A mathematical study on the spread of Cholera

Transcription

1 South Asian Journal of Mathematics 2014, Vol. 4 ( 2 ) : ISSN RESEARCH ARTICLE A mathematical study on the spread of Cholera Prabir Panja 1, Shyamal Kumar Mondal 1 1 Department of Applied Mathematics, Vidyasagar University, Midnapore ,W.B., India prabirpanja@gmail.com Received: July ; Accepted: September *Corresponding author Abstract In this paper, an epidemic model associated with Vibrio Cholerae has been considered. Here we study the dynamical behaviors of the system in which the interaction of two populations such as (i) Susceptible human, Infected human and Recovered human (ii) Vibrio Cholerae in human intestines and Vibrio Cholerae in the environment have been considered. It is assume that the disease will be spread only introduction of vibrio cholerae of the environment into the Susceptible human. The model exhibits two equilibriums: one is disease free equilibrium and another is endemic equilibrium. It is found that if the basic reproduction number R 0 < 1, the disease free equilibrium is always locally asymptotically stable but the endemic equilibrium does not exist. Again when R 0 > 1, it is shown that only the endemic equilibrium is globally asymptotically stable under certain condition. Finally, we describe how the socioeconomic status parameter play an important role on the spread of Cholera disease. The epidemic model has been controlled by a treatment depending on time and optimize the model by this control parameter in such a way that the disease can be controlled by recovering the infected human and to draw some important conclusion regarding the model. Key Words MSC 2010 Cholera disease; Epidemic model; Treatment; Local stability; Global stability 33B15, 26A48 1 Introduction A highly pathogenic gram-negative bacterium Vibrio Cholerae O1 (classical or EI Tor) is the causative agent of the water-borne diarrheal disease, cholera. Ingestion of the contaminated Water, Food and Feces is mainly responsible for cholera disease transmission. The main symptoms of cholera are watery diarrhea,vomiting, rapid dehydration, metabolic acidosis and hypovolemic shock. The world have acquainted and feared cholera for hundred of years. Several cholera epidemics have occurred worldwide during the 15th to 18th century. During the 19th and 20th centuries, seven cholera panademic have ravaged the humankind. Although there are many recent progresses in Medical sciences, cholera remains now as a global threat in some parts of the World. At first a mathematical model was developed by Citation: Prabir Panja, Shyamal Kumar Mondal, A mathematical study on the spread of Cholera, South Asian J Math, 2014, 4(2),

2 P. Panja, et al: A mathematical study on the spread of Cholera Capasso [16] to describe the dynamics of epidemic of cholera in Italy in It was consisted with two equations to follow the dynamics of Infected individuals and the number of free-living infective stages. More recently Codeco[3] developed a more general model of Cholera with an additional equation in the population. Prevalence of cholera is closely related to poor environmental conditions and lack of basic infrastructure in developing countries. Mathematical models have become more important tools for analyzing the spread of Cholera disease and controlling the procedure. Some mathematical models for different types of control strategies have been used by many researchers in [4, 11, 16]. Optimal control by L. S. Pontryagins [6] is the another most important parts in Mathematics that is used extensively in controlling the spread of infectious disease. It is a powerful method to make decisions involving complex biological situation. It has been a substantial burden in the developing world for decades and it is endemic in Africa, Asia, South and Central America. Severe outbreaks usually occurs in underdeveloped areas with inadequate sanitation, poor hygiene and limited access to safe water supplies. Cholera is a disease with a short incubation period caused by the bacterium Vibrio Cholerae and infection is acquired by ingestion of water and food contaminated with faeces. The organisms do not spread beyond the gastrointestinal tract, where they multiply to very high concentrations in the small and large intestines. Unlike Shigellas, they do not penetrate the epithelial layer but remain adhered to the intestinal mucosa and produces diarrhoea as a result of secretion of an enterotoxin, called choleragen. Cholera is most commonly transmitted through the fecal-oral route via contaminated water or food. Cholera transmission has been linked to contaminated drinking water drawn from shallow unprotected wells, rivers or streams and even to bottled water and ice. Also the consumption of high risk food, impure water and poor sanitation correlate with socio-economic status and poverty to promote cholera transmission. The most striking feature of severe cholera is the voluminous watery stool output, and the dehydration it causes, leading rapidly to hypotension, tachycardia and vascular collapse. The patient become lethargic, with sunken eyes, cheeks and dry mucous membranes. Urine flow is decreased or absent and 60 percent patients are died as a result of severe dehydration and loss of electrolytes. In this paper, we have developed a mathematical model of infectious Cholera disease by transmission of Vibrio Cholerae in a human population in an region. Here we have considered impacts of two populations such as (i) Susceptible human, Infected human and Recovered human (ii) Vibrio Cholerae that growth in human intestines and Vibrio Cholerae in the environment. Here transmission of Vibrio Cholerae microorganism in a human population to spread the Cholera disease, has been occurred through the ingestion of food and contaminated drinking water drawn from shallow, unprotected wells, rivers or streams and even to bottled water and ice, considering the economic status of human in that region. We have studied different equilibrium points and stability behaviors of the proposed model around these equilibrium points. After that we study the impact of socioeconomic status parameter on the spread of the Cholera disease. optimal control carried out by a treatment parameter to get maximum removal of the disease from the Infected human. 70

