Mathematical Model for Malaria and Meningitis Co-infection among Children

Transcription

1 Applied Mathematical Sciences, Vol. 5, 11, no. 7, Mathematical Model for Malaria and Meningitis Co-infection among Children Lawi G. O. 1, Mugisha J. Y. T. and N. Omolo - Ongati 3 1 Department of Mathematics, Masinde Muliro University of Science and Technology Box 19, Kakamega 51, Kenya Department of Mathematics, Makerere University Box 7, Kampala, Uganda 3 Department of Mathematics and Applied statistics, Maseno University Private Bag, Maseno, Kenya Abstract Disease and poverty are a major threat to child survival in the developing world, where access to good nutrition, sanitation and health care is poor. In this paper, a mathematical model for malaria and meningitis co-infection among children under five years of age is developed and analysed. We establish the basic reproduction number R mm for the model, which is a measure of the course of the co-infection. The analysis shows that the disease-free equilibrium of the model may not be globally asymptotically stable whenever R mm is less than unity. The Centre Manifold theorem is used to show that the model has a unique endemic equilibrium which is locally asymptotically stable when R mm < 1 and unstable when R mm > 1. We deduce further that a reduction in malaria infection cases either through protection or prompt effective treatment, which is dependent on the socio-economic status of a community, would reduce the number of new co-infection cases. Mathematics Subject Classification: 3D Keywords: Malaria-meningitis model, Co-infection, Basic reproduction number, Stability, Socio-economic 1 Correponding author s glawi@mmust.ac.ke

2 338 G. O. Lawi, J. Y. T. Mugisha, N. Omolo - Ongati 1 Introduction Millenium Development Goal and Goal target improved child survival and reversing the high prevalence of diseases such as HIV/AIDS and malaria [1].Disease and poverty are a major threat to child survival in the developing world, where access to good nutrition, sanitation and health care is poor. In such settings HIV/AIDS, malaria and many other preventable infectious diseases continue to kill millions of children. Malaria enjoys a cause-effect relationship with poverty and is therefore a major hindrance to economic development. It costs Africa some $1 billion to $1 billion every year in lost gross domestic product [1]. Importantly, people living in malaria-endemic areas are frequently exposed to other diseases typically affecting the poor. These diseases not only take advantage of the compromised immunity due to the prolonged malaria exposure, coupled with limited and untimely chemotherapy but also present with malaria-like symptoms. Humans acquire malaria following infective bites from infected Anopheles female mosquitoes during blood feeding. Plasmodium falciparum is the parasite species that largely causes human malaria infections in Africa. Each year 35-5 million cases of malaria occur worldwide, and over one million people die, most of them young children less than five years of age in sub-saharan Africa [3]. Malaria was the fourth cause of death in children in developing countries in. In Kenya it accounts for 19% of all hospital admissions, 3% of all outpatient visits, with an estimate of % of all deaths in children less than five years of age being attributed to the disease [11]. The incidence of malaria has been on the rise in the recent past due to increasing parasite drug-resistance and mosquito insecticide-resistance. This rise has also been associated with climate change []. Meningitis is an infectious disease characterized by inflammation of the meninges (the tissues that surround the brain or spinal cord), usually due to the spread of an infection into the cerebral spinal fluid (CSF). The cause of the infection may be bacterial, viral, fungal or parasitic. Some of the risk factors for the disease are a compromised immune system due to illness, such HIV/AIDS or use of immunosuppressant drugs. The symptoms of meningitis include neck and/or back pain, headache, high fever and a stiff neck. Bacterial meningitis may cause acute or chronic brain injury and thus leading to death or long-term disability (such as deafness, paralysis, seizure and even mental retardation) []. The seasonal outbreak of meningitis in the African meningitis belt, a band of sub-saharan Africa, usually results into a high disease mortality and morbidity[]. An outbreak in , documented as the largest

