Mathematical Model for Malaria and Meningitis Co-infection ... · 2338 G. O. Lawi, J. Y. T. Mugisha, N. Omolo - Ongati 1 Introduction ... 500 million cases of malaria occur worldwide,

Applied Mathematical Sciences, Vol. 5, 2011, no. 47, 2337 - 2359

Mathematical Model for Malaria and Meningitis

Co-infection among Children

Lawi G. O. 1, Mugisha J. Y. T. 2 and N. Omolo - Ongati3

1Department of Mathematics, Masinde Muliro University of Science and TechnologyBox 190, Kakamega 50100, Kenya

2 Department of Mathematics, Makerere UniversityBox 7062, Kampala, Uganda

3 Department of Mathematics and Applied statistics, Maseno UniversityPrivate Bag, Maseno, Kenya

Abstract

Disease and poverty are a major threat to child survival in the devel-oping world, where access to good nutrition, sanitation and health careis poor. In this paper, a mathematical model for malaria and menin-gitis co-infection among children under five years of age is developedand analysed. We establish the basic reproduction number Rmm for themodel, which is a measure of the course of the co-infection. The analysisshows that the disease-free equilibrium of the model may not be glob-ally asymptotically stable whenever Rmm is less than unity. The CentreManifold theorem is used to show that the model has a unique endemicequilibrium which is locally asymptotically stable when Rmm < 1 andunstable when Rmm > 1. We deduce further that a reduction in malariainfection cases either through protection or prompt effective treatment,which is dependent on the socio-economic status of a community, wouldreduce the number of new co-infection cases.

Mathematics Subject Classification: 34D20

Keywords: Malaria-meningitis model, Co-infection, Basic reproductionnumber, Stability, Socio-economic

1Correponding author’s email: [email protected]

2338 G. O. Lawi, J. Y. T. Mugisha, N. Omolo - Ongati

1 Introduction

Millenium Development Goal 4 and Goal 6 target improved child survivaland reversing the high prevalence of diseases such as HIV/AIDS and malaria[14].Disease and poverty are a major threat to child survival in the develop-ing world, where access to good nutrition, sanitation and health care is poor.In such settings HIV/AIDS, malaria and many other preventable infectiousdiseases continue to kill millions of children. Malaria enjoys a cause-effectrelationship with poverty and is therefore a major hindrance to economic de-velopment. It costs Africa some $10 billion to $12 billion every year in lostgross domestic product [14]. Importantly, people living in malaria-endemic ar-eas are frequently exposed to other diseases typically affecting the poor. Thesediseases not only take advantage of the compromised immunity due to the pro-longed malaria exposure, coupled with limited and untimely chemotherapy butalso present with malaria-like symptoms.

Humans acquire malaria following infective bites from infected Anopheles fe-male mosquitoes during blood feeding. Plasmodium falciparum is the parasitespecies that largely causes human malaria infections in Africa. Each year 350-500 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[23]. Malaria was the fourth cause of death in children in developing countriesin 2002. In Kenya it accounts for 19% of all hospital admissions, 30% of alloutpatient visits, with an estimate of 20% of all deaths in children less than fiveyears of age being attributed to the disease [11]. The incidence of malaria hasbeen on the rise in the recent past due to increasing parasite drug-resistanceand mosquito insecticide-resistance. This rise has also been associated withclimate change [2].

Meningitis is an infectious disease characterized by inflammation of the meninges(the tissues that surround the brain or spinal cord), usually due to the spreadof an infection into the cerebral spinal fluid (CSF). The cause of the infectionmay be bacterial, viral, fungal or parasitic. Some of the risk factors for thedisease are a compromised immune system due to illness, such HIV/AIDS oruse of immunosuppressant drugs. The symptoms of meningitis include neckand/or back pain, headache, high fever and a stiff neck. Bacterial menin-gitis may cause acute or chronic brain injury and thus leading to death orlong-term disability (such as deafness, paralysis, seizure and even mental re-tardation) [20]. The seasonal outbreak of meningitis in the African meningitisbelt, a band of sub-Saharan Africa, usually results into a high disease mortal-ity and morbidity[22]. An outbreak in 1996-1997, documented as the largest

Malaria and meningitis co-infection 2339

claimed more than more than 25 000 lives, with about 250 000 cases of illnessacross 10 countries. In contrast, during the non-epidemic year of 2008 therewere only 27 000 cases across the whole of the belt [12].