3 South Asian J. Math. Vol. 4 No. 2 2 Model Formulation In this paper we have considered an epidemic model for disease of Cholera in a human population. Here total human population is divided at time t into three mutually exclusive subpopulation such as Susceptible human (S), Infected human (I) and Recovered human (R). Total bacterial population is divided into two mutually exclusive subpopulations namely Vibrio Cholerae that grows in human intestines (V I ) and Vibrio cholerae in the environment (V E ). We consider that the cholera transmission only through human to environment contact only i.e., contact of susceptible human with the bacterium that grows in the environment. We consider the recruitment rate of susceptible human at time t be A. Susceptible human also increases due to those recovered individuals who lost their natural immunity to cholera at a rate (δ). Susceptible human decreases due to infection and moves to infected compartment and the natural death rate of susceptible at a rate (d 1 ). Infected compartment increases due to inflow of infectives from the susceptible population. Infected compartment decreases due to natural mortality at a rate (d 2 ) and disease related mortality at a rate (m). A proportion (p) of the infected individuals get treatment and their natural recovery rate (γ 1 ) increases by the relative rate of shedding λ. Recovered Individuals increases by the inflow of infected individuals who get recover from cholera infection (either by natural recovery or by recovery due to effect of treatment). Recover class decreases due to natural death rate (d 3 ) and those individuals who losses natural immunity to cholera at a rate (δ). Vibrio cholerae in infected human intestines (V I ) increases at a growth rate (α). It decreases due to natural death at a rate (d 4 ) and also due to human shedding at a rate (η). Those humans who get treated have their relative rate of shedding is reduced by a fraction ψ. Vibrio cholerae in the environment is increases due to natural shedding of the infected human at a rate (η) and decreases due to natural death in the environment at a rate (d 4 ). We consider the cholera transmission to be a bilinear functional interaction between susceptible human population and the vibrio cholerae population in the environment (V E ), which has the following form: T (S, V E ) = SV E 1 + γ (1) where () is the Cholera transmission rate and γ(> 0) is the socio economic status of the susceptible human. ds di = = A d 1 S SV E + δr SV E d 2 I mi [(1 p)γ 1 + pγ 1 λ]i dr = [(1 p)γ 1 + pγ 1 λ]i d 3 R δr dv I = αv I d 4 V I [ψp + (1 p)]ηi dv E = [ψp + (1 p)]ηi d 5 V E (2) The initial conditions are taken as S(0) 0, I(0) 0, R(0) 0, V I (0) 0, V E (0) 0. 71

4 P. Panja, et al: A mathematical study on the spread of Cholera 3 Boundedness of Solution Theorem 1. All solutions of the system (2) are bounded in R 5 +, provided that d 4 > α. Proof. Let us define a function X, as follows: X = S + I + R + V I + V E (3) Now, differentiating (3) with respect to t and simplifying it is obtained that dx = A d 1S (d 2 + m)i d 3 R (d 4 α)v I d 5 V E. (4) Therefore, for each σ > 0 the following inequality is obtained as dx + σx = A (d 1 σ)s (d 2 + m σ)i (d 3 σ) (d 4 α σ)v I (d 5 σ)v E. (5) Then we have where σ = min{d 1, d 2 + m, d 3, d 4 α, d 5 }. Now applying differential inequality, the eq.(6) reduces to the following ( ) A 0 < X σ + ce σt dx Now, taking the limit of above as t we get + σx A (6) 0 < X A σ which implies that the system (2) is bounded and the solution of the system enters into the region Ω = {(S, I, R, V I, V E ) R 5 + : 0 < X A σ }. 4 Basic Reproduction Number and Equilibrium Points The basic reproduction number denoted by R 0 is defined as the average number of secondary infections produced when one infected individual is introduced into a host population where the rest of the population is Susceptible. Now, we calculate R 0 by using next generation operator method used by Driessche and Watmough [10]. First we enumerate the compartments in our model from left to right i.e., Susceptible human(s)=compartment1, Infected human (I)=Compartment2 and so on. Then by this method the new infection generation terms and the remaining transition terms denoted by two matrices F and V are as follows: ( ) ( ) Fi (x) Vi (x) F = and V = x j x=x 0 x j where F i (x) denote the rate of appearance of new infection in compartment i and V i (x) is the net transfer rate (other than infection) of compartment i. In our model there are two stages for transmission x=x 0 72