3 Malaria and meningitis co-infection 339 claimed more than more than 5 lives, with about 5 cases of illness across 1 countries. In contrast, during the non-epidemic year of 8 there were only 7 cases across the whole of the belt [1]. Malaria and meningitis have a symptom overlap. Malaria endemicity coupled with poverty causes a challenge in diagnosis and prompt treatment of infections with malaria-like symptoms. This is because in such settings diagnosis is usually clinically done and most cases are thought to be malaria[13]. Failure or delay of correct diagnosis may result into severity of the infection or co-infection[1]. In the study by [9] carried out in Kenya, % of the children admitted in the hospital were found to be infected with both malaria and acute bacterial meningitis. The study noted that co-infection played a major role in the group of children with high mortality. In this paper we thus develop a mathematical model to study the dynamics of malaria-meningitis co-infection. We assume that there is no simultaneous infection of a host with the two diseases. Model Description and Formulation The total human population at any time t, denoted N H is subdivided into subpopulation susceptible humans (S H ), those exposed to malaria parasites only (E 1 ), individuals infected with malaria ( I 1 ), those infected with meningitis (I ), individuals exposed to malaria and infected with meningitis (E 1 ) and individuals infected with both malaria and meningitis (I c ). The total vector population at any time t, denoted N v is subdivided into subpopulation of susceptible (S v ), exposed (E v ) and infectious (I v ). This means that and N H = S H + E 1 + I 1 + I + E 1 + I c (1) N v = S v + E v + I v () The rates of infection of susceptible humans with malaria and meningitis are λ ma and λ me respectively, while that of susceptible vectors with malaria is λ v. Let ψ and γ be malaria and meningitis induced mortality in humans respectively, and suppose that μ H and μ v are per capita natural death rates of the human and mosquito populations respectively. The constant per capita recruitment rate into the susceptible human and vector populations are Λ H and Λ v respectively. The rates at which exposed human and vector populations develop malaria clinical symptoms are σ H and σ v respectively, while the rate at which humans progress from the E 1 class to the I c class is ɛσ H, where

4 3 G. O. Lawi, J. Y. T. Mugisha, N. Omolo - Ongati ɛ is a modification parameter representing the assumption that meningitis infected individuals exposed to malaria develop malaria symptoms at a faster rate than those who are not infected with meningitis. Define φ 1 as the rate at which individuals infected with malaria recover, φ as the recovery rate from meningitis and φ 3 as the recovery rate from both infections. The recovered individuals do not acquire temporary immunity to either or both diseases and thus become susceptible again. We assume that infection with meningitis when one is exposed to malaria takes place at an advanced stage of this exposure. The parameter θ accounts for the increased susceptibility to infection with meningitis for individuals infected with malaria, while the parameter ρ accounts for the decreased susceptibility to infection with malaria for individuals infected with meningitis because of decreased contact due to ill health. The individuals displaying symptoms of both malaria and meningitis suffer malaria-induced mortality at the rate ϑψ, where the parameter ϑ accounts for the assumed increase in malaria-related mortality due to the dual infection with meningitis and also suffer meningitisinduced mortality at the rate ηγ, where the parameter η accounts for the assumed increase in meningitis-related mortality due to the dual infection with malaria. Define α as the number of bites per human per mosquito (biting rate of mosquitoes), a as the transmission probability of malaria in humans per bite, b as the transmission probability of malaria in vectors from any infected human, β as the effective contact rate for infection with meningitis. This yields λ ma = αai v N H (3) λ v = αb(i 1 + δi c ) N H () λ me = β(i + E 1 + κi c ) N H, (5) where δ and κ model the relative infectiousness of the co-infected individual as compared to their counterparts.

5 Malaria and meningitis co-infection 31 From the above definitions and variables we have the following model ds H de 1 di 1 di de 1 di c ds v de v di v = Λ H λ ma S H λ me S H + φ 1 I 1 + φ I + φ 3 I c μ H S H, = λ ma S H λ me E 1 σ H E 1 μ H E 1, = σ H E 1 θλ me I 1 ψi 1 φ 1 I 1 μ H I 1, = λ me S H ρλ ma I φ I γi μ H I, = ρλ ma I + λ me E 1 (ɛσ H + γ + μ H )E 1, () = ɛσ H E 1 + θλ me I 1 (φ 3 + ϑψ + ηγ + μ H )I c, = Λ v λ v S v μ v S v, = λ v S v σ v E v μ v E v, = σ v E v μ v I v..1 Positivity and Boundedness of solutions Model () describes the human and mosquito populations and therefore it can be shown that the associated state variables are non-negative for all time t and that the solutions of the model () with positive initial data remains positive for all time t. We assume the associated parameters as nonnegative for all time t. We show that all feasible solutions are uniformlybounded in a proper subset Ψ = Ψ H Ψ v. Theorem.1. Solutions of the model () are contained in the region Ψ= Ψ H Ψ v. Proof. To show that all feasible solutions are uniformly-bounded in a proper subset Ψ, we split the model () into the human component (N H ) and the mosquito component (N v ), given by equations (1) and () respectively. Let (S H,E 1,I 1,I,E 1,I c ) R + be any solution with non-negative initial conditions. From the theorem by [8] on differential inequality it follows that lim sup S H (t) Λ H. t μ H

