ttp://www.sciencepub.net/report A Matematical Model for Assessing te Control of and Eradication strategies for Malaria in a Community ABDULLAHI MOHAMMED BABA DEPARTMENT OF MATHEMATICS, NIGER STATE COLLEGE OF EDUCATION, MINNA Kutigi_baba@yaoo.com ABSTRACT: Tis study discusses te Stability of equilibrium states of a model in an attempt to suggest control of and Eradication strategies for malaria in a community. Te model exibits two equilibria namely; te trivial equilibrium state and diseases free equilibrium state. Te Model analysis sows tat combination of bot treatment and preventive metod can eradicate malaria cases.[abdullai Moammed Baba: A Matematical Model for Assessing te Control of and Eradication strategies for Malaria in a Community. [ABDULLAHI MOHAMMED BABA. A Matematical Model for Assessing te Control of and Eradication strategies for Malaria in a Community. Report and Opinion 202;(2):7-2]. (ISSN: 553-9873). ttp://www.sciencepub.net/report. 2 Key words: Malaria, mosquitoes, equilibrium, stability, drugs. For centuries Malaria as out ranked warfare as a source of uman suffering over te pass generation it as killed millions of uman beings and sapped te strengt of undreds of millions more, it continues to be a eavy drag on man s efforts to advance is agriculture and industry Jon F. Kennedy (962) US former President. INTRODUCTION Te uman race as suffered from malaria since earliest times (Malaria 200) and malaria Protozoa may ave a uman patogen for te entire istory of te species. (Malaria 2009), Malaria is one of te most common infectious diseases and enormous public ealt problem. Te disease is caused by protozoan parasites of te genus plasmodium. Five species of te Plasmodium Parasite can infect umans; te most serious form of te disease is caused by Plasmodium parasite (Malaria 2009)(wic is tougt to ave evolved in tropical Africa about 2.5million years ago (Malaria 200)). Malaria caused by Plasmodium Vivax, Plasmodium Ovale and Plasmodium Malariae causes milder disease in umans tat is not generally fatal (Malaria 2009) wic is tougt to ave evolved between 2.5million and 3.0 million years ago. A fift species, Plasmodium Knowlesi causes malaria in macaques but can also infect umans tis group of uman patogenic plasmodium Species is usually referred to as malaria parasites. Malaria (2009). At present te disease affects more tan 300 million uman and kills.5 to 3 million people every years (Ngwa and Su (2000), malaria (200) and Malaria (2009), Citnis (200), Cittnis, Cusings and Hyman (2006), and Pongsum Pun and tang (200). An African cild dies of te Mosquito-borne disease every 20 seconds, according to some estimates, making it te number one killer of infants less tan five years of age on te continent. Ninety percent of te.5 to 3.0 million people wo die of malaria per year are in sub Saaran Africa and it is still a problem tat it did not attract as muc attention as te Human Immune Deficiency Virus/Acquire Immune Deficiency Syndrome (HIV/AIDS). People in poor remote Villages are usually unable to get treatment. Te yearly Mortality from malaria is about 9 times tat of HIV/AIDS. Eac year, if /3 of a million people die of HIV/AIDS, 3 million people die of malaria. Malaria ranked tird among te major infectious diseases causing deats, after pneumococca acute respiratory infections Collectively designated as pneumonia, and tuberculosis (Malaria 2009) People get malaria by being bitten by an infective female Anopeles Mosquito. Only anopeles Mosquitoes can transmit malaria because tey feed on blood meal. About 60 of te 390 Species of anopeles mosquito transmit te malaria parasite. Out of tese only a dozen species are important in te transmission of malaria worldwide. Usually just one or two species of anopeles play a role in Malaria transmission in a particular region were te disease prevalent as pointed out by Williams (2006). 7
ttp://www.sciencepub.net/report Scematic illustration below sows te interaction and flow from one class to anoter (Figure ). Human Population ( 0 + ) A 3 I S R 2 S v I v 5 Key 2 2 Mosquito Population Interaction Flow Susceptible uman, S, can be infected wen tey are bitten by infected mosquitoes. Tey ten progress troug te infected, in to recovered, R compartments, via intake of anti-malaria drugs. Te recovered compartment will ave immediate return pat to infected I wen bitten by infected mosquito I v, tose not bitten re-enter te susceptible compartment after losing immunity. Susceptible mosquitoes, S v can be infected wen tey bite infected or recovered umans. Birt, deat and recruitment rate into and out of te population are sown in te scematic illustration. wedefine te following parameters as follows. : Te recruitment rate : Natural mortality of umans 2 : Natural mortality of Mosquitoes 0 : Infectious mortality of umans : Te probability tat a susceptible individual becomes infected by one infected Mosquito per contact unit time. 2 :Te recovery rate of te infected compartment 3 : Te rate of te recovery individual becomes susceptible. : Te probability tat a recovery individual become infected by one infected mosquito per contact per unit time 5: Te probability tat a susceptible mosquito becomes infected : Natural birt rate of mosquitoes A: Measure of te effectiveness of anti-malaria drug : Proportion of te offspring of te infected mosquito tat are infected (- ): Proportion of te offspring of te infected mosquito tat are susceptible Figure. Scematic illustration below sows te interaction and flow from one class to anoter 8