5 South Asian J. Math. Vol. 4 No. 2 of infection through (i) Infected Persons and (ii) Vibrio Cholerae in the open environment. So here x = (x 2, x 5 ) where x 2 denotes I and x 5 denotes V E. The net transfer rate is given by V i = V i V + i, where V i is the rate of transfer of individuals out of compartment i, and V + i is the rate of transfer of individuals into compartment i by means other than infection i.e., F = S 1 and V = d 2 + m + [(1 p)γ 1 + pγ 1 λ] 0 [ψp + (1 p)]η d 5 Now by this technique, R 0 is the largest(dominant) eigenvalue of the matrix F V 1 where F V 1 = S 1 1 (d 2+m+[(1 p)γ 1+pγ 1λ]) 0 [ψp+(1 p)]η d 5(d 2+m+[(1 p)γ 1+pγ 1λ]) 1 d 5 i.e., F V 1 = S 1[ψp+(1 p)]η ()d 5(d 2+m+[(1 p)γ 1+pγ 1λ]) S 1 d 5() 0 0 Therefore the eigenvalues of the matrix F V 1 can be obtained from the following equation det S 1[ψp+(1 p)]η ()d 5(d 2+m+[(1 p)γ 1+pγ 1λ]) λ S 1 d 5() 0 0 λ = 0 Then the maximum eigenvalue called spectral radius of the matrix F V 1 is given by S 1 η[ψp + (1 p)] R 0 = d 5 (1 + γ)(d 2 + m + [(1 p)γ 1 + pγ 1 λ]) Aη[ψp + (1 p)] = d 1 d 5 (1 + γ)(d 2 + m + [(1 p)γ 1 + pγ 1 λ]) (7) The proposed epidemic model (2) gives the relation between Susceptible,Infected and Recovered humans and Vibrio Cholerae that grows in human intestines and Vibrio Cholerae in the environment. Now, the equilibrium points of the proposed system can be obtained as following two types such as (i) E 0 =(S 1, 0, 0, 0, 0), where S 1 = A d 1, known as a disease free equilibrium, since there is no disease in the population. So in the absence of disease the Susceptible population size approaches to A d 1. (ii) E =(S, I, R, VI, V E ), this equilibrium point is known as endemic equilibrium point. Since S, I, R, VI and VE are in endemic situation then these must be always positive and this will happen when where S > 0, I > 0, R > 0, V I > 0 > 0 and V E > 0 S (d2 + m + [(1 p)γ1 + pγ1λ])(1 + γ)d5 = η[ψp + (1 p)] I (d 3 + δ)[aη[ψp + (1 p)] (d 2 + m + [(1 p)γ 1 + pγ 1λ])(1 + γ)d 1d 5] = η[ψp + (1 p)][(d 2 + m)(d 3 + δ)(1 + γ) + [(1 p)γ 1 + pγ 1λ][d 3(1 + γ) + γ]] R [(1 p)γ1 + pγ1λ]i = d 3 + δ 73