6 3 G. O. Lawi, J. Y. T. Mugisha, N. Omolo - Ongati Taking the time derivative of N H along a solution path of the model () gives Then, dn H =Λ H μ H N H ψi 1 γi (ϑψ + ηγ)i c dn H Λ H μ H N H From the theorem by [8] on differential inequality it follows that N H Λ H μ H + N H ()e μ Ht where N H () represents the value of (1) evaluated at the initial values of the respective variables. Thus as t, we have N H Λ H μ H (7) This shows that N H is bounded and all the feasible solutions of the humanonly component of model () starting in the region Ψ H approach, enter or stay in the region, where Similarly,let Ψ H = {(S H,E 1,I 1,I,E 1,I c ):N H Λ H μ H } (S v,e v,i v ) R 3 + be any solution with non-negative initial conditions. Then lim sup S v (t) Λ v. t μ v Taking the time derivative of N v along a solution path of the model () gives dn v =Λ v μ v N v The mosquito-only component () has a varying population size. Therefore, dn v < Λ v μ v N v

7 Malaria and meningitis co-infection 33 From the theorem by [8] on differential inequality it follows that N v Λ v μ v + N v ()e μvt where N v () represents the value of () evaluated at the initial values of the respective variables. Thus as t, we have N v Λ v μ v This shows that N v is bounded and all the feasible solutions of the mosquitoonly component of model () starting in the region Ψ v approach, enter or stay in the region, where Ψ v = {(S v,e v,i v ):N v Λ v μ v } (8) Thus it follows from (7) and (8) that N H and N v are bounded and all the possible solutions of the model starting in Ψ will approach, enter or stay in the region Ψ = Ψ H Ψ v t. Thus Ψ is positively invariant under the flow induced by (). Existence, uniqueness and continuation results also hold for the model () in Ψ. Hence model () is well-posed mathematically and epidemiologically and it is sufficient to consider its solutions in Ψ. 3 Disease-free equilibrium point Disease-free equilibrium (DFE) points of a disease model are its steady-state solutions in the absence of infection or disease. We denote this point by E and define the diseased classes as the human or mosquito populations that are either exposed or infectious. Define the positive orthant in R 9 by R 9 + and the boundary of R 9 + by R 9 +. Lemma 3.1. For all equilibrium points on Ψ R 9 +, E 1 = I 1 = I = E 1 = I c = E v = I v = The positive DFE for human and mosquito populations for the model () are N H = Λ H μ H and N v = Λ v μ v. (9) Lemma 3.. The model () has exactly one DFE, E =( Λ H μh,,,,,,, Λv μ v,, )

8 3 G. O. Lawi, J. Y. T. Mugisha, N. Omolo - Ongati Proof. The proof of the lemma requires that we show that DFE is the only equilibrium point of () on Ψ R 9 +. Substituting E into () shows all derivatives equal to zero, hence DFE is an equilibrium point. From Lemma 3.1, the only equilibrium point for N H is Λ H μh Λ v μ v. Thus the only equilibrium point for Ψ R 9 + and the only equilibrium point for N v is is DFE. 3.1 Local stability of the disease-free equilibrium The global dynamics of the model () is highly dependent on the basic reproduction number. The basic reproduction number is defined as the expected number of secondary infections produced by an index case in a completely susceptible population [1]. We define the basic reproduction number, R mm as the number of secondary malaria (or meningitis) infections due to a single malaria (or a single meningitis-infective) individual. We determine R mm using the next generation operator approach [19]. The associated next generation matrices are αa β β βκ F = αbλvμ H αbδλ Λ H μ v vμ H Λ H μ v and V = h 1 σ H h h 3 h ɛσ H h 5 h σ v μ v where h 1 = σ H + μ H, h = ψ + φ 1 + μ H, h 3 = φ + γ + μ H, h = ɛσ H + γ + μ H h 5 = φ 3 + ϑψ + ηγ + μ H and h = σ v + μ v. The basic reproduction number R mm is the spectral radius of the matrix FV 1. The eigenvalues of the matrix FV 1 β are,,,, φ +γ+μ H and ± α abσ H σ vμ H Λ v. Λ H μ v (σ H+μ H )(σ v+μ v)(φ 1 +ψ+μ H ) Therefore R mm is given by R mm = max{ α abσ H σ v μ H Λ v Λ H μ v(σ H + μ H )(σ v + μ v )(φ 1 + ψ + μ H ), β }. (1) φ + γ + μ H

