Advances in Environmental Biology

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1 Adances in Enironmental Biology, 9() Special 5, Pages: 6- AENSI Journals Adances in Enironmental Biology ISSN EISSN Journal home page: Mathematical Model for the Transmission Dynamics of Denque Feer with Control Strategy Bundit Unyong Department of Mathematics, Faculty of Science and Technology, Phuket Rajabhat Uniersity, Phuket, 8, Thailand A R T I C L E I N F O Article history: Receied February 5 Accepted March 5 Aailable online April 5 Keywords: Denque feer, stability, equilibrium point, basic reproductie number, control strategy. A B S T R A C T Background: In this work the dynamics of transmission of denque feer was proposed and analyzed. The standard method is used to analyze the behaiors of the proposed model. In the present inestigation of dynamical transmission of denque feer, the effect of human accination and the cost of this strategy are taken into account. Results: There were two equilibrium points; disease free and endemic equilibrium point. The qualitatie results are depended on a basic reproductie number (R ). We obtained the basic reproductie number by using the next generation method and finding the spectral radius. Routh-Hurwitz criteria is used for determining the stabilities of the model. If R, then the disease free equilibrium point is local asymptotically stable: that is the disease will died out, but if R, then the endemic equilibrium is local asymptotically stable. Conclusion: We concluded that the infected human and the infected mosquitoes be decreased by the effect of accination. The numerical result of the models are shown and compared for supporting the analytic results. 5 AENSI Publisher All rights resered. To Cite This Article: Bundit Unyong., Mathematical model for the transmission dynamics of Denque feer with control strategy. Ad. Eniron. Biol., 9(), 6-, 5 INTRODUCTION Dengue is a mosquito-borne infection found in tropical and sub-tropical regions around the world. In recent years, transmission has increased predominantly in urban and semi-urban areas and has become a major international public health concern. Seere dengue (also known as Dengue Haemorrhagic Feer) was first recognized in the 95s during dengue epidemics in the Philippines and Thailand. Today, seere dengue affects most Asian and Latin American countries and has become a leading cause of hospitalization and death among children in these regions [] estimated about 5 to million cases reported. Around 5, people are estimated to be infected by hemorrhagic dengue feer each year. Four serotypes of the dengue irus exist and are called DEN, DEN, DEN and DEN4. Infection by one serotype of the irus confirms permanent immunity to further infections by the infecting strain and temporary immunity to the others. The disease is usually found in tropical region of the world. This disease can be transmitted to human by biting of infected Aedes Aegypti mosquitoes [-8]. This mosquito breeds in artificial water containers such as discarded tires, cans, barrels and flower ases, all of which are usually found in the domestic enironment. Dengue feer (DF) is characterized by the rapid deelopment of the illness that may last from fie to seen days with headache, joint and muscle pain and a rash [,6,7,8]. The symptom of dengue patients may occur from four to twele days after exposure to an infected mosquito. Current data suggests that the immune Mathematical models hae become an important tool for understanding the spread and control of disease. Because of this disease is caused by irus, therefore no drug can cure this disease specifically. This paper is organized as follows. In section, we present the dynamics of the model for denque feer. The standard method is used to analyze the behaiors of the proposed model. The analysis of control problem is presented in section. In section4, we gie a numerical appropriate method and the simulation corresponding results. Finally, the conclusions are summarized in section 5. Corresponding Author: Bundit Unyong, Department of Mathematics, Faculty of Science and Technology, Phuket Rajabhat Uniersity, Phuket 8.Thailand. bundit_@yahoo.com; Tel:

2 7 Bundit Unyong, 5 Adances in Enironmental Biology, 9() Special 5, Pages: 6-. Mathematical model: In this paper, we study the transmission of denque feer through mathematical modeling. By using the standard method to analyze the behaiors of the proposed model which was adopted [] by using the effect of accination rate for human and the cost of this strategy is taken into account. The population consist of two groups : human population N and population ector N.We assume that human population and mosquito population are constant. Human population diided into two disease-state compartments: susceptible indiiduals S, people who can catch the disease; infectious (infectie) indiiduals I, people who hae the disease and can transmit the disease. Population ector or mosquitoes N are diided into two groups of mosquitoes: the susceptible mosquitoes population S, and the infected mosquitoes population I. In this study, we assumed that there are numbers of people in the populations that hae already infected by the irus while others hae not. It is also assumed that the transmission of the irus continues in the population but number of mosquitoes as the ector is constant. People and mosquitoes are categorized in one group at a time. Then we obtained the transmission model as shown by a system of ordinary differential equations as follows. ds c( ) N SI S dt N m di c( ) SI ( r) I () dt N m ds c dt N m A SI S di c S I I dt N m Where; N=S I and N S I S(t) is the susceptible human population at time t I(t) is the infected human population at time t N is the total number of human population, is the birth rate of human popuration A is the recruitment rate of mosquito population S (t) I (t) is the susceptible mosquitoes population at time t is the infected mosquitoes population at time t N is the total number of mosquitoes population, c is the biting rate of the mosquito population, is the accination rate of human population, is the cost of accination strategy is the death rate of human population, is the death rate of mosquitoes population, m is the number of other animals that the mosquitoes can feed on, r is the recoery rates of human population At first we hae to check both the inariant region and the positiity of the solutions. Therefore; N+N S I S I, then d(n+n ) ds di ds di.we consider into three cases; such as N = N, N > N dt dt dt dt dt and N < N,for t,then we got N <.Next for the positiity of the solution; Let s consider the first equation; ds c(- )S N - I S ; dt N+m ds c(- ) - ( I )S ; dt N+m ds c(- ) - ( I )dt ; S N+m c(- ) - ( I )dt N+m S(t) S() e.

3 8 Bundit Unyong, 5 Adances in Enironmental Biology, 9() Special 5, Pages: 6- In the same method for I(t), S (t) all t >. We normalize the system of equations () by letting ; and I (t), we can see that the solution set of model system is positie for S I S=, I= and S S S N N N and I I I. Since the total human and mosquito populations are A/ N A/ constant, the time rate of change of the human population is equal to zero This means that the birth rate and the death rate of the human population is equal zero; that is. The total number of mosquito at equilibrium is equal to A,then we get; The system of equations () as fallow: ds Ac( ) ( S) SI dt (N m) di Ac( ) SI ( r) I dt (N m) di cn ( I ) I I dt N m () Analysis of the model:. Basic properties of the model: * * * The equilibrium points for (S,I,I ) are found by setting the right hand side of each equations () equal to zero. We obtained two equilibrium points as follows;.. Disease Free Equilibrium Point (E ) : In the absence of the disease in the community, there are I = and I, then we obtaineds =, So that the disease free equilibrium is E (,,)... Endemic Equilibrium Point (E ) : In case the disease is presented in the community, I > and I * *, we obtained, E (S,I,I ) where; (N m) S, (N m) Ac I ( ) AcI S( ) (N m)( r) I, cni I (c NI )(N m) And; R c c( )NA (N m) (r ) (). Basic Reproductie Number (R ) : We obtained a basic reproductie number by using the next generation method (an den Driessche and Watmough, )[9]. By rewriting the system of equations () in matrix form; dx F(X) V(X) dt (4) Where F(X) is the non-negatie matrix of new infection terms and V(X) is the non-singular matrix of remaining transfer terms. Then we get; Ac ( )SI (-S)+ S (N m) dx Ac ( )SI = I, F(X) =,V(X) = (r+μ)i dt (N m) (5) I cn( I )I I N m And setting;

