On Stability Equilibrium Analysis of Endemic Malaria

Transcription

1 IOSR Journal of Applied Pysics (IOSR-JAP) e-iss: Volume 5, Issue (o. - Dec. ), PP 7- On Stability Equilibrium Analysis of Endemic Malaria I.I Raji, A.A.Abdullai and M.O Ibraim Department of Matematics and Statistics, Federal Polytecnic, Ado-Ekiti, igeria. Department of Matematics, Uniersity of Ilorin, Ilorin, kwara state, igeria. Abstract:Tis paper considers te stability equilibrium of malaria in a arying population. We establised te Disease free equilibrium and te endemic equilibrium and carried out te stability analysis of te Disease free equilibrium state (te state of complete eradication of malaria from te population). Keywords: malaria, matematical model, disease-free equilibrium, endemic equilibrium, basic reproduction number. I. Introduction Seeral insects are known to be ectors of uman disease. Mosquitoes in particular enjoy te questionable onour of aing been te first insects to be associated wit te transmission of a disease. Malaria is an infectious disease caused by te Plasmodium parasite and transmitted between mosquitos.in 898, te Italian Zoologist G.B. Grassi and is co-workers first described te complete cycle of te uman malaria parasites and pointed to a species of te genus Anopeles as responsible of malaria transmission. Of more tan 48 species of anopeles, only about 5 species transmit malaria. Te abits of most of te anopeles mosquitoes ae been caracterized as antropopagic (prefer uman blood meal), endotermic (bite indoors), and nocturnal (bite at nigt) wit peak biting at midnigt, between pm and am. Te epidemiology of malaria in a gien enironment is te result of a complex interplay among man, plasmodia, and anopeline mosquitoes. Tese tree elements ae to be present for malaria transmission to occur in nature. Te incidence of malaria as been growing recently due to increasing parasite drug- resistance and mosquito insecticide resistance. It remains one of te most difficult epidemiological, parmacological and immunological callenges in sub-saara Africa. Its influence as been profound and sustained oer te centuries and malaria remains a major cause of morbidity and mortality. Moreoer, te epidemiological situation appears to be canging for te worse in many countries (Offosu, Amaa, 99). An estimated 4% of te world s populations lie in malaria endemic areas. Eac year to 5 million people deelop malaria. It kills about 7, to.7 million people a year 75% of wom are Africa cildren Malaria is te nint largest cause of deat and disability globally.malaria episodes lead to direct cost associated wit ealt care and ealt care related transport and indirect costs associated wit lost income for patients, lost income for careers and scool days missed by cildren. In areas were malaria transmissions are ig, te igest number of cases is concentrated among young cildren. Howeer, malaria illness affects all age groups wic leads to a iger loss of income and decreased productiity [8]. We deelop a matematical model to better understand te transmission and spread of malaria. Tis model is used to determine wic factors are most responsible for te spread of malaria. II. Model Formulation We formulate a model similar to tat of [,] describing te transmission of malaria.te equations for model is as described below ds P R S de S i E di ie I qi dr qi R 7 Page

2 ds V de S i di ie I S E. Table : Te state ariables for te malaria model equation. Parameter Description S (t) - umber of susceptible uman osts at time t E (t) - umber of exposed uman osts at time t I (t) - umber of infections uman osts at time t R (t) - umber of recoered uman wit temporary immunity of time t S (t) - umber of susceptible mosquito ectors at time t E (t) - umber of exposed mosquito ectors at time t (t) - Total uman population (t) - Total mosquito population Parameter P i Q V i Table II: Model parameters and teir interpretations for te malaria model equation. Description Birt atural deat rate for umans Force of infection for umans from susceptible state to exposed state. Incident rate for umans Treatment of infected umans Disease- induced deat rate for umans Rate of loss of immunity for umans Recruitment rate of mosquitoes Force of infection of mosquitoes from susceptible state to exposed state. Incident rate for mosquitoes atural deat rate for mosquitoes TOTAL POPULATIO DESITY Te total population sizes are Probability of transmission of infection from an infectious mosquito to a susceptible uman proided tere is a contact. Probability of transmission of infection from an infectious uman to a susceptible mosquitoes proided tere is a contact also d = S + E + I + R = S + E + I d = P μ η I = V φ. wereδ = ᴧ, δ = wereᴧ denotes te rate at wic te susceptible uman S, become infected by infectious female Anopeles mosquitoes I and represents te rate at wic te susceptible mosquitoes S are infected by infectious uman I. Te model is analysed to examine te critical factors wic determine te persistence or eradication of malaria. We ence determine if te mode is well posed. Teorem (.): Te solution of (.) are feasible for all t> and is defined in te subset [, ) of R Page