9 Malaria and meningitis co-infection 35 α Denoting R ma = abσ H σ vμ H Λ v and R β Λ H μ v(σ H +μ H )(σ v+μ v)(φ 1 +ψ+μ H ) me = φ +γ+μ H, we have R mm = max{r ma,r me }. R ma is a measure of the average number of secondary malaria infections in human or mosquito population caused by a single infective human or mosquito introduced into an entirely susceptible population. Similarly, R me is a measure of the average number of secondary meningitis infections in humans caused by a single infective human introduced into an entirely susceptible population. The following lemma follows from Theorem of [19]. Lemma 3.3. The the disease-free equilibrium E of the model () is locally asymptotically stable whenever R mm < 1 and unstable when R mm > Global stability of the disease-free equilibrium The global asymptotic stability (GAS) of the disease-free state of the model is investigated using the theorem by Castillo-Chavez et.al []. We rewrite the model as dx dz = H(X, Z), = G(X, Z),G(X, ) = (11) where X =(S H,S v ) and Z =(E 1,I 1,I,E 1,I c,e v,i v ), with the components of X R denoting the uninfected population and the components of Z R 7 denoting the infected population. The disease-free equilibrium is now denoted as E =(X, ),X =( Λ H μ H, Λ v μ v ). (1) The conditions in (13) must be met to guarantee a local asymptotic stability: dx = H(X, ),X is globally asymptotically stable (GAS) G(X, Z) = PZ Ĝ(X, Z), Ĝ(X, Z) for (X, Z) Ω (13) where P = D z G(X, ) is an M-matrix (the off-diagonal elements of P are non-negative) and Ω is the region where the model makes biological sense. If the system (11) satisfies the conditions of (13) then the theorem below holds. Theorem 3.. The fixed point E =(X, ) is a globally asymptotically stable equilibrium of system (11) provided that R mm < 1 and the assumptions in (13) are satisfied.

10 3 G. O. Lawi, J. Y. T. Mugisha, N. Omolo - Ongati Proof. From the model system () and (11), we have ( ) ΛH μ H(X, ) = H S H Λ v μ v S v where and Ĝ(X, Z) = P = G(X, Z) =PZ Ĝ(X, Z) h αa σ H h 7 h 8 β βκ h 9 ɛσ H h 1 αb αbδ h 11 σ v μ v Ĝ 1 (X, Z) Ĝ (X, Z) Ĝ 3 (X, Z) Ĝ (X, Z) Ĝ 5 (X, Z) Ĝ (X, Z) Ĝ 7 (X, Z) = αai v (1 S H N H )+λ me E 1 θλ me I 1 ρλ ma I + β(i + E 1 + κi c )(1 S H NH ) (λ me E 1 + ρλ ma I ) θλ me I 1 αb(i 1 + δi c )(1 Sv N H ) where h = σ H +μ H, h 7 = ψ+φ 1 +μ H, h 8 = β (φ +γ+μ H ), h 9 = ɛσ H +γ+μ H, h 1 = φ 3 + ϑψ + ηγ + μ H and h 11 = σ v + μ v. Ĝ (X, Z) <, Ĝ5(X, Z) < and so the conditions in (13) are not met so E may not be globally asymptotically stable when R mm < 1. Endemic equilibrium of the Model A disease is endemic in a population if it persists in the population. The endemic equilibrium of the model is studied using the Centre Manifold Theorem [1, 3]. Theorem.1. Castillo-Chavez and Song [3] Consider the following general system of ordinary differential equations with a parameter φ dx = f(x, φ), f : Rn R R n and f C (R n R) where is an equilibrium point of the system (i.e.f(,φ) for all φ) and

11 Malaria and meningitis co-infection A = D x f(, ) = ( f i x j (, )) is the linearization matrix of the system around the equilibrium point with φ evaluated at ;. Zero is a simple eigenvalue of A and all other eigenvalues of A have negative real parts; 3. Matrix A has a right eigenvector w and a left eigenvector v corresponding to the zero eigenvalue. Let f k be the kth component of f and s = r = n k,i,j=1 f k v k w i w j (, ), x i x j n f k v k w i (, ) x i a k,i=1 then the local dynamics of the system around the equilibrium point is totally determined by the signs of s and r. Particularly, (i) s >, r >, when a < with a 1, (, ) is locally asymptotically stable and there exists a positive unstable equilibrium; when <a 1, (, ) is unstable and there exists a negative and locally asymptotically stable equilibrium. (ii) s <, r <, when a < with a 1, (, ) is unstable; when <a 1, (, ) is asymptotically stable and there exists a positive unstable equilibrium. (iii) s >, r <, when a < with a 1, (, ) is unstable, and there exists a negative and locally asymptotically stable equilibrium; when < a 1, (, ) is stable and there exists a positive unstable equilibrium. (iv) s <, r >, when a < changes from negative to positive, (, ) changes its stability from stable to unstable. Correspondingly a negative equilibrium becomes positive and locally asymptotically stable. To apply this theorem we make the following change of variables. Let S H = x 1,E 1 = x,i 1 = x 3,I = x,e 1 = x 5,I 1 = x,s v = x 7,E v = x 8,I v = x 9 so that N H = x 1 +x +x 3 +x +x 5 +x and N v = x 7 +x 8 +x 9. The model () can be rewritten in the form dx = F (x) where X =(x 1,x,x 3,x,x 5,x,x 7,x 8,x 9 )

