A Mathematical Model of Malaria. and the Effectiveness of Drugs

Transcription

1 Applied Matematical Sciences, Vol. 7,, no. 6, HIKARI Ltd, A Matematical Model of Malaria and te Effectiveness of Drugs Moammed Baba Abdullai Scool of Matematical Sciences Universiti Sains Malaysia 8 USM Penang Malaysia Kutigi_baba@yaoo.com Yaya Abu Hasan Scool of Matematical Sciences Universiti Sains Malaysia 8 USM Penang Malaysia ayaya@cs.usm.my Fara Aini Abdulla Scool of Matematical Sciences Universiti Sains Malaysia 8 USM Penang Malaysia faraaini@usm.my Copyrigt Moammed Baba Abdullai et al. Tis is an open access article distributed under te Creative Commons Attribution License, wic permits unrestricted use, distribution, and reproduction in any medium, provided te original work is properly cited.

2 8 Moammed Baba Abdullai et al. Abstract We propose a new deterministic model for te dynamics of malaria and te effectiveness of drugs and te positivity of te model investigated. Te uman population was portioned into tree distinct compartments of susceptible, infected and te recovered, wile te mosquito population was portioned into two distinct Compartments of susceptible and te infected, giving rise to a set of five ordinary differential equations. A parameter ( A ) is introduced to measure te effectiveness of anti malarial drugs in te infected compartment. Te dynamics of te model was obtained. It was observed tat bot te disease free equilibrium states will be stable if ( A ) > 6% and if te mosquitoes population is reduced te uman population will also be stable. Numerical simulations were presented to sow some results Keywords: Malaria, stability, Drug. INTRODUCTION In certain parts of te world, malaria is a major ealt worries for te population [8]. Malaria is caused by te Plasmodium parasite, its transmission to uman troug bites by te female Anopeles mosquito [], it can also be transmitted from an infected moter to er unborn baby (congenitally) or troug blood transfusion [5, ]. Tis disease is treatening umanity to te same level as te world s major infectious diseases suc as colera, tuberculosis and HIV/AIDS [9]. In te African continent, tis disease is endemic and a uge number of uman lives, mostly cildren, are lost eac year []. One of te ways to understand te transmission and spread of te disease is troug properly developed matematical model. Essentially te model involves te interactions between uman and mosquito and matematically written as a system of ordinary differential equations. One of te first models

3 Matematical model of malaria 8 was described by Ross in 9 and major extensions given by MacDonald in 957 []. Since ten, various models ave proposed to take into account various scenarios and regions. A major question tat tese models oped to answer is te effect of drug and treatment. As a result of drug resistant strains of te parasite, treating malaria and controlling it is increasingly difficult. One factor for te lack of vaccine is te complex adaptability of te malaria patogens [8]. However, drug is still te main biological defense against te disease. Wat is needed is a proper strategy in te administration of te drug togeter wit te drugs efficiency. In tis paper, we propose a new model for te spread of malaria, wic is an extension of te existing matematical models of malaria see for example [,, 4, and ]. Te extension of model includes transmission from blood transfusion and congenitally. It also include rejecting te return pat from infected to te susceptible and recovered to susceptible, it is assumed tat te recovered individuals acquired permanent immunity against re-infection. We sall analyze te models dynamically and ten look at te effectiveness of drug. Te paper is organized as follows, in section ; we described te model and its basic properties. In section, equilibrium state and it stability. In section 4, numerical simulation and finally, conclusion in section 5.

4 8 Moammed Baba Abdullai et al.. MATHEMATICAL MODEL HUMANS MOSQUITOES µ Λ π S S m β β β I m η ( α + µ ) γ R µ FIGURE scematic illustration of te model equation.. Susceptible, infected and recovered umans Te population of susceptible umans is increased via recruitment of umans (by birt or immigration) into te community at a constant rate Λ. It is decreased by infection acquired troug contact wit infected mosquitoes (at a rate β ) or by troug blood transfusion (at a rate β ) and natural deat (at a rate µ ). Tis gives ds ( β β µ ) = Λ I + I + S () m

