which exists if ðm þ a þ dþ > kl ða Þ s s d þ s 1 Q 0 Poof. The existence of the fist fou equilibia is obvious. The existence of the fifth equilibium E 5 is shown as follows. Setting the ight-hand side of system (11) to zeo and simplifying we get Y ¼ a 1 N; ð12þ whee N hn by 2 b ð1 aþ C ¼ L s Q 0 þ ln s d þ s 1 : : o i N ðmþaþd þ kcþ Y kcn ¼ 0; ð13þ

10 394 M. Ghosh et al. / Appl. Math. Comput. 152 (2004) As befoe, we see that in N Y plane (12) is a paabola and (13) is a hypebola unless b ¼ð1 aþ= when it is a paabola passing though oigin and a banch in the fist quadant fo k > 0. Fom (13), we note that the slope " dy ¼ Y k # 1Y þ kc þ klls 1 s ðn Y Þ > 0 fo Y > 0; N > 0; dn ðby 2 þ kcnþ whee k 1 ¼ b ð1 aþ=, assumed positive. Also kl s d þ s 1 s Q 0 dy ¼ dn ð0;0þ m þ a þ d þ kl s ðs dþþks 1 Q 0 > 0: Using these aspects and plotting (12) and (13) in the fist quadant (see Fig. 4), we see that fo the existence of nontivial by and bn, the slope of (12) at ð0; 0Þ must be geate than the slope of (13) at ð0; 0Þ, i.e. ðm þ a þ dþ > kl s ða Þ s d þ s 1 Q 0 ; ð14þ which is same as (6) fo Q 0 ¼ Q a. Thus, afte knowing by and bn, coesponding values of bc and be can be calculated as follows: bc ¼ L s fs d þ s 1 b Eg and be ¼ Q 0 þ l bn : Hee the inequality (14) is the sufficient condition fo existence of the fifth equilibium point E 5 ðby ; bn ; bc; beþ. Fig. 4. Existence of equilibium point.

11 M. Ghosh et al. / Appl. Math. Comput. 152 (2004) Remak. In both the cases when caie population is absent, fo disease to gow we must have a theshold condition as ½b ð1 aþš=ðm þ a þ dþ > 1, which is same as mentioned in [5] Stability analysis Now we discuss the linea stability of these equilibia and nonlinea stability only of the nontivial equilibium E 5. The local stability esults of all equilibia ae stated in the following theoem. Theoem 4. The equilibia E 1, E 2 and E 3 ae unstable. The fouth equilibia E 4 is stable if kc þ m þ a þ d > othewise if and ða Þ kc > ðm þ a þ dþ; ða Þ kc þ m þ a þ d < o kc < ðm þ a þ dþ; it is unstable and the fifth equilibium E 5 exists. The fifth equilibium is locally asymptotically stable povided a 3 a 1 a 3 a a 2 > 0 and 1 a 2 a 0 0 a 3 a 1 > 0; whee a 0, a 1, a 2 and a 3 ae given in the poof of the theoem. Poof. The vaiational matices M 1, M 2, M 3, M 4 and M 5 coesponding to system (11) at equilibium points E 1, E 2, E 3, E 4 and E 5 espectively ae given by 0 1 ðm þ a þ dþ a 0 0 M 1 ¼ Q B s d þ s 1 0 A ; 0 l b fmþaþd þð1 aþg 0 k 0 a 0 0 M 2 ¼ QðÞ B 0 0 s d þ s 1 0 A ; 0 l 0