12 38 G. O. Lawi, J. Y. T. Mugisha, N. Omolo - Ongati and F =(f 1,f,f 3,f,f 5,f,f 7,f 8,f 9 )as dx 1 dx dx 3 dx dx 5 dx dx 7 dx 8 dx 9 = f 1 =Λ H λ c ma x 1 λ c me x 1 + φ 1 x 3 + φ x + φ 3 x μ H x 1, = f = λ c max 1 λ c mex (σ H + μ H )x, = f 3 = σ H x θλ c mex 3 (ψ + φ 1 + μ H )x 3, = f = λ c mex 1 ρλ c max (φ + γ + μ H )x, = f 5 = ρλ c ma x + λ c me x (ɛσ H + γ + μ H )x 5, (1) = f = ɛσ H x 5 + θλ c me x 3 (φ 3 + ϑψ + ηγ + μ H )x, = f 7 =Λ v (λ c v + μ v)x 7, = f 8 = λ c v x 7 (σ v + μ v )x 8, = f 9 = σ v x 8 μ v x 9. where λ c ma = αax 9,λ c NH c v = αb(x 3+δx ) and λ c NH c me = β(x +x 5 +κx ). NH c The jacobian of (1) at the DFE EMT o is given by J(E o )= μ H φ 1 β + φ β βκ + φ 3 αa K 1 αa σ H K K 3 β βκ K ɛσ H K 5 αbp αδbp μ v αbp αδbp K σ v μ v where K 1 = σ H + μ H, K = ψ + φ 1 + μ H, K 3 = β (φ + γ + μ H ), K = ɛσ H + γ + μ H, K 5 = φ 3 + ϑψ + ηγ + μ H, K = σ v + μ v and p = μ HΛ v Λ H μ v. To analyze the dynamics of (1), we compute the eigenvectors of the jacobian of (1) at the DFE. It can be shown that this jacobian has a right eigenvector given by W =(w 1,w,w 3,w,w 5,w,w 7,w 8,w 9 ) T, where w =,w 5 =,w = and w 1 = φ 1w 3 αaw 9 μ H,w = αaw 9 K 1,w 3 = αaσ H w 9 K 1 K,w 7 = αbpw 3 μ v,w 8 = μvw 9 σ v,w 9 = w 9 > and a left eigenvector given by V =(v 1,v,v 3,v,v 5,v,v 7,v 8,v 9 ) T where v 1 =

13 Malaria and meningitis co-infection 39,v 9 = and v = σ Hv 3 K 1,v 3 = v 3 >,v = α abpσ H v 3 K 1 K 3 K,v 5 = βv +ɛσ H v K,v = βκv αδbpv 7 K 5,v 7 = α δabpσ H v 3 μ vk 1 K,v 8 = αaσ H v 3 K 1 K. Consider the case when R mm = 1 (assuming that R me <R ma ) and choose a = a as a bifurcation parameter. Solving for a from R mm = R ma = 1 gives a = a = Λ Hμ v(σ H + μ H )(σ v + μ v )(ψ + φ 1 + μ H ) α bσ H σ v μ H Λ v (15) It can be shown after some manipulation involving the evaluation of the associated non-vanishing partial derivatives of f that s = μ H (v w 1 w 9 μ v αa + v w w 9 μ v αa + v w 3 w 9 μ v αa + Λ H μ v v w 7 w 9 μ v αa + v w 8 w 9 μ v αa + v 7 w 1 w 3 Λ v αb + v 7 w w 3 Λ v αb + v 7 w3 Λ vαb + v 7 w 3 w 7 Λ v αb v 8 w 1 w 3 Λ v αb v 8 w w 3 Λ v αb v 8 w3λ v αb v 8 w 3 w 7 Λ v αb) < and r = v w 9 α>. Thus we have established the following theorem Theorem.. The model () has a unique endemic equilibrium which is locally asymptotically stable when R mm < 1 and unstable when R mm > 1. 5 Disease Control, Socio-economic Challenges and Implications The applicability or importance of an epidemiological model lies in its ability to provide biologically meaningful interpretations and the possible disease control measures. Possible disease control strategies would be to reduce or guard against incidences of co-infection by keeping the prevalence of each disease at low levels or complete eradication of either disease. This could be achieved through prompt recognition of symptoms, correct diagnosis, effective treatment (and quarantine where possible) and prevention as we illustrate here. β From R me = φ +γ+μ H, we could rightly claim that R me is directly proportional to the mean time spent in the infective class given by 1 φ +γ+μ H. Clearly in the presence of prompt and effective treatment of meningitis infectives φ as γ. The implication of this is that R me and thus no new 1 meningitis infections, since φ +γ+μ H. Unfortunately, the treatment costs for meningitis infection are relatively higher as observed in the study conducted in Kenya by [18]. Besides, recent major advances in vaccine developments may