5 Matematical model of malaria 8 Infected umans are generated eiter by birt as a result of infected moter to er unborn baby (congenitally) at a rate η or troug te infection of susceptible umans follows contact wit infected mosquitoes. It is decreased eiter by natural deat (at a rate µ ), and malaria induced deat (at a rateα ) and by recovery from infection at a rateγ wit drug efficiency (at a rate A ). Tus di = ηi + ( β Im + βi) SH ( µ + α + Aγ ) I () Te recovered umans are generated by te recovery of infected umans (at a rateγ ) as result of intake of drugs (anti-malaria) wit efficiency (at a rate A ). It decreased by natural deat (at a rate µ ). So tat dr = Aγ I µ R (). Susceptible and infected mosquitoes Te susceptible female mosquito population is generated by te birt or immigration of susceptible mosquitoes (at a rateπ ). It is diminised eiter by infection, acquired wen susceptible mosquitoes feed from te blood of infected umans (at a rate β ), or by natural deat (at a rate ). Tis yield dsm = π ( β I ) Sm (4) Te infected female mosquito population is generated via te infection of susceptible mosquitoes by umans and diminised by natural deat (at a rate ). It is assumed tat vertical transmission in mosquitoes is negligible and tus omitted. Hence dim = βism Im (5) Te above assumptions and derivations leads to te following system of ordinary differential equations ds = Λ ( βim + β I + µ ) S di = ηi + ( β Im + βi) S ( µ + α + Aγ ) I dr = Aγ I µ R (6)

6 84 Moammed Baba Abdullai et al. ds di m m = π ( β S I + ) = β I S I m m m. Basic properties of te model. Since te model (6) monitors uman and mosquito populations, all its associated parameters are non-negative. Furtermore, te following positivity results olds. Teorem Te variables of te model (6) are positive for all timet. In oter words, solutions of te model equation (6) wit positive initial data will remain positive for all timet >. Proof: Let te initial data be S (), I (), R (), S, I () (7) m m From te first equation of te model (6) we ave ds ( β β µ ) ( β + β µ ) = Λ I + I + S I I + S (8) m m ( ) t Terefore, S ( t) S () e β + β + µ > (9) Similarly, it can be sown tat IM ( t ) > and I H ( t ) > far all time t >.. EQUILIBRIUM STATES AND ITS STABILITY Te equilibrium states are obtained by setting te rigt and side of equation (6) equal to zero. We get two equilibrium points. Te disease free equilibrium state (DFE) of te model (6) is given by Λ π E = (,,,,) () µ Te endemic equilibrium (EE) f te model (6) is given by E = ( S, I, R, Sm, Im) ()

7 Matematical model of malaria 85. Basic reproduction number. In epidemiological models, te basic reproduction number, some time called basic reproduction rate or basic reproductive ratio, denoted byr, is one of te most useful tresold parameters, it is unit-less tresold quantity for te disease control wic defines te number of secondary infections produced by a single infected individual in a completely susceptible population. It elps determine were an infectious disease will spread troug a population. Te most important equilibrium state for disease control is te disease free equilibrium state (DFE) and its linear stability is governed by te basic reproduction R [6]. Te dynamics of te model is analyzed by R given by R ˆ = β Λβ π ( Aγ + α + µ ) µ () β Λβ π R = ( Aγ + α + µ ) µ () Equation () is te basic reproduction number of uman - mosquito interaction. Hence, using Teorem of [], we establised te following result: Teorem Disease free equilibrium ( DFE ) of te system (6) is asymptotically stable ifr < and unstable if R > Proof Local stability of E is govern by te eigenvalues of te matrix at ( DFE )

8 86 Moammed Baba Abdullai et al. βλ βλ µ µ µ βλ ( µ + α + Aγ ) µ J ( E ) = a γ µ β π β π Te caracteristics equation of te matrix is given by 5 4 λ + (µ + + α η + Aγ ) λ + (µ + 6 µ + µ α µ η + µ Aγ + µα µη + µ + µ Aγ βπλ) λ µ + ( µ + ( µ + µ )( µα µη + µ + µ Aγ ) + ( µ + )( µα βπλ+ µ µη + µ Aγ )) λ µ ( µ α µ η+ µ + µ + µ Aγ µβ πλ+ µα β πλ µη + µ Aγ ) λ + + µ ( µα β πλ + µ µη + µ Aγ ) (4) Ten eigenvalues are λ = µ, λ = µ, λ =. Te remaining eigenvalues are found by solving λ + a + =. We can see tat λ a a a 4a λ4 = and a + a 4a λ5 =. Te eigenvalue λ 4 and λ 5 as a negative real part wen a 4a a < or a 4a < a or a >. It is seen tat λ, λ, λ, λ 4 and λ 5