12 396 M. Ghosh et al. / Appl. Math. Comput. 152 (2004) m 0 11 m 0 12 kn 1 a 1 N 1 0 M 3 ¼ a 2N 0 0 ; B QðN Þ 0 0 s d þ s 1 0 A 0 l 0 whee m 0 11 ¼ b a N 1 N n and m 0 12 ¼ b 0 1 ðkc þ m þ a þ dþ kc 0 0 a 0 0 M 4 ¼ B 0 0 s 1 C C L A 0 l 0 and 0 M 5 ¼ bby þ k bn bc Y ð1 aþ o a l N n b ð1 aþ o Y b þ k bc kð bn by Þ 0 a 2 bn s bc s 1 bc L 0 l 0 N ; Since the matices M 1, M 2 and M 3 have positive eigenvalues so the equilibium points coesponding to these matices ae unstable. The chaacteistic polynomial coesponding to matix M 4 is ðw þ Þ w þ sc fw 2 þðkc þ m þ a þ d Þw L ðkc þ m þ a þ dþþakcg ¼0: Clealy two oots ae negative. Using the Routh Huwitz citeia [17], this equilibium point is locally asymptotically stable if the following conditions ae satisfied:. kc þ m þ a þ d > othewise if kc þ m þ a þ d < and o ða Þ kc > ðm þ a þ dþ; ða Þ kc < ðm þ a þ dþ; 1 : C A ð15þ

13 it is unstable and the fifth equilibium exists as mentioned ealie in condition (14). It is clea fom above that the second inequality in (15) may not be satisfied even if the fist is satisfied, but when the fist inequality is not satisfied, the second will not be satisfied. So violation of any one of the above inequalities gives existence of the fifth equilibium. The chaacteistic polynomial coesponding to matix M 5 is whee w 4 þ a 3 w 3 þ a 2 w 2 þ a 1 w þ a 0 ¼ 0; a 3 ¼ kð bn by ÞC b by þ bn a 2 ¼ M. Ghosh et al. / Appl. Math. Comput. 152 (2004) by þðmþdþ ð bn by Þ þ aby 2 þð1 aþþ bn by bn ð bn by Þ ð bn by Þ þ s C L b þ > 0;! bby þ k bn bc hn by ð2 bn o Þ þ s C L b i þ þ n ð2 s bn Þ C L b o þ þ s Cd L b hn 0 þ a b ð1 aþ o i Y b þ k bc ;! a 1 ¼ bby þ k bn bc by ð2 s bn Þ C L b þ þ ð2 bn Þ s bc L þ bby þ k bn C b! s b! Cd hn 0 by L þ a b ð1 aþ o Y b þ k bci sc b L þ ;! a 0 ¼ bby þ k bn bc by ð2 bn Þ s bc L þ a s bc hn L b ð1 aþ o Y b þ kc b i þ akð bn by Þs 1 bcl: By the Routh Huwitz citeia, conditions fo local stability of the system ae a a 3 > 0; 3 a 1 a 3 a 1 0 a 3 a a 2 > 0; 1 a 2 a 0 0 a 3 a 1 > 0 and 1 a 2 a a 3 a 1 0 > 0: 0 1 a 2 a 0 Clealy the fist inequality is obvious. If the second and the thid inequalities ae satisfied, so is the fouth one. Hence the equilibium point E 5 is locally asymptotically stable if the second and the thid inequalities ae satisfied. Remak. It is seen that second is satisfied fo bn P =2 and b > ð1 aþ=. So in this case only thid inequality is the condition fo local stability of E 5.

14 398 M. Ghosh et al. / Appl. Math. Comput. 152 (2004) Nonlinea analysis and simulation. Hee too we speculate that the nontivial equilibium point E 5 of the model (11) may be globally stable unde the local stability conditions. To illustate this and to see the effects of vaious paametes on the spead of the disease, the system (11) is integated using the fouth ode Runge utta method by taking QðNÞ ¼Q 0 þ ln and using the following set of paametes in the simulation, which satisfies the local stability condition mentioned above. b ¼ 0: ; k ¼ 0: ; m ¼ 0:012; a ¼ 0:0005; d ¼ 0:6; ¼ 0:001; ¼ 0:0003; d ¼ 0:0004; a ¼ 0:3; ¼ 50000; s ¼ 0:9; Q 0 ¼ 20; s 1 ¼ 0:000002; l ¼ 0:00005 and L ¼ : All the paametes ae in units of pe day except the caying capacity L, which has the same dimension as C. The equilibium values of by, bn, bc and be have been found as by ¼ 6121:797; bn ¼ 35716:819; bc ¼ 38174:626; be ¼ 21785:839: In Fig. 5, the infected population is plotted against the susceptible population and fom the solution cuves, it is concluded that the system appeas to be globally stable fo this set of paametes. In Figs the effects of vaious paametes, i.e. Q 0, L, s, s 1, and l on the infective population have been shown. It is noted fom these figues that as these paamete values incease, the infective population inceases and we have simila conclusions egading the spead of the infectious disease as discussed ealie. Fig. 5. Vaiation of infective population with susceptible population.