4 9 Bundit Unyong, 5 Adances in Enironmental Biology, 9() Special 5, Pages: 6- F i(e ) F Xi and V i(e ) V X (6) i for all i, j,,, be the Jacobean matrix of F(X) and V(X) at E. The basic reproductie number (R ) is the number of secondary case generate by a primary infectious case (Andeson and May, 99; an den Driessche and Watmough, ) or basic reproductie number is a measure of the power of an infectious disease to spread in a susceptible population. It can be ealuated through the formula ρ FV. (7) Where FV is called the next generation matrix and ρfv For our model, the Jacobean matrix at E can be written as; Ac ( ) (N m) Ac ( ) F(E ) and V(E (N m) ) ( r) cn N m Hence, ac ( )N Ac ( ) (N m) ( r) (N m) ( r) cn (N m)( r) V (E ) Ac ( )N Ac ( ) FV (N m) ( r) (N m) Thus, Ac ( )N FV E (N m) ( r) Then we get the reproduction number R where; R Ac ( )N (N m) ( r) is the spectral radius of FV. (8). Stability Analysis of the model: In this section, we show the stability of both the disease-free equilibrium and endemic equilibrium point. First we hae to show the system of equations () is locally asymptotically stable. To support this reason we use the disease-free equilibrium point E about the system, the following Jacobean matrix J,is presented. Theorem.: For R <,the disease-free equilibrium of the system of equations (),about the equilibrium point E locally asymptotically stable and unstable wheneer R >.,is Proof: To show the system of equations () is locally asymptotically stable,we use the Jacobean matrix of the system of equations () ealuated at equilibrium point E. Then we get the following Jacobean matrix J, J ( r) cn N m Ac ( ) (N m) Ac ( ) (N m) (9)

5 Bundit Unyong, 5 Adances in Enironmental Biology, 9() Special 5, Pages: 6- The eigenalues of We obtained characteristic equation as follow. Ac ( ) (N m) Ac ( ) J I ( r) (N m) cn N m det J I. J are obtained by soling equation ; () The characteristic equation of the aboe Jacobean matrix is; (λ+μ)(λ +aλ+ b ) = () Where; c NA ( ) a ( r), b ( r) ( N m) From the characteristic equation (), we can see that, and the rest of the two eigenalues are obtained by the Routh - Hurwitz criteria for stability. Next we hae to show that, a > and b > for the pair of eigenalues that we can get from equation (). Here a is clearly positie and if c NA ( ), then we get b >. ( r) ( N m) So that all eigenalues associated to the system of equations () are haing negatie real parts. Thus we concluded that the system of equations is locally asymptotically stable, when R <. Local Stability of Endemic Equilibrium: Next, we hae to show the local asymptotic stability of the system of equations () about an endemic equilibrium point E. Theorem.: If R >, then the system of equations () about an endemic equilibrium point. E is locally asymptotically stable, and unstable when, R <. Proof: For the system of equations (),the Jacobean matrix J at E,is gien by; Ac ( )I Ac ( )S (N m) (N m) Ac ( )I Ac ( )S (N m) (N m) c N ( I ) c NI N m N m J ( r) ( ) () The eigenalues of the J are obtained by soling det J I. Where; Ac ( )I Ac ( )S (N m) (N m) Ac ( )I Ac ( )S J I ( r) (N m) (N m) () cn ( I ) cni ( ) N m N m The characteristic equation of the aboe Jacobean matrix is; det(j λi )= (4) λ +bλ +bλ+b = (5)