9 Matematical model of malaria 87 ave negative real parts. Tis implies tat te disease free equilibrium point is locally stable for R <. Teorem implies tat wen R < malaria can be eradicated from uman mosquito population. Teorem If R, ten te disease free equilibrium E of (6) is globally asymptotically stable (GAS) in D In establising te global stability of te disease free equilibrium point E, we construct te following Lyapunov function for uman mosquito interaction βλ π L = µ I + Im. Clearly, L along te long te solution of te system () and is zero if and only if bot I and Te time derivate of (,,,, ) m m I m are zero. L S I R S I computed along te solution of te model equation (6) is βλ L = µ [ ηi + ( β Im + βi) S ( µ + α + Aγ ) I] + [ βism Im] Λ βλ βiπ βλ I = µη I + µ ( β Im + βi) µ ( µ + α + Aγ ) I + µ µβ Λ βλβ π I + ( + + A ) + µ [ µη µ µ α γ ] βλ = I[ µ ( η + ( µ + α + Aγ ))( R )] µ We neglect te first inequality βimλ βλ Im using tat fact it is in. Tat last inequality follows from assumption tat R. If R <, te derivative L = if and only if compact invariant set in I = and I =. Terefore, te largest ( S, I, R, Sm, Im ) D : L were E is te disease free equilibrium point. m = is te singleton { E }, From LaSalle s invariant principle [7, ] tis implies tat E is globally asymptotically stable ind. Tis implies tat malaria can be eradicated in te uman- mosquito population. m

10 88 Moammed Baba Abdullai et al. 4. NUMERICAL SIMULATION In tis section we use te caracteristic equation (4) in te form H ( λ ) and apply te result of Bellman and Cooke, as used by [], to test te stability or oterwise of te disease free equilibrium state wit te effectiveness of te drug application. H ( λ) λ (µ α η γ ) λ 5 4 = A + (µ + 6 µ + µ α µη + µ Aγ + µα µη + µ + µ Aγ β πλ ) λ + µ ( µ + ( µ + µ )( µα µη + µ + µ Aγ ) + ( µ + )( µα β πλ+ µ µη + µ Aγ )) λ µ ( µ α µη+ µ + µ + µ Aγ µβ πλ+ µα β πλ µη + µ Aγ ) λ + + µ ( µα β πλ + µ µη + µ Aγ ) We set λ = ip H ( ip) ( ip) (µ α η Aγ )( ip) 5 4 = (µ + 6 µ + µ α µ η+ µ Aγ + µα µη + µ + µ Aγ β πλ)( ip) + µ ( µ + ( µ + µ )( µα µη + µ + µ Aγ ) + ( µ + )( µα β πλ+ µ µη + µ Aγ ))( ip) µ ( µ α µη+ µ + µ + µ Aγ µβ πλ+ µα β πλ µη + µ Aγ )( ip) + + µ ( µα β πλ + µ µη + µ Aγ ) Resolving into real and imaginary parts, we ave H ( ip) = F( p) and ig( p ) Were F ( p ) and ig( p ) are given respectively F ( p ) = (µ + + α η + Aγ )( ip) 4 + µ ( µα β πλ + µ µη + µ Aγ )

11 Matematical model of malaria 89 ( µ + ( µ + µ )( µα µη + µ + µ Aγ ) + ( µ + )( µα β πλ+ µ µη + µ Aγ ))( ip) µ G( p ) = p 5 + (µ + 6 µ + µ α µ η+ µ Aγ + µα µη+ µ + µ Aγ βπλ )( p) + µ ( µ α µη+ µ + µ + µ Aγ µβ πλ+ µα β πλ µη + µ Aγ )( p) + Differentiating wit respect to p and setting p =, we ave F() = µ ( µα β πλ + µ µη + µ Aγ ) (5) G () = (6) F '() = (7) ( µα µη+ µ + µ + µ Aγ µβ πλ+ µα β πλ µη + µ Aγ ) G'() = Te condition for stability according to Bellman and Cooke teorem is given by ' ' ( ) ( ) ( ) ( ) (8) F G F G > (9) Since G () and F '() = equation (9) becomes ' ( ) ( ) F G > () ' Let j ( A) = F ( ) G ( ) Hence te disease free equilibrium state will be stable if j ( A ) > () Using ypotetical values for parameters in (5) and (8), values were generated for j ( A) so as to verify te result of te analysis. Some of te values obtained are presented in te table 5. below