Malaria and meningitis have a symptom overlap. Malaria endemicity coupledwith poverty causes a challenge in diagnosis and prompt treatment of infec-tions with malaria-like symptoms. This is because in such settings diagnosisis usually clinically done and most cases are thought to be malaria[13]. Fail-ure or delay of correct diagnosis may result into severity of the infection orco-infection[21]. In the study by [9] carried out in Kenya, 4% of the childrenadmitted in the hospital were found to be infected with both malaria and acutebacterial meningitis. The study noted that co-infection played a major role inthe group of children with high mortality.In this paper we thus develop a mathematical model to study the dynamicsof malaria-meningitis co-infection. We assume that there is no simultaneousinfection of a host with the two diseases.

2 Model Description and Formulation

The total human population at any time t, denoted NH is subdivided into sub-population susceptible humans (SH), those exposed to malaria parasites only(E1), individuals infected with malaria ( I1), those infected with meningitis(I2), individuals exposed to malaria and infected with meningitis (E12) andindividuals infected with both malaria and meningitis (Ic). The total vectorpopulation at any time t, denoted Nv is subdivided into subpopulation of sus-ceptible (Sv), exposed (Ev) and infectious (Iv). This means that

NH = SH + E1 + I1 + I2 + E12 + Ic (1)

and

Nv = Sv + Ev + Iv (2)

The rates of infection of susceptible humans with malaria and meningitis areλma and λme respectively, while that of susceptible vectors with malaria isλv. Let ψ and γ be malaria and meningitis induced mortality in humansrespectively, and suppose that μH and μv are per capita natural death ratesof the human and mosquito populations respectively. The constant per capitarecruitment rate into the susceptible human and vector populations are ΛH andΛv respectively. The rates at which exposed human and vector populationsdevelop malaria clinical symptoms are σH and σv respectively, while the rateat which humans progress from the E12 class to the Ic class is εσH , where


ε is a modification parameter representing the assumption that meningitisinfected individuals exposed to malaria develop malaria symptoms at a fasterrate than those who are not infected with meningitis. Define φ1 as the rate atwhich individuals infected with malaria recover, φ2 as the recovery rate frommeningitis and φ3 as the recovery rate from both infections. The recoveredindividuals do not acquire temporary immunity to either or both diseases andthus become susceptible again.

We assume that infection with meningitis when one is exposed to malaria takesplace at an advanced stage of this exposure. The parameter θ accounts for theincreased susceptibility to infection with meningitis for individuals infectedwith malaria, while the parameter ρ accounts for the decreased susceptibilityto infection with malaria for individuals infected with meningitis because ofdecreased contact due to ill health. The individuals displaying symptoms ofboth malaria and meningitis suffer malaria-induced mortality at the rate ϑψ,where the parameter ϑ accounts for the assumed increase in malaria-relatedmortality due to the dual infection with meningitis and also suffer meningitis-induced mortality at the rate ηγ, where the parameter η accounts for theassumed increase in meningitis-related mortality due to the dual infection withmalaria. Define α as the number of bites per human per mosquito (biting rateof mosquitoes), a as the transmission probability of malaria in humans perbite, b as the transmission probability of malaria in vectors from any infectedhuman, β as the effective contact rate for infection with meningitis.

This yields

λma =αaIvNH

(3)

λv =αb(I1 + δIc)

NH

(4)

λme =β(I2 + E12 + κIc)

NH, (5)

where δ and κ model the relative infectiousness of the co-infected individualas compared to their counterparts.


From the above definitions and variables we have the following model

dSHdt

= ΛH − λmaSH − λmeSH + φ1I1 + φ2I2 + φ3Ic − μHSH ,

dE1

dt= λmaSH − λmeE1 − σHE1 − μHE1,

dI1dt

= σHE1 − θλmeI1 − ψI1 − φ1I1 − μHI1,

dI2dt

= λmeSH − ρλmaI2 − φ2I2 − γI2 − μHI2,

dE12

dt= ρλmaI2 + λmeE1 − (εσH + γ + μH)E12, (6)

dIcdt

= εσHE12 + θλmeI1 − (φ3 + ϑψ + ηγ + μH)Ic,

dSvdt

= Λv − λvSv − μvSv,

dEvdt

= λvSv − σvEv − μvEv,

dIvdt

= σvEv − μvIv.