ttp://www.sciencepub.net/report DEFFECTS OF EARLIER MODELS A number of Matematical Models ad been developed and publised. Significantly of interest are te Matematical Models formulated by Ngwa and Su (2000), Citnis, Cusings and Hyman (2006) and Moammed Model (2008). Te former was a system of seven ordinary differential equations i. e Susceptible-Exposed-Infections-Recovered- Susceptible(SEIRS) pattern for umans and Susceptible-Infectious (SEI) for mosquito a susceptible-exposed-infections-recovered- Susceptible (SEIRS) for Mosquito. Citnis et al (2006) analysis a similar model wit same number of differential equations, Moammed (2008), analysed a similar model for malaria transmission wit five ordinary differential equations i. e Susceptible- Exposed-Infections-Recovered-Susceptible (SEIRS) pattern for uman and a Susceptible-Infectious (SI) for Mosquito. In tis paper we extend te Moammed Model (2008). Te new model (Figure ) divide te uman population to tree classes; susceptible, S, Infected I, and Recovered, R. We divide te mosquito population into two classes; Susceptible S v, and Infected I v. we consider only Female Mosquitoes (because only female Mosquitoes feed on blood). Te extension of te Ngwa and Su model include uman immigration, excludes direct uman recovery from te infectious to te Susceptible class. Te birt of mosquitoes is not all to susceptible but some portion belongs to te infectious class. Te extension of Citniset al Model (2006) include recruitment rate (te immigration and uman birt) sould be presented wit a parameter. Te birt rate of Mosquitoes as in Ngwa and Su (2000). Tere is no interaction in bot uman and mosquito exposed classes, wic sows tat tere is no need for te class. Te extension of Moammed s Model (2008), include recruitment rate as in above a return part from Recovered class, R to infected class if tere is interaction between te infected Mosquitoes wit te Recovered individuals. Design of te Matematical Model Variable and parameters: we use te following notations and/or definitions for te umans population and mosquito population Susceptible (S ): te number of individuals wo can be infected but ave not yet contacted te malaria but may contact it if exposed to any mode of its transmission. Infected (I ): Te number of individuals wo ave contacted te malaria and are actively or capable of transmitting it to oters. Recovery (R ): te number of individuals wo are recovered after treatment and are immune to te disease. Assuming in te given population at time say t. S (t), I (t) and R (t) denote susceptible, infected and recovery, respectively and mosquito population S v (t), I v (t). Susceptible Mosquito (S v ): te number of mosquitoes wic can infected if exposed to te mode of transmission. Infected Mosquito (I v ): te number of mosquito tat can transmit malaria. Assumptions of te Model: te following assumptions are taken into te consideration in te construction of te Model. All te people in te community are likely to be infected individually in case of contact. All immigrations and new born in te community are uninfected, ence tey join te susceptible compartment Birt in te mosquito population are grouped into two susceptible and infected mosquitoes Te community as fixed area size and only te population size will be varying Te uman population size is N S I R. Te mosquito population is Nv Sv Iv Compartmentalization of te model: te dynamics of malaria model can be described as in te compartment model fig Equation of te model: In view of te above assumption and interrelations between te variables and parameters as described in te Compartmentalized model in fig, te followingsystem of differential equations were obtained. ds IvS S 3R di IvS 2 A 0 I IvR 2 9
ttp://www.sciencepub.net/report dr 2IvS 3 Iv R 3 dsv Sv Iv 2Sv 5SvI di v Iv 2Iv 5SvI 5 Analysis of te Model Normalization S I Iv X, Y, Z N N Nv S I R, Sv Iv 6 R X Y, Sv z Subsequently substituting equation (6) into te system, (), (2) and (5) respectively gives. 3 3 3 7 X z X y Y x y x z A z y 2 0 8 Z z y z 2 2 5 9 Subsequently, we sall consider te equation (7), (8) and (9) for te purpose of our analysis Equilibrium States of te Model At te equilibrium State, let,, X t x Y t y Z t z So tat from equation (7), (8) and (9), we ave ten te system of equation become 3 z 3 x 3y 0 0 z y z x y x z 2 A 0 z y 0 2 2 5 0 2 We solve tis system of equations simultaneously to obtain te following values for (x, y, z) at critical points a. Te existence of te trivial equilibrium point; for as long as te recruitment term is not zero; te population will not be extinct. Tis implies tat tere is no trivial equilibrium state i.e (x, y, z) (0, 0, 0) b. Te existence of te disease free equilibrium State, (x, y, z): At te disease free equilibrium, we ave,,,0,0 3 3 X is te asymptotic carrying capacity of te population 3 x y z 3 Stability of te Equilibra Now to determine te stability of stability of trivial equilibrium state and disease free equilibrium, te following variational matrix is computed corresponding to equilibrium states as used by Lenka (2007) and Akinwande (995) 0