15 M. Ghosh et al. / Appl. Math. Comput. 152 (2004) Fig. 6. Vaiation of infective population with time fo diffeent cumulative envionmental dischage ates. Fig. 7. Vaiation of infective population with time fo diffeent caying capacities of caie population. Fig. 8. Vaiation of infective population with time fo diffeent intinsic gowth ates of caie population.

16 400 M. Ghosh et al. / Appl. Math. Comput. 152 (2004) Fig. 9. Vaiation of infective population with time fo diffeent gowth ate coefficients of caie population due to the cumulative envionmental dischages. Fig. 10. Vaiation of infective population with time fo diffeent intinsic gowth ates of human population. Fig. 11. Vaiation of infective population with time fo diffeent l.

17 M. Ghosh et al. / Appl. Math. Comput. 152 (2004) Conclusions In this pape an SIS model fo caie-dependent infectious diseases, like cholea, diahea, etc. caused by diect contact of susceptibles with infectives as well as by caies is poposed and analyzed. It is assumed that both the human and the caie populations ae gowing logistically. The density of caie population is futhe assumed to incease with the incease in the cumulative density of dischages by the human population into the envionment. The mathematical model is analyzed fo the following two cases: (i) the ate of cumulative envionmental dischages Q is a constant, and (ii) the ate of cumulative envionmental dischages Q is a function of the population density. Equilibium analysis is pesented fo both the cases and it is seen that the local stability of the nontivial equilibia in both the cases is guaanteed only unde cetain conditions. By compute simulation it is shown that unde local stability conditions, the nontivial equilibium appeas to be globally stable in both the cases. It is concluded fom the analysis that if the gowth of caie population caused by conducive household dischages inceases, the spead of the infectious disease inceases. Also when the gowth ate of the human population inceases due to demogaphic changes, the infectious disease speads even futhe and becomes moe endemic. Refeences [1] R.M. Andeson, R.M. May, Vaccination against ubella and measles: quantitative investigations of diffeent policies, J. Hyg. Camb. 90 (1983) [2] N.T.J. Bailey, Intoduction to the modelling of veneeal disease, J. Math. Biol. 8 (1979) [3] N.T.J. Bailey, Spatial models in the epidemiology of infectious diseases, Lectue Notes in Biomath. 38 (1980) [4] S. Caincoss, R.G. Feachem, Envionmental Health Engineeing in the Topics, John Wiley, New Yok, [5] L.Q. Gao, H.W. Hethcote, Disease tansmission models with density dependent demogaphics, J. Math. Biol. 32 (1992) [6] D. Geenhalgh, Some theshold and stability esults fo epidemic models with a density dependent death ate, Theo. Pop. Bio. 42 (1990) [7] D. Geenhalgh, Some esults fo an SEIR epidemic model with density dependence in the death ate, IMA J. Math. Appl. Med. Biol. 9 (1992) [8] D. Geenhalgh, R. Das, Modelling epidemic with vaiable contact ates, Theo. Pop. Bio. 47 (1995) [9] J. Gonzalez-Guzman, An epidemiological model fo diect and indiect tansmission of Typhoid Feve, Math. Biosci. 96 (1989) [10] D.P. Hay, S.L. ent, Ticks of public health impotance and thei contol, US Depatment of Health, Education and Welfae, Communicable Disease Cente, Atlanta, Geogia, [11] D.P. Hay, S.L. ent, Lice of public health impotance and thei contol, US Depatment of Health, Education and Welfae, Communicable Disease Cente, Atlanta, Geogia, 1961.

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