3 Proof: Let On Stability Equilibrium Analysis of Endemic Malaria 7 = (S, E, I, R, S, E, I )εr + be any solution of te system (.) wit non-negatie initial conditions. In absence of te disease (malaria), I =, d P μ d + μ P d e μ t Pe μ t. Were c is a constant of integration e μ t Peμ t + c μ P + ce μ t μ at t= P + c μ P μ c Wit P + ( μ P )e μ t. μ P μ att Terefore, as t in (.), te uman population. approaces P, were P = kcalled te carrying capacity. μ μ Hence all feasible solution set of te uman of te uman population of te model (.) enter te region. = S, E, I, R εr 4 + : S >, E, I, R and P In like manner, te feasible solutions set of te mosquito population enters te region = S, E, I εr + : S >, E, I, a V φ ds dx = P + εr δ S μ S δ S μ S ds δ dx S + μ S δ S μ S ds δ + μ S dus δ + μ t + C S e δ +μ t+c e δ +μ e C S t Ae δ +μ t Att = S A S e δ +μ t S t S e δ +μ t Also te remaining equations of system (.) are all positie fort >. Hence, te domain is positiely inariant, because no solution pats leae troug any boundary. Since pats cannot leae, solutions exist for all positie time. Tus te model is matematically and epidemiologically well-posed. Disease free equilibrium: We now sole te model equations to obtain te equilibrium states. At te equilibrium state in te absence of te disease, we ae E =I = E =I =. From (.) P μ = μ 9 Page

4 = P -.5 μ Also V φ = = V -.6 φ Terefore, te disease free equilibrium point of te malaria mode (.) is E o = S, E, I, R, S, E, I = ( P μ,,,, V ψ,,). Tis is te state in wic tere is no infection (in te absence of malaria) in te society. THE BASIC REPRODUCTIO RATIO R o Tis is te expected number of secondary cases, produced, in a completely susceptible population, by a typical infected indiidual during its entire period of infectiousness, and matematically as te dominant eigen alue of positie linear operator. In te original parameters R o = Vi i μ Pψ i +μ q+μ+η (i +ψ )ψ LOCAL STABILITY OF THE DISEASE-FREE EQUILIBRIUM Teorem.: Te disease-free equilibrium point is locally asymptotically stable if R o < and is unstable if R o >. Proof: Te matrix of te system of equation is μ i i μ + η + q q μ ε p φ i i ψ Te Jacobian matrix is gien as (A I) T + T J = T T 4 T 5 T 6 + = T T 7 + T 8 T 9 T = μ + i, T = ᴧ φ, T = i, T 4 = μ + η + q T 5 = q, T 6 = μ ε, T 7 = ψ + i, T 8 = i V T 9 = ψ, T = p From te tird column, wic as a diagonal entry, one of te eigenalues is T 6 +. Te matrices terefore reduces to Page