12 9 Moammed Baba Abdullai et al. Table 4. A µ α η β Λ γ π F ( ) G ' () Remark E-5 7.5E-4 unstable e-5 6.5E-5 unstable e-5 6.E-5 unstable e-4.5e-5 unstable e 7.E-4 neutral E-5.E- stable E-5.7E- stable From table 4. above, it can be seen tat. j ( A ) < : Te efficiency of te drug is between. to 4% wic implies instability of te population.. j ( A ) > : Te efficiency of te drug is from 6% and above wic implies stability of te population.. j ( A ) > : Te natural deat of mosquitoes increases. From te analysis, it can be seen tat te disease free equilibrium state will be unstable if te anti malarial drug is less tan 6% efficient and stable from 6% and above. However it was also observed tat wen te deat rate of te mosquito is ig te population will be stable. We can conclude tat once malaria is in a population, te application of efficient anti malaria drugs and reducing mosquito s can reduce infectious deat and te population will be stable.

13 Matematical model of malaria 9 Figure, sows te unstable situation in a population, wenever te efficiency of te drug A < 6%. Figure, sows te stable situation in a population, wenever te efficiency of te drug A > 6%.

14 9 Moammed Baba Abdullai et al. Figure 4 sows te stability wen j ( A ) > Figure 5 sows instability wen j ( A ) <

15 Matematical model of malaria 9 5. CONCLUSION Te model tat we proposed in tis paper explicitly includes drug efficiency as a parameter. It was sown tat if te anti-malarial drug is above certain tresold efficiency, te population will be stable, tat is tere will not be an epidemic. Tis can be one of te strategies in te administration of drug to contain malaria. ACKNOWLEDGMENT Tis researc is supported by te Fundamental Researc Grant Sceme from Universiti Sains Malaysia (USM) Pulau Pinang, Malaysia. REFERENCES [] S. Abdulraman, Matematical Model of HIV/AIDS Pandemic wit te Effect of Drug Application. In THE PROCEEDINGS OF NMC- COMSATS conference on matematics modeling of global callenging problems. (9), [] A. M. BABA, A Matematical Model for Assessing te Control of and Eradication strategies for Malaria in a Community,Science.pub.net 4(), (), 7-. [] N. Citnis, J. M. Cusing, & J. M. Hyman, Bifurcation analysis of a matematical model for malaria transmission. SIAM Journal on Applied Matematics, 67(), (6), 4-45.

16 94 Moammed Baba Abdullai et al. [4] C. Ciyaka, J. M.Tcuence, W. Garira, & S. Dube, A matematical analysis of te effects of control strategies on te transmission dynamics of malaria. Applied Matematics and Computation, 95(), (8), [5] R. M. Fairurst, T. E. Wellems, Plasmodium species (Malaria). In: Mandell GL, Bennett JE, Dolin R, eds. Principles and Practice of Infectious Diseases. 7t ed. Piladelpia, Pa: Elsevier Curcill Livingstone; 9: cap 75. [6] H. W. Hetcote, Te matematics of infectious diseases. SIAM review, 4(4), (), [7] B. S. E. Iyare, F. E. U. Osagiede, Analysis of a malaria disease model wit direct transmission.journal of matematics and tecnology, (), -6. [8] K. Mars, Malaria disaster in Africa. Lancet, 5(9), (998), 94. [9] G. A. Ngwa, Modelling te dynamics of endemic malaria in growing populations. Discrete and Continuous Dynamical Systems Series B, 4,( 4), 7-. [] G. A. Ngwa, W. S. Su, A matematical model for endemic malaria wit variable uman and mosquito populations. Matematical and computer modelling, (7), (), [] P, Van den Driessce, J. Watmoug, Reproduction number and subtresold endemic equilibria for compartment models of disease transmission. Matematical biosciences, 8(), (), [] C. Vargas de León, J. A. Castro Hernández, Local and Global Stability of Host-Vector Disease Models. Foro-Red-Mat: Revista electrónica de contenido matemático, 5(), (8), -8.

17 Matematical model of malaria 95 [] ttp:// Received: April,