2.1 Positivity and Boundedness of solutions

Model (6) describes the human and mosquito populations and therefore itcan be shown that the associated state variables are non-negative for all timet ≥ 0 and that the solutions of the model (6) with positive initial data remainspositive for all time t ≥ 0. We assume the associated parameters as non-negative for all time t ≥ 0. We show that all feasible solutions are uniformly-bounded in a proper subset Ψ = ΨH × Ψv.

Theorem 2.1. Solutions of the model (6) are contained in the region Ψ =ΨH × Ψv.

Proof. To show that all feasible solutions are uniformly-bounded in a propersubset Ψ, we split the model (6) into the human component (NH) and themosquito component (Nv), given by equations (1) and (2) respectively.Let

(SH , E1, I1, I2, E12, Ic) ∈ R6+

be any solution with non-negative initial conditions. From the theorem by [8]on differential inequality it follows that

lim supt→∞

SH(t) ≤ ΛH

μH.


Taking the time derivative of NH along a solution path of the model (6) gives

dNH

dt= ΛH − μHNH − ψI1 − γI2 − (ϑψ + ηγ)Ic

Then,

dNH

dt≤ ΛH − μHNH

From the theorem by [8] on differential inequality it follows that

0 ≤ NH ≤ ΛH

μH+NH(0)e−μHt

where NH(0) represents the value of (1) evaluated at the initial values of therespective variables. Thus as t→ ∞, we have

0 ≤ NH ≤ ΛH

μH(7)

This shows that NH is bounded and all the feasible solutions of the human-only component of model (6) starting in the region ΨH approach, enter or stayin the region, where

ΨH = {(SH , E1, I1, I2, E12, Ic) : NH ≤ ΛH

μH}

Similarly,let

(Sv, Ev, Iv) ∈ R3+

be any solution with non-negative initial conditions. Then

lim supt→∞

Sv(t) ≤ Λv

μv.

Taking the time derivative of Nv along a solution path of the model (6) gives

dNv

dt= Λv − μvNv

The mosquito-only component (2) has a varying population size.Therefore,

dNv

dt< Λv − μvNv


From the theorem by [8] on differential inequality it follows that

0 ≤ Nv ≤ Λv

μv+Nv(0)e−μvt

where Nv(0) represents the value of (2) evaluated at the initial values of therespective variables. Thus as t→ ∞, we have

0 ≤ Nv ≤ Λv

μv

This shows that Nv is bounded and all the feasible solutions of the mosquito-only component of model (6) starting in the region Ψv approach, enter or stayin the region, where

Ψv = {(Sv, Ev, Iv) : Nv ≤ Λv

μv} (8)

Thus it follows from (7) and (8) that NH and Nv are bounded and all thepossible solutions of the model starting in Ψ will approach, enter or stay inthe region Ψ = ΨH × Ψv ∀t ≥ 0.

Thus Ψ is positively invariant under the flow induced by (6). Existence, unique-ness and continuation results also hold for the model (6) in Ψ. Hence model(6) is well-posed mathematically and epidemiologically and it is sufficient toconsider its solutions in Ψ.

3 Disease-free equilibrium point

Disease-free equilibrium (DFE) points of a disease model are its steady-statesolutions in the absence of infection or disease. We denote this point by E0

and define the ”diseased” classes as the human or mosquito populations thatare either exposed or infectious. Define the positive orthant in R9 by R9

+ andthe boundary of R9

+ by ∂R9+.

Lemma 3.1. For all equilibrium points on Ψ∩∂R9+, E1 = I1 = I2 = E12 =

Ic = Ev = Iv = 0

The positive DFE for human and mosquito populations for the model (6)are

NH =ΛH

μHand Nv =

Λv

μv. (9)

Lemma 3.2. The model (6) has exactly one DFE, E0 = (ΛH

μH, 0, 0, 0, 0, 0, 0, Λv

μv, 0, 0)


Proof. The proof of the lemma requires that we show that DFE is the onlyequilibrium point of (6) on Ψ∩∂R

9+. Substituting E0 into (6) shows all deriva-

tives equal to zero, hence DFE is an equilibrium point. From Lemma 3.1, theonly equilibrium point for NH is ΛH

μHand the only equilibrium point for Nv is

Λv

μv. Thus the only equilibrium point for Ψ ∩ ∂R

9+ is DFE.