14 35 G. O. Lawi, J. Y. T. Mugisha, N. Omolo - Ongati not benefit the poor due to high costs and poor infrastructure for their delivery. This translates to increased susceptibility to meningitis infection. Since bacterial meningitis is highly infectious, the situation is compounded by the fact that in low socio-economic settings people reside in crowded places such as slums thus increasing the contact rates. α The reproduction number R ma = abσ H σ vμ H Λ v Λ H μ v(σ H +μ H )(σ v+μ v)(φ 1 +ψ+μ H on the other ) hand can be written as R ma = AB where A = aασv and μ v(σ v+μ v) αbσ B = H μ H Λ v Λ H μ v(σ H +μ H )(φ 1 +ψ+μ H. A represents the total number of secondary malaria ) infections in humans caused by one infected mosquito. This number is highly dependent on the mosquito biting rate α and the probability of mosquito survival till the infectious stage v σ σ v+μ v. Thus a reduction of the biting rate through such means as use of insecticide treated nets and indoor residual spraying would reduce infections in humans. A reduction in the probability of mosquito survival till the infectious stage would probably be the most effective control strategy. This can be achieved by prompt and effective treatment of infected humans or an intensive campaign for global vaccination to produce herd immunity. This makes a strong case for research in malaria vaccinology. The expression B, on the other hand, represents the total number of malaria infections in mosquitoes caused by a single infected human. It is directly proportional to the biting rate α, the probability of survival till infectious stage for σ humans H 1 σ H +μ H and the mean time spent in the infective class φ 1 +ψ+μ H. The life expectancy of the human is relatively longer and this survival probability may not be be reduced as in the latter case and thus prevention of infection would be desirable. Therefore a combination of optimal control strategies that would lead to A,B and thus R ma would guarantee no new malaria infections. Some of the cost-effective interventions used in malariaendemic areas include insecticide-treated nets (ITNs), intermittent preventive treatment in pregnancy (IPTp)and infancy (IPTi) and artemisinin-based combination therapy (ACT) [3]. People living in malaria-endemic areas are exposed to other diseases typically affecting the poor. These diseases not only take advantage of the compromised immunity due to the prolonged malaria exposure, coupled with limited and untimely chemotherapy but also present with malaria-like symptoms. This means that for an acute febrile patient that is infected with malaria, laboratory diagnosis for infections such as meningitis, pneumonia and diarrhea should be done so as to rule out or confirm co-infection. Failure to diagnose other coinfections means a delay in the initiation of their therapy and possibly ensuing sever complications to the patient [1]. As noted above malaria is endemic in low socio-economic settings. We also observe that in such settings the health

15 Malaria and meningitis co-infection 351 facilities are usually few and inadequate in terms of equipment and personnel. This could possibly lead to non performance of comprehensive laboratory tests. Consequently, most patients with fever resort to buying cheap and ineffective over-the-counter drugs, thus fueling the spread of disease. Parameter symbol Value Source Recruitment rate of humans Λ H day 1 [5] Recruitment rate of mosquitoes Λ v.71 day 1 [1] Natural death rate of humans μ H day 1 [5] Natural death rate of mosquitoes μ v.19 day 1 [15] Malaria-induced death rate ψ day 1 [] Meningitis-induced death rate γ.85 1 day 1 [7] Transmission probability a.8333 day 1 [15] for malaria in humans Transmission probability b V ariable day 1 Variable for malaria in mosquitoes Contact rate for β V ariable day 1 Variable meningitis infection Biting rate of mosquitoes α (.15, 1) day 1 Assumed Modification parameters δ, κ 1.5, 1.5 Assumed Modification parameters ɛ, ρ, θ 1.5, 1.5,.8 Assumed Modification parameters η,ϑ 1.5, 1.5 Assumed Recovery rate from malaria φ 1.55 Estimate Recovery rate from meningitis φ.5 Estimate Recovery rate from co-infection φ 3.75 Estimate Rate at which humans exposed to σ H.8333 [15] malaria develop symptoms Rate at which vectors exposed to σ v.1 [15] malaria develop symptoms Table1: Parameter Values Numerical simulations To illustrate some of the theoretical results arrived at in this paper, simulations of the model using various initial conditions were carried out using parameter values in Table 1. Since the study targeted children under the age of five years, some of the parameters values used in the simulation are specific to this age. For instance, the natural death rate of humans [5], malaria-induced death rate [] and meningitis-induced death rate [7], while others are estimated or assumed to vary within realistic limits. Whenever the respective reproduction