ttp://www.sciencepub.net/report z 3 3 x 2 0 0 z y M z A z x x y 5 2 2 5 We can easily see tat te trace of te matrix is T z 3 2 A 0 z 2 2 5 y 0 And te determinant of te matrix is z [ A z y x x y 3 2 0 2 5 5 z] 3[ 2 A 0 z5 z x z] 0 5 5 We ave T 0 and 0 is te condition for linearized stability. According to te principle of linearised stability te trivial equilibrium states is locally asymptotically stable i.er 0 < 3 3 3 3 M 0 2 A 0 3 0 5 2 We can easily see tat te trace of te matrix is T A And tat its determinant 3 2 0 2 0 3 2A 0 2 5 0 3 According to te principle of linearized stability te disease free equilibrium is locally and asymptotically unstable i.er 0 > Conclusion In te study we used te analytical metod to determine te population equilibrium of a community aving malaria, wit reference to trivial equilibrium state and disease free equilibrium state. Furtermore we investigated te conditions of existence and stability of te equilibrium. By analysing te model, we found a tresold parameter R 0 ; It is noted tat wen R 0 < ten te epidemic will die out and wen R 0 > te disease will persist in te population and become endemic. Te Model as two non-negative equilibria namely: 3 x, y, z,0,0 3 And(x, y, z) (0, 0, 0) Conclusively, it is observed tat te non-trivial equilibrium is unstable tis indicates tat it is very possible to control, or minimize or eradicate te disease in a community. Tis can be acieved if te recommendations are strictly adered to.
ttp://www.sciencepub.net/report Recommendations Te two main control strategies against te spread of malaria in umans and mosquito reductionstrategies and personal protection against exposure to Mosquitoes. Mosquito reduction strategies include te elimination of mosquito breeding sites (troug improved drainage and prevention of standing water). Larvaciding (killing Mosquito Larva before tey become adult) and adulticiding (killing adult Mosquitoes Via fogging). Using appropriate biological agents. Personal protection, on te oter and entails te use of cloting protection. Insect repellents [containing dietyl-meta-toluamide (DEET)] and avoiding places were mosquitoes bite. Increasing te community awareness on te mode of transmission of malaria Ministry of Healt sould sensitize te public troug te malaria control programmes and oter NGO s wo provide ealt care services about malaria Corresponding Address Abdullai Moammed Baba Department Of Matematics, Niger State College Of Education, Minna E mail: kutigi_baba@yaoo.com References [] Akinwande, N. Ilocal stability analysis of equilibrium state of a Matematical model of yellow fever epidemics, J. Nig. Matematical society, 995: (2) 73 79 [2] Citnis, N. Modelling te spread of malaria. ttp://mat. an.gov 200 [3] Citnis, N; Cusings, J. M., and Hyman, J. M. Bifurcation analysis of Matematical model for malaria transmission: SIAM Journal on Applied Matematics 2006: 67() 2 5 [] ttp://encarta.msn..com [5] Lenka, B. Te matematics of infectious diseases. M.sc tesis publised, Comenius university, Bratislava. 2007 [6] Koella, j.c and Anita, R Epidemiological models for te spread of anti-malaria resistance. Malaria Journal 2003: 2(3):76-79 [7] Malaria.Encyclopaedia Britannica, Ultimate reference suite, Cicago encyclopaedia Britannica Malaria.Wikipedia, te free encyclopaedia.http://en.wikipedia.org/wiki/malaria. 200 [8] Moammed, B.A. A matematical model for malaria wit effect of drug application M. Sc. Tesis unpublised, bayero University Kano, Nigeria 2008:5-57 [9] Ngwa, G. A and Su, W.S, A matematical Model for endemic malaria wit variable uman And Mosquito populations. Matematical and Computer Modelling, 2000:32 :77 763 [0] Pongsumpon, P and Tang I. M. Te transmission Model of Plasmodium felaprm and Plasmodium Vivax malaria between Tai and Burmese. International journal of Matematical models and metods in applied Sciences. 2009: 3() [] Sonkwiler, R and Aneke, S. J. Some matematical modelsformalaria.ttp://www.mat.gatec.edu/seik/malaria.po :200 [2] Tumwiine, J. Luckaus, M and Luboobi, L. S. An-age Structured matematical model for witin Host dynamics of malaria and te immune system. J. mat model Algor:2008: 7: 79 97 [3] Williams, S. Statistical analysis witin ost dynamic of Plasmodium falciparum infection, a P. D tesis publised. Swiss tropical institute base Swizerland. :2006. /3/202 2