3.1 Local stability of the disease-free equilibrium

The global dynamics of the model (6) is highly dependent on the basic repro-duction number. The basic reproduction number is defined as the expectednumber of secondary infections produced by an index case in a completelysusceptible population [16]. We define the basic reproduction number, Rmm

as the number of secondary malaria (or meningitis) infections due to a singlemalaria (or a single meningitis-infective) individual. We determine Rmm usingthe next generation operator approach [19]. The associated next generationmatrices are

F=

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 αa0 0 0 0 0 0 00 0 β β βκ 0 00 0 0 0 0 0 00 0 0 0 0 0 0

0 αbΛvμH

ΛHμv0 0 αbδΛvμH

ΛHμv0 0

0 0 0 0 0 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠and

V=

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

h1 0 0 0 0 0 0−σH h2 0 0 0 0 0

0 0 h3 0 0 0 00 0 0 h4 0 0 00 0 0 −εσH h5 0 00 0 0 0 0 h6 00 0 0 0 0 −σv μv

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠where h1 = σH + μH, h2 = ψ+ φ1 + μH, h3 = φ2 + γ +μH , h4 = εσH + γ + μHh5 = φ3 + ϑψ + ηγ + μH and h6 = σv + μv.The basic reproduction number Rmm is the spectral radius of the matrixFV −1. The eigenvalues of the matrix FV −1 are 0, 0, 0, 0, β

φ2+γ+μHand ±√

α2abσHσvμHΛv

ΛHμ2v(σH+μH)(σv+μv)(φ1+ψ+μH)

.

Therefore Rmm is given by

Rmm = max{√

α2abσHσvμHΛv

ΛHμ2v(σH + μH)(σv + μv)(φ1 + ψ + μH)

,β

φ2 + γ + μH}. (10)


Denoting Rma =√

α2abσHσvμHΛv

ΛHμ2v(σH+μH)(σv+μv)(φ1+ψ+μH)

and Rme = βφ2+γ+μH

, we have

Rmm = max{Rma, Rme}. Rma is a measure of the average number of secondarymalaria infections in human or mosquito population caused by a single infec-tive human or mosquito introduced into an entirely susceptible population.Similarly, Rme is a measure of the average number of secondary meningitisinfections in humans caused by a single infective human introduced into anentirely susceptible population. The following lemma follows from Theorem 2of [19].

Lemma 3.3. The the disease-free equilibrium E0 of the model (6) is locallyasymptotically stable whenever Rmm < 1 and unstable when Rmm > 1.

3.2 Global stability of the disease-free equilibrium

The global asymptotic stability (GAS) of the disease-free state of the modelis investigated using the theorem by Castillo-Chavez et.al [4]. We rewrite themodel as

dX

dt= H(X,Z),

dZ

dt= G(X,Z), G(X, 0) = 0 (11)

where X = (SH , Sv) and Z = (E1, I1, I2, E12, Ic, Ev, Iv), with the componentsof X ∈ R2 denoting the uninfected population and the components of Z ∈ R7

denoting the infected population.The disease-free equilibrium is now denoted as

E0 = (X∗, 0), X∗ = (ΛH

μH,Λv

μv). (12)

The conditions in (13) must be met to guarantee a local asymptotic stability:

dX

dt= H(X, 0), X∗ is globally asymptotically stable (GAS)

G(X,Z) = PZ − G(X,Z), G(X,Z) ≥ 0 for (X,Z) ∈ Ω (13)

where P = DzG(X∗, 0) is an M-matrix (the off-diagonal elements of P arenon-negative) and Ω is the region where the model makes biological sense. Ifthe system (11) satisfies the conditions of (13) then the theorem below holds.

Theorem 3.4. The fixed point E0 = (X∗, 0) is a globally asymptoticallystable equilibrium of system (11) provided that Rmm < 1 and the assumptionsin (13) are satisfied.