5 J = T + T T T 4 + = T T 7 + T 8 T 9 + wic forms te caracteristics equation T + T 4 + T 7 + T 9 + T T T 8 T = = T T 4 + T + T 4 + T 7 T 9 + T 7 + T 9 + T T T 8 T = 4 + γ + γ + γ + γ = For γ = T + T 4 + T 7 + T 9 γ = T T 7 + T T 9 + T 4 T 9 + T 7 T 9 + T T 4 γ = T T 4 T 7 + T T 4 T 9 + T T 7 T 9 + T 4 T 7 T 9 + T 4 T 7 γ = T T 4 T 7 T 9 T T T 8 T To ealuate te signs of te roots of, we use te Rout Hurwitz criterion and Descartes Rule of sign. Teorem.: Rout-Hurwitz Criteria Gien te polynomial P = n + γ n + γ n + γ n were te coefficients γ i are real constants, i =,, n; define te n Hurwitz matrices is equal to te number of sign canges γ i of te caracteristic polynomial. F o, F, F, 5 4 F, F 4 were γ i = ifi > n. All of te roots of te polynomial P are negaties or ae negatie real parts if and only if te determinants of all Hurwitz matrices are positie: det F j >, j =,,.., n. We sow tat wen R o <, all te coefficients, γ i, of te caracteristics equation andf j are positie. From te caracteristic equation, wit n = 4, te Rout Hurwitz criteria are γ >, γ >, γ >, γ 4 > anddet F = γ >, detf = γ γ >, 5 4 det F and = γ γ γ γ > i.e. = γ γ γ > = γ γ γ γ γ 4 γ > = γ γ γ γ γ γ 4 > Since all te determinants of te Hurwitz are positie, wic implies tat all te Eigen alues of te Jacobian matrix ae negatie real part. Hence disease free equilibrium point is asymptotically stable and R o >. Page

6 Te Endemic Equilibrium Point Endemic equilibrium points are steady state situations were te disease persist in te population (all state ariables are positie). Te endemic equilibrium of te model is gien as E = S, E, I, R, S, E, I >. Consider equation * * * * * ds = P + εr I S μs de = I S μ E i E = di = i E μ + ᴧ I qi = dr = qi μ R εr = de = I = i E ψ ds = ψ S I S I S ψ E i E = di = i E ψ E = from i E ψ I = ψ I = i E (i) I = Also E ψ +i = I E I S = (ii) i Substitute (ii) and (i) I I S = (iii) i from ds S (ψ + = ψ S I I S ^φ φi S μ S ) = = S = (i) I Substitute (i) in (iii) i I (i) i I III. Conclusion Page

7 From our analysis, we found tat te disease free equilibrium is stable wen te tresold parameter is less tan unity. Howeer, te disease could be reduced if te contact rate is aoided. Tis could be acieed by te use of insecticide bed treated net and or indoor spraying.. References []. Herbert.W. Hetcote (). Te Matematics of Infectious Diseases. Society for Industrial and Applied Matematics. Vol.4(4), []. Ibraim, M.O. and Egbetade, S.A. (). A Matematical Model for te Epidemiology of Tuberculosis wit estimate of te basic reproduction number. Arcimedes Journal Mats. Vol.. []. Jibril. L.and Ibraim, M.O. (). Matematical Modeling on te CDTI Prospects for elimination of Oncocerciasis: A deterministic model Approac. Matematics and Statistics. Vol.(4), 6-4. [4]. Jon, G. Ayisi, Anna. M.VanEijk, Robert D. ewman (). Maternal Malaria and Perinatal HIV Transmission, Western Kenya Centers for Disease Control and Preention. Vol.4(). [5]. JulienArino, Connell. C. Mccluskey, and Van Den Driessce (). Global results for an epidemic model wit Vaccination tat exibits Backward Bifurcation. Society for Industrialand Applied Matematics. 64(), [6]. Kakkilaya, B.S. (). Can Cild Be Affected by Moter s Malaria. Malariasite.com. [7]. Kakkilaya, B.S. (). Transmission of malaria. Malariasite.com [8]. Kakkilaya, B.S. (). Pregnancy and Malaria. Malariasite.com [9]. Labadin. J., C. Kon, M.L and Juan, S.F.S.(9). Deterministic Malaria Transmission Model wit Acquired Immunity. Proceedings of te World Congress on Engineering and Computer Science. Vol., - []. Lawi, G.O, Mugisa, J.Y.T and Omolo-ongati, (). Matematical Model for Malaria and Meningitis Co-infection among Cildren. Applied Matematical Sciences. Vol.5 (47), []. akul R.C (5). Using matematical models in controlling te spread of malaria. P.D. tesis. []. George, T. A (). A matematical model to control te spread of malaria in Gana Page