6 P. Panja, et al: A mathematical study on the spread of Cholera V I V E [ψp + (1 p)]ηi = α d 4 = [ψp + (1 p)]ηi d 5 (8) Now since I is feasible i.e., I > 0 so we have R 0 > 1. If R 0 < 1 then the endemic equilibrium point does not exit. Here R 0 is the threshold parameter that determines the existence and local stability of the disease free equilibrium of a compartmental infectious disease model. If R 0 < 1, then there exist a locally asymptotically stable equilibrium. In biology it means that on average an infected individual produces less than one new infected individual over the course of its infectious period. Hence the infection cannot persist and the model will eventually reach a locally stable disease free equilibrium. If R 0 1 then the disease free equilibrium point becomes locally unstable and the infection will persists because each infected individual will spread the disease to at least one susceptible individual on average. Again, we can write the expression of R 0 as follows R 0 = k 1 + γ dr 0 dγ = k (1 + γ) 2 d 2 R 0 dγ 2 = k (1 + γ) 3 > 0 where k = Aη[ψp+(1 p)] d 1d 5(d 2+m+[(1 p)γ 1+pγ 1λ]) > 0, since all the parameters are positive. So it is concluded that the basic reproduction number is decreased if the socio-economic status parameter value increased. It will be a most important factor for controlling the spread of Cholera disease. 5 Stability Analysis of the System The stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical system under small perturbations of initial conditions. More generally, a system is stable if small changes in the hypothesis lead to small variations in the conclusion. Stability means that the trajectories do not change too much under small perturbations. One of the key ideas in stability theory is that the qualitative behavior of an orbit under perturbations can be analyzed using the liberalization of the system near the orbit. In particular, at each equilibrium of a smooth dynamical system with an n-dimensional phase space, there is a certain n n matrix whose eigenvalues characterize the behavior of the nearby points (Hartman- Grobman theorem). More precisely, if all eigenvalues are negative real numbers or complex numbers with negative real parts then the point is a stable attracting fixed point. The stability of the proposed system around each of the equilibrium point have been discussed as follows. Now, the jacobian matrix of the 74

7 South Asian J. Math. Vol. 4 No. 2 system at any point (S, I, R, V I, V E ) is given by d 1 V E 0 δ 0 S V E (d 2 + m + [(1 p)γ 1 + pγ 1 λ]) 0 0 S J = 0 [(1 p)γ 1 + pγ 1 λ] d 3 δ [ψp + (1 p)]η 0 α d [ψp + (1 p)]η 0 0 d 5 Theorem 3. When (d 4 α) > 0 then the disease free equilibrium point E 0 is locally asymptotically stable or unstable according to R 0 < 1 or R 0 > 1. Proof. The jacobian matrix denoted by J E0 of the two population epidemic model (2) at the disease free equilibrium point E 0 ( A d 1, 0, 0, 0, 0) is obtained by replacing the point (S, I, R, V I, V E ) by E 0 (S 1, 0, 0, 0, 0) in jacobian J and then following is obtained as: J E0 = d 1 0 δ 0 S 1 0 (d 2 + m + [(1 p)γ 1 + pγ 1 λ]) 0 0 S 1 0 [(1 p)γ 1 + pγ 1 λ] d 3 δ [ψp + (1 p)]η 0 α d [ψp + (1 p)]η 0 0 d 5 i.e., (9) Now, the characteristic equation of the Jacobian Matrix J E0 E 0 is given by in (9) at its disease free equilibrium point (δ + d 3 + x)(d 4 α + x){x 3 + A 1 x 2 + A 2 x + A 3 } = 0 (10) where (i) A 1 =d 2 + m + [(1 p)γ 1 + pγ 1 λ] + d 1 + d 5 (ii) A 2 =d 1 d 5 + (d 1 + d 5 )(d 2 + m + [(1 p)γ 1 + pγ 1 λ]) (iii) A 3 =(d 5 (d 2 + m + [(1 p)γ 1 + pγ 1 λ])) S1[ψp+(1 p)]η Now, we know that the disease free equilibrium point E 0 will be locally asymptotically stable only if all eigenvalues of the characteristic equation (10) are negative. The eigenvalues of the variational matrix J E0 are x = (d 4 α) and the roots of the biquadratic equation will be negative, by Routh-Hurwith criteria we have A i > 0 for i = 1, 2, 3 and A 1 A 2 > A 3 holds. The above condition will be satisfied if (d 5 (d 2 + m + [(1 p)γ 1 + pγ 1 λ])) S 1[ψp + (1 p)]η 1 + γ S 1 η[ψp + (1 p)] i.e., d 5 (1 + γ)(d 2 + m + [(1 p)γ 1 + pγ 1 λ]) < 1 i.e., R 0 < 1 > 0 75