Proof. From the model system (6) and (11), we have

H(X, 0) =

(ΛH − μHSHΛv − μvSv

)

G(X,Z) = PZ − G(X,Z)

where

P =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

−h6 0 0 0 0 0 αaσH −h7 0 0 0 0 00 0 h8 β βκ 0 00 0 0 −h9 0 0 00 0 0 εσH −h10 0 00 αb 0 0 αbδ −h11 00 0 0 0 0 σv −μv

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠and

G(X,Z) =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

G1(X,Z)

G2(X,Z)

G3(X,Z)

G4(X,Z)

G5(X,Z)

G6(X,Z)

G7(X,Z)

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠=

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

αaIv(1 − SH

NH) + λmeE1

θλmeI1ρλmaI2 + β(I2 + E12 + κIc)(1 − SH

NH)

−(λmeE1 + ρλmaI2)−θλmeI1

αb(I1 + δIc)(1 − Sv

NH)

0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠where h6 = σH+μH , h7 = ψ+φ1+μH , h8 = β−(φ2+γ+μH), h9 = εσH+γ+μH ,h10 = φ3 + ϑψ + ηγ + μH and h11 = σv + μv.G4(X,Z) < 0, G5(X,Z) < 0 and so the conditions in (13) are not met so E0

may not be globally asymptotically stable when Rmm < 1.

4 Endemic equilibrium of the Model

A disease is endemic in a population if it persists in the population. The en-demic equilibrium of the model is studied using the Centre Manifold Theorem[10, 3].

Theorem 4.1. Castillo-Chavez and Song [3]Consider the following general system of ordinary differential equations with aparameter φ

dx

dt= f(x, φ), f : R

n × R → Rn and f ∈ C

2(Rn × R)

where 0 is an equilibrium point of the system (i.e.f (0, φ) ≡ 0 for all φ) and


1. A = Dxf(0, 0) = ( ∂fi

∂xj(0, 0)) is the linearization matrix of the system

around the equilibrium point 0 with φ evaluated at 0;

2. Zero is a simple eigenvalue of A and all other eigenvalues of A havenegative real parts;

3. Matrix A has a right eigenvector w and a left eigenvector v correspondingto the zero eigenvalue.

Let fk be the kth component of f and

s∗ =

n∑k,i,j=1

vkwiwj∂2fk∂xi∂xj

(0, 0),

r∗ =n∑

k,i=1

vkwi∂2fk∂xi∂a∗

(0, 0)

then the local dynamics of the system around the equilibrium point 0 is totallydetermined by the signs of s∗ and r∗. Particularly,

(i) s∗ > 0, r∗ > 0, when a∗ < 0 with |a∗| 1, (0, 0) is locally asymptoticallystable and there exists a positive unstable equilibrium; when 0 < a∗ 1,(0, 0) is unstable and there exists a negative and locally asymptoticallystable equilibrium.

(ii) s∗ < 0, r∗ < 0, when a∗ < 0 with |a∗| 1, (0, 0) is unstable; when0 < a∗ 1, (0, 0) is asymptotically stable and there exists a positiveunstable equilibrium.

(iii) s∗ > 0, r∗ < 0, when a∗ < 0 with |a∗| 1, (0, 0) is unstable, and thereexists a negative and locally asymptotically stable equilibrium; when 0 <a∗ 1, (0, 0) is stable and there exists a positive unstable equilibrium.

(iv) s∗ < 0, r∗ > 0, when a∗ < 0 changes from negative to positive, (0, 0)changes its stability from stable to unstable. Correspondingly a negativeequilibrium becomes positive and locally asymptotically stable.

To apply this theorem we make the following change of variables. Let SH =x1, E1 = x2, I1 = x3, I2 = x4, E12 = x5, I12 = x6, Sv = x7, Ev = x8, Iv = x9 sothat NH = x1 +x2+x3+x4+x5+x6 and Nv = x7 +x8+x9. The model (6) canbe rewritten in the form dX

dt= F (x) where X = (x1, x2, x3, x4, x5, x6, x7, x8, x9)


and F = (f1, f2, f3, f4, f5, f6, f7, f8, f9) as

dx1

dt= f1 = ΛH − λcmax1 − λcmex1 + φ1x3 + φ2x4 + φ3x6 − μHx1,

dx2

dt= f2 = λcmax1 − λcmex2 − (σH + μH)x2,

dx3

dt= f3 = σHx2 − θλcmex3 − (ψ + φ1 + μH)x3,

dx4

dt= f4 = λcmex1 − ρλcmax4 − (φ2 + γ + μH)x4,

dx5

dt= f5 = ρλcmax4 + λcmex2 − (εσH + γ + μH)x5, (14)

dx6

dt= f6 = εσHx5 + θλcmex3 − (φ3 + ϑψ + ηγ + μH)x6,

dx7

dt= f7 = Λv − (λcv + μv)x7,

dx8

dt= f8 = λcvx7 − (σv + μv)x8,

dx9

dt= f9 = σvx8 − μvx9.