6 Bundit Unyong, 5 Adances in Enironmental Biology, 9() Special 5, Pages: 6- Where; ( ) ( ) b = MR R M M, ( M ) MR ( ) ( )( ) b =, M MR M R MR ( M) MR b = M(R -) (8) And cn( ) ( r ), M ( N m) These eigenalues are negatie when the coefficients b, b and b are satisfy the Routh-Hurwitz criteria. ) b > ) b > ) bb b > It is easily to see that ( MR) R( M ) b = M ( M ) MR b = M(R -) > then we consider the next conditions. For the last condition,we hae to show that b b b >.Therefore b,b and R then bb MR b. So that all three conditions satisfied the Routh-Hurwitz condition. We deduce that E is local asymptotically stable when R >.. 4. Numerical simulations: In this section we present the results obtained by soling numerically from the control strategy system; The simulations at endemic state were carried out using MATLAB and the following parameter alues taken from table,with initial conditions ; S().5, I()., I ().5, (). and().. And results show below; Table: Parameters alues used in numerical simulation at endemic state. Parameters Description Value Death rate of human populations.456 day - Death rate of mosquitoes populations.5 day - c The biting rate of the mosquito population.5 day - A The recruitment rate of mosquitoes 4-5 m The number of other animals that the mosquitoes can feed on,. r The recoery rates of human population.48 day - The sufficient rate of correlation from human to ector.75 The sufficient rate of correlation from ector to human. N Number of human populations Stability of disease-free state: We used the alues of the parameters listed in table,and calculated eigenalues, and the basic reproductie number are;.456,.7776,.56andr.8467.therefore all eigenalues are negatie and the basic reproductie number less than one, the equilibrium state will be the disease-free state E (,,). Stability of endemic state: For this state, the effect of human accination ( ) and the cost ( ) of this strategy is taken into account. We change the alue of the recruitment rate of mosquito A form 4 to 5 and keep the other alues of the parameters the same. With these alues,we obtain;.69, i, iandr.. Therefore all eigenalues are negatie and the basic reproductie number greater than one. So that the equilibrium state will be the endemic state E and this state will be locally asymptotically stable. The result at endemic state for each state ariable are show below; (6) (7)

7 INFECTED MOSQUITOES (IV) INFECTED HUMAN SUSCEPTIBLE HUMAN (S) Bundit Unyong, 5 Adances in Enironmental Biology, 9() Special 5, Pages: 6-.8 S & T.7 SWC SWO TIME-DAY Fig. : Represent time series of susceptible indiiduals (S) with control(blue line) and without control(green line). It s show that the number of susceptible human are increased after control.. x - I & T IWO IWC TIME DAY Fig. : Represent time series of infected indiiduals (I) with and without controls. It s show the number of infected indiiduals with control (green line) are decreased rapidly. 5 x - IV & T IVWC IVWO TIME-DAY Fig. : Represent time series of infectious mosquitoes (I ) with and without controls.

8 Bundit Unyong, 5 Adances in Enironmental Biology, 9() Special 5, Pages: 6- It s show that the number of infected mosquitoes (I ) with controls(blue line) are decreased more than without control. 5. Conclusion: In this paper, the modified model for transmission of denque feer with control strategy were proposed and analyzed. To reduce the infected human and the infected ector and the other, minimizing the cost of accination human, the control strategy has been applied. Simulation results indicate that the number of infected human and the infected ector are decreased after we used the control strategy. ACKNOWLEDGEMENT The authors are grateful to the Department of Mathematics, Faculty of Science and Technology and Phuket Rajabhat Uniersity, Thailand for proiding the facilities to carry out the research. REFERENCES [] WHO. Fact sheets: Dengue and dengue haemorrhagicfeer. (4). [] Naowarat, S., Thanon Korkiatsakul and I. Tang,. Dynamical model for determining human susceptibility to denque feer. American Journal of Applied Sciences, 8(): -6. [] Ang, K.C. and Z. Li, 999. Modeling the spread of dengue in Singapore. In Conference Proceedings for The International Congress on Modeling and Simulation 999, Hamilton, New Zealand, : [4] Derouich, M. and A. Boutayeb, 6. Dengue Feer: Mathematical modelling and computer simulation. Applied Mathematics and Computation, 77(): [5] Derouich, M.A., Boutayeb and E.H. Twizell,. A Model of Dengue Feer. Brunel Uniersty, England. pp: -. [6] Pongsumpun, P., 6. Transmission model for dengue disease with and without the effect of extrinsic incubation period. KMITL Sci. Tech. J. Thailand, 6: [7] Pongsumpun, P. and K. Patanarapelert, 4. Infection risk to traelers going to dengue feer endemic regions. Mahidol Uniersity, Thailand, 5: [8] Pongsumpun, P. and I. Tang,. Transmission of dengue hemorrhagic feer in age structured population. Mathematical and Computer Modelling, 7: [9] Van den Driessche, P. and J. Watmough,. Reproductie numbers.