8 P. Panja, et al: A mathematical study on the spread of Cholera Hence, the system (2) will be locally asymptotically stable at disease free equilibrium point E 0 if (d 4 α) > 0 and R 0 < 1. Similarly, the disease free equilibrium point will be unstable provided R 0 > 1. Theorem 4. The endemic equilibrium point E (S, I, R, VI, V E ) of the system (2) will be locally asymptotically stable provided that B 1 B 2 > B 3, B 4 < (B 1 B 2 B 3 )B 3 /B1 2 and (d 4 α) > 0 holds. Proof. The stability of the system (2) about the endemic equilibrium point E can be analyzed using the characteristic equation of the jacobian J at E (S, I, R, VI, V E ). Now, the characteristic equation at E is given by where (i) B 1 =d 1 + d 2 + d 3 + d 5 + δ + (d 4 α + x){x 4 + B 1 x 3 + B 2 x 2 + B 3 x + B 4 } = 0. (11) V E + m + [(1 p)γ 1 + pγ 1 λ] (ii) B 2 =(d 3 + δ)(d 1 + V E + d 5(d 2 + m + [(1 p)γ 1 + pγ 1 λ]) S η )[ψp + (1 p)] (iii) B 3 =(d 1 +d 3 +δ + V E )[d 5(d 2 +m+[(1 p)γ 1 +pγ 1 λ]) S η[ψp+(1 p)] +(d 3 +δ)(d 1 + V E )(d 2 + d 5 + m + [(1 p)γ 1 + pγ 1 λ])] V E δ [(1 p)γ 1 + pγ 1 λ] 2 S V E η () [ψp + (1 p)] 2 (iv) B 4 =[(d 3 + δ)(d 1 + V E )[d 5(d 2 + m + [(1 p)γ 1 + pγ 1 λ]) S η[ψp+(1 p)] d5δv E [(1 p)γ1+pγ1λ] ] + (d 3 + δ) S η[ψp+(1 p)] ] Using Routh-Hurwith criteria it can be shown very easily that the endemic equilibrium point (E ) is locally asymptotically stable if B 4 < (B 1 B 2 B 3 )B 3 /B 2 1 and (d 4 α) > 0 holds. Otherwise it will be unstable. Theorem 5. There exists an global stability around the disease free equilibrium point E 0 provided that R 0 < 1. Proof. From the system (2) we get di(t) = (F V ) I(t) V E (t) dv E (t) 0 S 1 S 0 0 I(t) V E (t) i.e., di(t) dv E (t) (F V ) I(t) V E (t) provided that (S 1 > S) t 0 in R 5. (12) Now, the system (12) is stable when the eigenvalues of the matrix (F V ) are negative. We have the matrix (F V ) = (d S 2 + m + [(1 p)γ 1 + pγ 1 λ]) 1 η[ψp + (1 p)] d 5 Then the characteristic equation of (F V ) is given by x 2 + x(d 2 + d 5 + m + [(1 p)γ 1 + pγ 1 λ]) + d 5 (d 2 + d 5 + m + [(1 p)γ 1 + pγ 1 λ]) 76

9 South Asian J. Math. Vol. 4 No. 2 Therefore, the roots of the equation are given by S 1[ψp + (1 p)]η 1 + γ x = (d 2 + d 5 + m + [(1 p)γ 1 + pγ 1 λ]) ± 2 (d 2 + d 5 + m + [(1 p)γ 1 + pγ 1 λ]) 2 4(d 5 (d 2 + d 5 + m + [(1 p)γ 1 + pγ 1 λ]) S1[ψp+(1 p)]η ) Hence all eigenvalues will be negative if S 1 [ψp + (1 p)]η d 5 (1 + γ)(d 2 + d 5 + m + [(1 p)γ 1 + pγ 1 λ] < 1 Then the solution of (12) will be (I(t), V E (t)) 0 as t. 2 i.e., R 0 < 1 Then substituting I(t) = 0, V E (t) = 0 in the system (1) then R(t) 0 as t. Thus (S(t), I(t), R(t), V I (t), V E (t)) ( A d 1, 0, 0, 0, 0). Thus if, R 0 < 1 then the system (2) will be Globally asymptotically stable. Theorem 6. There exists global stability around the endemic equilibrium point E provided that R 0 > 1. Proof. Given that R 0 > 1, then there exits surely an endemic equilibrium. Let us construct a Lypunov function Y (t) in the following form Then the time derivative of Y, we have Now, let us assume that = 0. Y = S + I + R + V I + V E (13) dy = A + αv I d 1 S (d 2 + m)i d 3 R d 4 V I d 5 V E (14) q max = d 1 S max + (d 2 + m)i max + d 3 R max + d 4 V Imax + d 5 V Emax where S max, I max, R max, V Imax and V Emax are respectively the largest value of S, I, R, V I and V E provided that they are finite. Also let V Imin =inf V I (t). Then from the above equation it is easily concluded that if α (q max A)/V Imin holds. Then we have dy 0 although Y 0. From this theorem we can concluded that the system (2) will be globally asymptotically stable around its endemic equilibrium point. 6 The Optimal Control of the Proposed Model Optimal control theory is an extension of the calculus of variation and it is a mathematical optimization method for deriving control policies. Optimal control deals with the problem of finding a 77