where λcma = αax9

NcH, λcv = αb(x3+δx6)

NcH

and λcme = β(x4+x5+κx6)Nc

H.

The jacobian of (14) at the DFE EoMT is given by

J(Eo)=

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

−μH 0 φ1 −β + φ2 −β −βκ + φ3 0 0 −αa0 −K1 0 0 0 0 0 0 αa0 σH −K2 0 0 0 0 0 00 0 0 K3 β βκ 0 0 00 0 0 0 −K4 0 0 0 00 0 0 0 εσH −K5 0 0 00 0 −αbp 0 0 −αδbp −μv 0 00 0 0 αbp 0 0 αδbp 0 −K6

0 0 0 0 0 0 0 σv −μv

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠where K1 = σH + μH , K2 = ψ + φ1 + μH , K3 = β − (φ2 + γ + μH),K4 = εσH + γ + μH , K5 = φ3 + ϑψ + ηγ + μH , K6 = σv + μv and p = μHΛv

ΛHμv.

To analyze the dynamics of (14), we compute the eigenvectors of the jacobianof (14) at the DFE. It can be shown that this jacobian has a right eigenvectorgiven by

W = (w1, w2, w3, w4, w5, w6, w7, w8, w9)T , where w4 = 0, w5 = 0, w6 = 0 and

w1 = φ1w3−αaw9

μH, w2 = αaw9

K1, w3 = αaσHw9

K1K2, w7 = −αbpw3

μv, w8 = μvw9

σv, w9 = w9 > 0

and a left eigenvector given by V = (v1, v2, v3, v4, v5, v6, v7, v8, v9)T where v1 =


0, v9 = 0 and v2 = σHv3K1

, v3 = v3 > 0, v4 = −α2abpσHv3K1K3K6

, v5 = βv4+εσHv6K4

, v6 =βκv4−αδbpv7

K5, v7 = α2δabpσHv3

μvK1K6, v8 = αaσHv3

K1K6.

Consider the case when Rmm = 1 (assuming that Rme < Rma) and choosea = a∗ as a bifurcation parameter. Solving for a from Rmm = Rma = 1 gives

a = a∗ =ΛHμ

2v(σH + μH)(σv + μv)(ψ + φ1 + μH)

α2bσHσvμHΛv

(15)

It can be shown after some manipulation involving the evaluation of the asso-ciated non-vanishing partial derivatives of f that

s∗ =−2μHΛHμv

(v2w1w9μvαa+ v2w2w9μvαa+ v2w3w9μvαa+

v2w7w9μvαa+ v2w8w9μvαa+ v7w1w3Λvαb+ v7w2w3Λvαb+

v7w23Λvαb+ v7w3w7Λvαb− v8w1w3Λvαb−

v8w2w3Λvαb− v8w23Λvαb− v8w3w7Λvαb) < 0 and

r∗ = v2w9α > 0.

Thus we have established the following theorem

Theorem 4.2. The model (6) has a unique endemic equilibrium which islocally asymptotically stable when Rmm < 1 and unstable when Rmm > 1.

5 Disease Control, Socio-economic Challenges

and Implications

The applicability or importance of an epidemiological model lies in its abilityto provide biologically meaningful interpretations and the possible disease con-trol measures. Possible disease control strategies would be to reduce or guardagainst incidences of co-infection by keeping the prevalence of each disease atlow levels or complete eradication of either disease. This could be achievedthrough prompt recognition of symptoms, correct diagnosis, effective treat-ment (and quarantine where possible) and prevention as we illustrate here.From Rme = β