16 35 G. O. Lawi, J. Y. T. Mugisha, N. Omolo - Ongati numbers are less than unity, the infections reduce in time (Fig.1). However when the reproduction numbers are greater than unity the infections become endemic (Fig.3). We observe from the simulation that increase in malaria cases in humans has the effect of increasing the number of co-infections, probably due to the immunosuppressive effect of malaria (Fig.a, Fig.a). However, an increase in the number of meningitis infectives reduces the number of coinfected individuals, due to its high mortality and morbidity (Fig.b, Fig.b). The rise of malaria cases especially in the developing nations is thus a cause for worry. In these low resource settings poor diagnosis, incomplete and/or ineffective treatment have contributed to increasing parasite drug-resistance. Assuming that the recovery rate from infection is proportional to treatment levels, it is evident from (Fig.5a) that in the presence of effective(and prompt) malaria treatment, and no treatment for meningitis infectives or co-infected individuals (i.e φ = φ 3 = ), the number of human malaria infectives would drastically reduce and as a result there would be no appreciable rise in the number of co-infected humans (Fig.a) and meningitis infectives(fig.c). If effective treatment(or quarantine) is administered for meningitis infection, in the absence of treatment for malaria or co-infection cases, then the number of new meningitis infections would quickly reduce in time (Fig.7b). A reduction in the number of meningitis patients would slightly increase the number of co-infected individuals probably due to increased mobility (Fig.8b). When effective treatment is done for malaria, meningitis and co-infection cases, the number of co-infections is observed to greatly reduce (Fig.9). 7 Discussion A deterministic model for the dynamics of malaria and meningitis co-infection is presented and analysed. We establish the basic reproduction number R mm, which is the expected number of secondary infections produced by an infective in a completely susceptible population. The analysis shows that the diseasefree equilibrium of the model may not be globally asymptotically stable whenever R mm is less than unity. The Centre Manifold theorem is used to show that the model has a unique endemic equilibrium which is locally asymptotically stable when R mm < 1 and unstable when R mm > 1. These results are consistent with those of other co-infection models such as [17]. From the numerical simulations, we deduce that a reduction in malaria infection cases either through protection or prompt effective treatment, which is dependent on the socio-economic status of a community, would reduce the number of new co-infection cases. In the absence of good nutrition, sanitation and health care, preventable infectious diseases such as malaria and meningitis continue to thrive. These diseases are not only a threat to child survival but also the

17 Malaria and meningitis co-infection c a 5 1 Infected mosquitoes b 5 1 d 5 1 Figure 1: Simulation of model (), with α =.15,b =.15 and β =.3, giving R ma =.31895,R me =.,R mm =.31895, with varying initial conditions c a 8 b....8 Figure : Simulation of model (), with α =.15,b =.15 and β =.3, giving R ma =.31895,R me =.,R mm =.31895, with varying initial conditions.

18 35 G. O. Lawi, J. Y. T. Mugisha, N. Omolo - Ongati c a 5 1 Infected mosquitoes b 5 1 d Figure 3: Simulation of model (), with α =.,b =. and β =.15, giving R ma =3.3539,R me =1.131,R mm =3.3539, with varying initial conditions. 8 a c b Figure : Simulation of model (), with α =.,b =. and β =.15, giving R ma =3.3539,R me =1.131,R mm =3.3539, with varying initial conditions.

19 Malaria and meningitis co-infection c... a d Infected mosquitoes b 5 1 Figure 5: Simulation of model (), with φ 1 =.3,φ =,φ 3 =,α=.15,b=.15 and β =.3, with varying initial conditions. 8 a....8 c b Figure : Simulation of model (), with φ 1 =.3,φ =,φ 3 =,α=.15,b=.15 and β =.3, with varying initial conditions.

20 35 G. O. Lawi, J. Y. T. Mugisha, N. Omolo - Ongati c a d Infected mosquitoes b 5 1 Figure 7: Simulation of model (), with φ 1 =,φ =.3,φ 3 =,α=.15,b=.15 and β =.3, with varying initial conditions c a 8 b Figure 8: Simulation of model (), with φ 1 =,φ =.3,φ 3 =,α=.15,b=.15 and β =.3, with varying initial conditions.