10 P. Panja, et al: A mathematical study on the spread of Cholera control law for a given system such that a certain optimality criteria is achieved. One of the primary reasons for study model of infectious disease is to control the disease in such a way that ultimately the infection can be eradicated from the population. Now the question is how exactly to control this element in order to produce the best outcome, as determined by some predetermined goal or goals. Since the proposed model is epidemic, so it is very useful to control the disease by this method. In this section we use the optimal control theory to analyze the behavior of the system (2). Our objective is to reduce the infected individuals, the cost of treatment and to increase the number of the treated human among Infected human. For this purpose we use Pontrygin s maximum principal. We first consider the objective function as follows: J(p) = min p tf 0 (C 1 mi(t) + C 2 p(t)i(t) + C 3 p 2 (t)) (15) subject to the equations (2). Here C 1 is the cost of death due to Cholera, C 2 be the cost of antibiotic and rehydration treatment and C 3 be the cost of medicine and hospital admission for severe Cholera causes. The square of the control variable p is taken here to remove the severity of the side effect and overdose of treatment and vaccination respectively. Our objective is to find a control p such that J(p ) = min p P J(p) where P ={p is measurable and 0 p 1 and for t [0, t f ]} is the set of controls. Now, the lagrangian (L) of this problem can be written as L(I, p) = C 1 mi(t) + C 2 p(t)i(t) + C 3 p 2 (t) Again the Hamiltonian (H) for our system can be constructed as follows: H(S, I, R, V I, V E, p, λ 1, λ 2, λ 3, λ 4, λ 5 ) where λ 1, λ 2,..., λ 5 be adjoint variables. = L + λ 1 (t) ds + λ 2(t) di + λ 3(t) dr + λ 4(t) dv I +λ 5 (t) dv E i.e., H = C 1 mi(t) + C 2 p(t)i(t) + C 3 p 2 (t) + λ 1 {A 1 + γ SV E d 1 S + δr} +λ 2 { 1 + γ SV E d 2 I mi [(1 p)γ 1 + pγ 1 λ]i} +λ 3 {[(1 p)γ 1 + pγ 1 λ]i d 3 R δr} +λ 4 {αv I d 4 V I [ψp + (1 p)]ηi} + λ 5 {[ψp + (1 p)]ηi d 5 V E } (16) Theorem 7. There exists an optimal control p for which the optimal solution of the system is (S, I, R, VI, V E ) which minimizes the objective function J(p) over P provided that the adjoint functions λ 1, λ 2, λ 3, λ 4 and λ 5 satisfy the following equations λ 1(t) = λ 1 d γ V E(λ 1 (t) λ 2 (t)) λ 2(t) = C 1 m C 2 p(t) + (d 2 + m + [(1 p)γ 1 + pγ 1 λ])λ 2 (t) λ 3 (t)([(1 p)γ 1 + pγ 1 λ]) 78

11 South Asian J. Math. Vol. 4 No. 2 +(λ 4 (t) λ 5 (t))η[ψp + (1 p)] λ 3(t) = (d 3 + δ)λ 3 (t) λ 1 (t)δ λ 4(t) = (d 4 α)λ 4 (t) λ 5(t) = S 1 + γ (λ 1(t) λ 2 (t)) + λ 5 (t)d 5 and the transversality conditions λ i (t f ) = 0, i = 1, 2, 3, 4, 5. Furthermore, the optimal control p is given by p = min{1, max{0, ((λ 3(t) λ 2 (t))γ 1 (1 λ) + (λ 4 (t) λ 5 (t))η(ψ 1) C 2 )I 2C 3 }} Proof. The adjoint equations and the transversality conditions are obtained from the Pontrygin s Maximum Principle such that λ 1(t) = H S = λ 1d γ V E(λ 1 (t) λ 2 (t)) λ 2(t) = H I = C 1m C 2 p(t) + (d 2 + m + [(1 p)γ 1 + pγ 1 λ])λ 2 (t) λ 3 (t)([(1 p)γ 1 + pγ 1 λ]) +(λ 4 (t) λ 5 (t))η[ψp + (1 p)] λ 3(t) = H R = (d 3 + δ)λ 3 (t) λ 1 (t)δ λ 4(t) = H V I = (d 4 α)λ 4 (t) λ 5(t) = H V E = S 1 + γ (λ 1(t) λ 2 (t)) + λ 5 (t)d 5 The optimal control p can be solved from the optimality conditions H p = 0 i.e., C 2 I(t) + 2C 3 p(t) + λ 2 (t)γ 1 (1 λ)i λ 3 (t)γ 1 (1 λ)i λ 4 (t)(ψ 1)ηI + λ 5 (t)(ψ 1)ηI = 0 i.e., p(t) = [(λ 3(t) λ 2 (t))γ 1 (1 λ) + (λ 4 (t) λ 5 (t))η(ψ 1) C 2 ]I 2C 3 Putting p = p and I = I, we get p = [(λ 3 λ 2 )γ 1 (1 λ) + (λ 4 λ 5 )η(ψ 1) C 2 ]I 2C 3 Since the bounds of p and 0 p 1. Hence optimum control have the following form. [(λ 3 (t) λ 2 (t))γ 1 (1 λ) + (λ 4 (t) λ 5 (t))η(ψ 1) C 2 ]I if 0 < p < 1 2C 3 p = 0 if p 0 1 if p 1 Hence the theorem. 79