φ2+γ+μH, we could rightly claim that Rme is directly propor-

tional to the mean time spent in the infective class given by 1φ2+γ+μH

. Clearlyin the presence of prompt and effective treatment of meningitis infectivesφ2 → ∞ as γ → 0. The implication of this is that Rme → 0 and thus no newmeningitis infections, since 1

φ2+γ+μH→ 0. Unfortunately, the treatment costs

for meningitis infection are relatively higher as observed in the study conductedin Kenya by [18]. Besides, recent major advances in vaccine developments may


not benefit the poor due to high costs and poor infrastructure for their deliv-ery. This translates to increased susceptibility to meningitis infection. Sincebacterial meningitis is highly infectious, the situation is compounded by thefact that in low socio-economic settings people reside in crowded places suchas slums thus increasing the contact rates.

The reproduction number Rma =√

α2abσHσvμHΛv

ΛHμ2v(σH+μH )(σv+μv)(φ1+ψ+μH)

on the other

hand can be written as Rma =√AB where A = aασv

μv(σv+μv)and

B = αbσHμHΛv

ΛHμv(σH+μH)(φ1+ψ+μH). A represents the total number of secondary malaria

infections in humans caused by one infected mosquito. This number is highlydependent on the mosquito biting rate α and the probability of mosquito sur-vival till the infectious stage σv

σv+μv. Thus a reduction of the biting rate through

such means as use of insecticide treated nets and indoor residual sprayingwould reduce infections in humans. A reduction in the probability of mosquitosurvival till the infectious stage would probably be the most effective controlstrategy. This can be achieved by prompt and effective treatment of infectedhumans or an intensive campaign for global vaccination to produce herd im-munity. This makes a strong case for research in malaria vaccinology. Theexpression B, on the other hand, represents the total number of malaria infec-tions in mosquitoes caused by a single infected human. It is directly propor-tional to the biting rate α, the probability of survival till infectious stage forhumans σH

σH+μHand the mean time spent in the infective class 1

φ1+ψ+μH. The

life expectancy of the human is relatively longer and this survival probabilitymay not be be reduced as in the latter case and thus prevention of infectionwould be desirable. Therefore a combination of optimal control strategies thatwould lead to A → 0, B → 0 and thus Rma → 0 would guarantee no newmalaria infections. Some of the cost-effective interventions used in malaria-endemic areas include insecticide-treated nets (ITNs), intermittent preventivetreatment in pregnancy (IPTp)and infancy (IPTi) and artemisinin-based com-bination therapy (ACT) [23].

People living in malaria-endemic areas are exposed to other diseases typicallyaffecting the poor. These diseases not only take advantage of the compro-mised immunity due to the prolonged malaria exposure, coupled with limitedand untimely chemotherapy but also present with malaria-like symptoms. Thismeans that for an acute febrile patient that is infected with malaria, laboratorydiagnosis for infections such as meningitis, pneumonia and diarrhea should bedone so as to rule out or confirm co-infection. Failure to diagnose other co-infections means a delay in the initiation of their therapy and possibly ensuingsever complications to the patient [21]. As noted above malaria is endemic inlow socio-economic settings. We also observe that in such settings the health


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 ineffectiveover-the-counter drugs, thus fueling the spread of disease.

Parameter symbol Value SourceRecruitment rate of humans ΛH 9.6274 × 10−5 day−1 [5]Recruitment rate of mosquitoes Λv 0.071 day−1 [1]Natural death rate of humans μH 2.537 × 10−5 day−1 [5]Natural death rate of mosquitoes μv 0.1429 day−1 [15]Malaria-induced death rate ψ 4.49312 × 10−4 day−1 [6]Meningitis-induced death rate γ 6.8445 × 10−4 day−1 [7]Transmission probability a 0.8333 day−1 [15]for malaria in humansTransmission probability b V ariableday−1 Variablefor malaria in mosquitoesContact rate for β V ariableday−1 Variablemeningitis infectionBiting rate of mosquitoes α (0.125, 1) day−1 AssumedModification parameters δ, κ 1.0005, 1.05 AssumedModification parameters ε, ρ, θ 1.0025, 1.0025, 0.80 AssumedModification parameters η, ϑ 1.0005, 1.00025 AssumedRecovery rate from malaria φ1 0.00556 EstimateRecovery rate from meningitis φ2 0.00065 EstimateRecovery rate from co-infection φ3 0.00075 EstimateRate at which humans exposed to σH 0.08333 [15]malaria develop symptomsRate at which vectors exposed to σv 0.1 [15]malaria develop symptoms