21 Malaria and meningitis co-infection c a b Figure 9: Simulation of model (), with φ 1 =.55,φ =.5,φ 3 =.3,α =.15,b =.15 and β =.3, giving R ma =.31895,R me =.,R mm =.31895, with varying initial conditions. associated economic burden is a major hindrance to poverty reduction. References [1] A. Gemperli, P. Vounatsou, N. Sogoba and T. Smith, Malaria mapping using transmission models:application to survey data. American Journal of Epidemiology, 13(), [] C. Akukwe, Malaria and Tuberculosis: Forgotten Diseases, (), [3] C. Castillo-Chavez and B. Song, Dynamical models of tuberculosis and their applications, Mathematical Biosciences and Engineering, 1(), 31-. [] C. Castillo-Chavez, Z. Feng, and W. Huang, On the computation of R o and its role on global stability,in: Castillo-Chavez C., Blower S., van den Driessche P., Krirschner D. and Yakubu A.A.(Eds), Mathematical Approaches for Emerging and Reemerging Infectious Diseases: An Introduc-

22 358 G. O. Lawi, J. Y. T. Mugisha, N. Omolo - Ongati tion. The IMA Volumes in Mathematics and its Applications. Springer- Verlag, New York, 15(), 9-5. [5] Central Intelligence Agency-The World Factbook. on August 1, 1) [] C. Francesco, et al. Malaria Epidemics and intervention, Kenya, Burundi, Southern Sudan and Ethiopia Emerging Infectious Diseases, 1(), [7] D.R. Feikin, et al. Rapid Assessment Tool for Haemophilus Influenza type b Diseases in Developing countries. Emerging Infectious Diseases, 1(7)(), [8] G. Birkhoff and G.C. Rota, Ordinary Differential Equations, th Edition, John Wiley and Sons, Inc., New York, (1989). [9] J.A. Berkley, I. Mwangi, F. Mellington, S. Mwarumba, K. Marsh, Cerebral malaria versus bacterial meningitis in children with impaired consciousness. International Journal of Medicine,9(1999), [1] J. Carr, Applications of Centre Manifold Theory, Springer-Verlag, New York, (1981). [11] Kenya. Ministry of Health. National Guidelines for Diagnosis, Treatment and Prevention of Malaria for health workers in Kenya. Division of Malaria Control, Ministry of Health 8 edition, (8). [1] L. Roberts, An Ill Wind Bringing Meningitis. Science, 3 (8), 171. [13] M. English, J. Berkley, I. Mwangi et al, Hypothetical performance of syndrome-based management of acute pediatric admissions of children aged more than days in a Kenyan district hospital. Bull WHO. 81,(3), [1] Millenium Development Goals. (Accessed on nd February 11) [15] N. Chitnis, J.M. Cushing and J.M. Hyman, Bifurcation analysis of a mathematical model for malaria transmission SIAM J.Appl.Math., 7(1)(), -5. [1] O. Diekmann and J.A.P. Heesterbeek, Mathematical epidemiology of infectious diseases, Wiley series in Mathematical and Computational biology, John Wiley and Sons, West Sussex, England,.

23 Malaria and meningitis co-infection 359 [17] O. Sharomi, C.N. Podder, A.B. Gumel and B. Song, Mathematical analysis of the transmission dynamics of HIV/TB coinfection in the presence of treatment. Mathematical Biosciences and Engineering. 5(8), [18] P. Ayieko, A.O. Akumu, U.K. Griffiths and M. English, The economic burden of inpatient paediatric care in Kenya: household and provider costs for treatment of pneumonia, malaria and meningitis, Cost Eff Resour Alloc. 7,(9), [19] P. van den Driessche, and J. Watmough, Reproduction numbers and the sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci., 18(), 9-8. [] R.D. Snyder, Bacterial Meningitis:Diagnosis and Treatment. Current Neurology and Neuroscience Reports, 3 (3), 1-9. [1] S. Gwer, C.R. Newton and J.A. Berkley, Over-diagnosis and co-morbidity of severe malaria in African children: A guide for clinicians, American journal of Tropical Medicine and Hygiene 77, (7), -13. [] T. Irving, K. Blyuss, C. Colijn and C. Trotter, Does Resonance Account for the Epidemiology of Meningococcal Meningitis in the African Meningitis Belt? University of Bristol, Bristol Centre for Complexity Sciences,(9). [3] World Health Organization, World Malaria Report 8, WHO/HTM/GMP (8).1. Received: February, 11