12 P. Panja, et al: A mathematical study on the spread of Cholera 7 Numerical Simulation Table 1. Parameter Value Unit References A 10 P erson day 1 Senelani D. et al (2011) (Cell day) 1 Misra, A.K. and Singh,V. (2012) d day 1 Misra, A.K. and Singh,V. (2012) γ day 1 Assumed δ day 1 Misra, A.K. and Singh,V. (2012) d day 1 Misra, A.K. and Singh,V. (2012) m day 1 Misra, A.K. and Singh,V. (2012) γ day 1 Andrews, JR and Basu, S (2011) λ 2.3 day 1 Andrews, JR and Basu, S (2011) α 0.73 day 1 Senelani D. et al (2011) d day 1 Misra, A.K. and Singh,V. (2012) d day 1 Andrews, JR and Basu, S (2011) η 50.0 day 1 Miller Neilan, R. L. et al (2010) ψ 0.52 day 1 Andrews, JR and Basu, S (2011) d 5 1/30 day 1 Andrews, JR and Basu, S (2011) Reproduction Number Reproduction Number Socio economic Status (γ) Fig.1. Reproduction number with different values of socioeconomic status parameter. Using the above parametric values in Table 1, a graph of the reproduction number with respect to socio-economic status parameter has been drawn. From the Fig.1. it is concluded that if the socioeconomic status i.e., the education, health, economic condition of a country are high, then the spread of the Cholera disease will be decreased. The spread of the disease can be controlled so easily if the 80

13 South Asian J. Math. Vol. 4 No. 2 socioeconomic status is high x 10 7 Transmission Rate Transmission Rate Socio economic Status (γ) Fig.2. Transmission rate with different values of socioeconomic status parameter. Again, we also study the impact of socioeconomic status parameter on the transmission of disease from Susceptible human to the Infected human. From the above tabular values a graph has been drawn and it is also seen that the rate of transmission is decreased if the socioeconomic status of a country is increased. Next we solve the optimality of the system by using Rounge-Kutta method of fourth order. First we solve the state equation by forward Rounge-Kutta method of fourth order for the time t f = 100 days starting with an initial guess for the adjoint variables. The we use backward Rounge-Kutta method of fourth order to solve the adjoint variables in the same time interval. For this solution, we consider S(0) = 400, I(0) = 200, R(0) = 400, V I (0) = 500, V E (0) = 300 by taking the time interval [0 100] where t 0 = 0 and t f = 100 days. We compare the results with treatment and without treatment control. It is assume that C 1 =cost of death due to Cholera (= USD per human death), C 2 =cost of antibiotic and rehydration treatment of Infected people (= 2.00 USD per day) and C 3 =cost of treatment of Infected human (= USD per (portion of Infected) 2 ). From Fig.3. it is observed that the number of infected human decreases due to the application of the treatment on the infected human. On the other hand, if we cannot use treatment controls then the number of infected human will be gradually increased. Then the control of the disease will be very difficult. From Fig.4. it is seen that the number of recovered human increases very quickly with time for the application of treatment on the Infected human. In the other case the number of recovered human decreases if we cannot used the treatment controls on the infected human. From Fig.5. it is also seen that Vibrio Cholerae population in the environment increases very much when no treatment control is used on the infected human. In the same time when we use the treatment on the infected human then Vibrio Cholerae population in the environment does not increases quickly. Because some bacteria is killed by antibiotic due to treatment on the infected human. Fig.6. represents the optimal treatment control applied to the infected human to eradicate the cholera 81