Table1: Parameter Values

6 Numerical simulations

To illustrate some of the theoretical results arrived at in this paper, simulationsof the model using various initial conditions were carried out using parametervalues in Table 1. Since the study targeted children under the age of fiveyears, some of the parameters values used in the simulation are specific to thisage. For instance, the natural death rate of humans [5], malaria-induced deathrate [6] and meningitis-induced death rate [7], while others are estimated orassumed to vary within realistic limits. Whenever the respective reproduction


numbers are less than unity, the infections reduce in time (Fig.1). Howeverwhen the reproduction numbers are greater than unity the infections becomeendemic (Fig.3). We observe from the simulation that increase in malaria casesin humans has the effect of increasing the number of co-infections, probablydue to the immunosuppressive effect of malaria (Fig.2a, Fig.4a). However,an increase in the number of meningitis infectives reduces the number of co-infected individuals, due to its high mortality and morbidity (Fig.2b, Fig.4b).The rise of malaria cases especially in the developing nations is thus a causefor worry. In these low resource settings poor diagnosis, incomplete and/orineffective treatment have contributed to increasing parasite drug-resistance.

Assuming that the recovery rate from infection is proportional to treatmentlevels, it is evident from (Fig.5a) that in the presence of effective(and prompt)malaria treatment, and no treatment for meningitis infectives or co-infectedindividuals (i.e φ2 = φ3 = 0), the number of human malaria infectives woulddrastically reduce and as a result there would be no appreciable rise in thenumber of co-infected humans (Fig.6a) and meningitis infectives(Fig.6c). Ifeffective treatment(or quarantine) is administered for meningitis infection, inthe absence of treatment for malaria or co-infection cases, then the number ofnew meningitis infections would quickly reduce in time (Fig.7b). A reductionin the number of meningitis patients would slightly increase the number ofco-infected individuals probably due to increased mobility (Fig.8b). Wheneffective treatment is done for malaria, meningitis and co-infection cases, thenumber of co-infections is observed to greatly reduce (Fig.9).

7 Discussion

A deterministic model for the dynamics of malaria and meningitis co-infectionis presented and analysed. We establish the basic reproduction number Rmm,which is the expected number of secondary infections produced by an infectivein a completely susceptible population. The analysis shows that the disease-free equilibrium of the model may not be globally asymptotically stable when-ever Rmm is less than unity. The Centre Manifold theorem is used to showthat the model has a unique endemic equilibrium which is locally asymptot-ically stable when Rmm < 1 and unstable when Rmm > 1. These resultsare consistent with those of other co-infection models such as [17]. From thenumerical simulations, we deduce that a reduction in malaria infection caseseither through protection or prompt effective treatment, which is dependenton the socio-economic status of a community, would reduce the number ofnew co-infection cases. In the absence of good nutrition, sanitation and healthcare, preventable infectious diseases such as malaria and meningitis continueto thrive. These diseases are not only a threat to child survival but also the


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Figure 1: Simulation of model (6), with α = 0.125, b = 0.125 and β =0.0003, giving Rma = 0.318905, Rme = 0.22062, Rmm = 0.318905, with vary-ing initial conditions.

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Figure 2: Simulation of model (6), with α = 0.125, b = 0.125 and β =0.0003, giving Rma = 0.318905, Rme = 0.22062, Rmm = 0.318905, with vary-ing initial conditions.


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Figure 3: Simulation of model (6), with α = 0.6, b = 0.6 and β =0.0015, giving Rma = 3.35369, Rme = 1.1031, Rmm = 3.35369, with varyinginitial conditions.

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Figure 4: Simulation of model (6), with α = 0.6, b = 0.6 and β =0.0015, giving Rma = 3.35369, Rme = 1.1031, Rmm = 3.35369, with varyinginitial conditions.


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Figure 9: Simulation of model (6), with φ1 = 0.00556, φ2 = 0.00065, φ3 =0.3, α = 0.125, b = 0.125 and β = 0.0003, giving Rma = 0.318905, Rme =0.22062, Rmm = 0.318905, with varying initial conditions.

associated economic burden is a major hindrance to poverty reduction.

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Received: February, 2011

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