14 P. Panja, et al: A mathematical study on the spread of Cholera Infected human With Treatment 60 Time Infected human Without Treatment 200 Time Fig.3. Infected human with and without Treatment. 480 With Treatment 450 Without Treatment Recovered human Recovered human Time 400 Time Fig.4. Recovered human with and without Treatment. disease from the human population. It is observed that it is most effective from the starting period to the thirty days and therefore it decreases slowly and becomes zero at the end of the hundred days. 8 Conclusion In this paper, first we examine the dynamical behaviors of an epidemic system: Cholera disease considering relationship between (i) Susceptible human, Infected human and Recovered human (ii) Vibrio Cholerae in human intestine and Vibrio Cholerae in the environment in a locality of a region. Here we use treatment as control parameter for reducing the Vibrio Cholerae from the infected persons. There are two possible equilibrium points, one is disease free E 0 which is always exists and locally asymptotically stable for R 0 < 1 and other is endemic equilibrium point which exists and locally asymptotically stable for R 0 > 1 where R 0 is the basic reproduction number. Further E 0 will be conditionally global asymptotically stable for R 0 < 1 and in this case the disease vanishes permanently from the system. Again, for R 0 > 1 the endemic equilibrium will also be conditionally global asymptotically stable. Then we study the impact 82

15 South Asian J. Math. Vol. 4 No With Treatment Without Treatment Vibrio Cholerae in the environment Vibrio Cholerae in the environment Time Time Fig.5. Vibrio Cholerae population in the environment Optimal Treatment Time Fig.6. Treatment control. of socioeconomic status on the spread of cholera disease and it is observed that if the socioeconomic status of a country or region is high then the control of the disease will be very easy. A numerical simulation has been given for optimal control to reduce the number of infected persons from the Cholera disease using treatment. From this simulation it is observed that the use of treatment on the infected human can reduce infected person from the cholera disease and the number of vibrio Cholerae in the environment will be decreased. References 1 Misra, A. K., Sharma, A., Shukla, J. B., Modelling and analysis of effects of awareness programns by media on the spread of infectious diseases, Math. Comput. model Misra, A.K., Singh,V.,2012. A delay mathematical model for the spread and control of water borne diseases, J.theor.biol Codeco, C. T.,2001. Endemic and epidemic dynamics of cholera:the role of the aquatic reservoir, BMC Infect.Dis Hartley, D. M., Morris, J.G., Smith,D.L.,2006. Hyperinfectivity: a critical element in the ability of V. cholerae to cause 83

16 P. Panja, et al: A mathematical study on the spread of Cholera epidemics?, PLoS Med Hethcote, H. W.,1976. Qualitative analysis of communicable disease models, Math. Biosci Shukla, J.B., Singh, V., Misra, A.K.,2011. Modeling the spread of an infectious disease with bacteria and carriers in the environment, Nonlinear Anal. RWA. 12(5) Pontryagins, L. S., et al The Mathematical Theory of Optimal Processes, Wiley,New York. 8 Esteva, L., Matias, M.,2001. A model for vector transmitted diseases with saturation incidence, J. Biol. Sys. 9 (4) Pascual, M., Bouma, M.J., Dobson, A.P.,2002. Cholera and climate: revisiting the quantitative evidence, Microb. Infect Ghosh, M., Chandra, P., Sinha, P., Shukla, J.B.,2005. Modeling the spread of bacterial disease: effect of service providers from an environmentally degraded region, Appl. Math. Comput Driessche, P., Watmough, J.,2002. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci Miller Neilan, R. L.,et al Modelling Optimal Intervention Strategies for Cholera, Bull.Math.Biol Lenhart, S., Workman, J. T., Optimal control Applied to Biological Model, Mathematical and Computational Biology Series Chapman and Hall/CRC. 14 Hove-Mosekwa, S. D. et al., Modelling and analysis of the effects of malnutrition in the spread of Cholera, Math.Comput.Model Singh, S., Chandra, P., Shukla, J.B.,2003. Modeling and analysis of the spread of carrier dependent infectious diseases with environmental effects, J. Biol. Sys. 11(3) Sardar, T., et al An optimal cost effectiveness study on Zimbabwe cholera seasonal data from ,PLOS ONE.8(12). 17 Capasso, V., Paveri-Fontana, S.L., A mathematical model for the 1973 cholera epidemic in the European Mediterranean region, Rev. Epidemiol. Sante Publ Wang, W.D., Zhao, X.Q., Threshold dynamics for compartmental epidemic models in periodic environments, J. Dyn. Differ. Equat. 20(3) Andrews J.R., Basu. S., Transmission Dynamics and Control of Cholera in Haiti: An Epidemic Model. Lancet 377(9773):