A co-infection model of malaria and cholera diseases with optimal control.

ARTICLE IN PRESS

JID: MBS

[m5G;September 23, 2014;14:20]

Mathematical Biosciences 000 (2014) 1–14

Contents lists available at ScienceDirect

Mathematical Biosciences journal homepage: www.elsevier.com/locate/mbs

A co-infection model of malaria and cholera diseases with optimal control K.O. Okosun a,∗, O.D. Makinde b

Q1

a b

Department of Mathematics, Vaal University of Technology, X021, Vanderbijlpark, 1900, South Africa Faculty of Military Science, Stellenbosch University, Private Bag X2, Saldanha, 7395, South Africa

a r t i c l e

i n f o

Article history: Received 6 May 2013 Revised 30 May 2014 Accepted 13 September 2014 Available online xxx Keywords: Malaria Cholera Stability Centre manifold theorem Optimal control

Q2

a b s t r a c t In this paper we formulate a mathematical model for malaria--cholera co-infection in order to investigate their synergistic relationship in the presence of treatments. We first analyze the single infection steady states, calculate the basic reproduction number and then investigate the existence and stability of equilibria. We then analyze the co-infection model, which is found to exhibit backward bifurcation. The impact of malaria and its treatment on the dynamics of cholera is further investigated. Secondly, we incorporate time dependent controls, using Pontryagin’s Maximum Principle to derive necessary conditions for the optimal control of the disease. We found that malaria infection may be associated with an increased risk of cholera but however, cholera infection is not associated with an increased risk for malaria. Therefore, to effectively control malaria, the malaria intervention strategies by policy makers must at the same time also include cholera control. © 2014 Published by Elsevier Inc.

1

1. Introduction

2

Malaria is a preventable and curable vector borne disease. The strategy for reducing malaria transmission is to protect individuals from mosquito bites by the distribution of inexpensive mosquito nets and insect repellents or by mosquito control measures such as indoor spraying of insecticides and draining of stagnant water where mosquitoes breed [27]. The recent flooding across Africa and Asia continents posed a serious challenge to good environmental sanitation and availability of clean water thereby providing breeding condition for malaria and cholera to strive [2]. The study of the epidemiology of cholera was heralded by John Snow and this began the modern epidemiology research [13]. The link between contaminated drinking water and cholera was long established. Cholera is a severe bacterial infection that produces profuse watery diarrhea and vomiting and can lead to severe dehydration and electrolyte imbalance and finally death if this is not corrected. Cholera is transmitted through ingestion of water contaminated (usually from feces or effluent) with the bacterium Vibrio cholerae. Human colonization of cholera creates a hyper infectious state that is maintained after dissemination and this contributes to epidemic disease. Prevention of cholera is achieved by good sanitation and water treatment [33]. Mathematical modeling has been an important tool in understanding the dynamics of disease transmission and also in decision making

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

∗

Corresponding author. Tel.: +27 795718766/169509539. E-mail address: [email protected] (K.O. Okosun).

processes regarding intervention mechanisms for disease control. For example, Ross [32] developed the first mathematical models of malaria transmission. His focus was on mosquito control and he showed that for the disease to be eliminated the mosquito population should be brought below a certain threshold. Other studies include Koella and Anita [16] who included a latent class for mosquitoes. They considered different strategies to reduce the spread of resistance and studied the sensitivity of their results to the parameters. Anderson and May [5] derived a malaria model with the assumption that acquired immunity in malaria is independent of exposure duration. Different control measures and the role of transmission rate on the disease prevalence were further examined. Nikolaos et al. [26] proposed a detailed analysis of a dynamical model to describe pathogenesis of HIV infection. Christopher and Jorge [8] derived a simple two-dimensional SIS (susceptible--infected--susceptible) model with vaccination and multiple endemic states. Guihua and Zhen [11] studied the global dynamics of an SEIR (susceptible--exposed--infected--recovered) epidemic model in which latent and immune states were infective. However, a few studies have been carried out on the formulation and application of optimal control theory to cholera models. To the best of our knowledge no work has been done to investigate the malaria--cholera co-infection dynamics or the application of optimal control methods. Only recently, the authors in [20] proposed and examined a deterministic model for the co-infection of HIV and malaria in a community. Also, the authors in [19] examined a deterministic model for the co-infection of tuberculosis and malaria, while in [23] the authors proposed a model for schistosomiasis and HIV/AIDS co-dynamics. The authors in [25] formulated a mathematical model

http://dx.doi.org/10.1016/j.mbs.2014.09.008 0025-5564/© 2014 Published by Elsevier Inc.

Please cite this article as: K.O. Okosun, O.D. Makinde, A co-infection model of malaria and cholera diseases with optimal control, Mathematical Biosciences (2014), http://dx.doi.org/10.1016/j.mbs.2014.09.008

24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51

JID: MBS 2

ARTICLE IN PRESS K.O. Okosun, O.D. Makinde / Mathematical Biosciences 000 (2014) 1–14

84

for cholera to include essential components such as hyper-infectious, short-lived bacterial state, a separate class for mild human infections and waning disease immunity. Also in [33], the authors presented a model for cholera epidemics which comprises seasonality, loss of host immunity and control mechanisms acting to reduce cholera transmission. A mathematical model of cholera transmission was examined in [24] to study the impact of public health educational campaigns, vaccination and treatment as control strategies in curtailing the disease. The authors in [36] carried out global stability analysis for deterministic cholera epidemic models. In [34], a general compartmental model for cholera was formulated that incorporates two pathways of transmission. A simple mathematical model was presented in [23] to assess whether HIV infection is associated with an increased risk for cholera or not. Authors in [35] studied a mathematical model for cholera that incorporates hyper infectivity and temporary immunity using distributed delays. In this paper, we formulate an SIR (susceptible, infected and recovered) model for malaria--cholera co infection with the optimal control problem. Our model include five different control strategies, namely malaria prevention (treated bednets), cholera prevention (sanitations and proper hygiene), malaria treatment, cholera treatment and combined therapy for malaria--cholera infection as time dependent control strategies, in order to determine the optimal strategy for the control of the diseases. The paper is organized as follows: Section 2 is devoted to the model description and the underlying assumptions. In Section 3, we analyze the cholera only model. Section 4 is devoted to the analysis of the malaria only model while in Section 5 the co-infection model is analyzed. In Section 6 we use Pontryagin’s Maximum Principle to investigate analysis of control strategies and to determine the necessary conditions for the optimal control of the disease. In Sections 7 and 8 we discuss the numerical methods used and the numerical results respectively. Our conclusion is presented in Section 9.

85

2. Model formulation

86

The model sub-divides the total human population, denoted by Nh , into sub-populations of susceptible humans Sh , individuals infected with malaria only Im , individuals infected with cholera only Ic , individuals infected with both malaria and cholera Gmc , individuals who recovered from malaria only Rm , individuals who recovered from cholera only Rc , individuals who recovered from both malaria and cholera Rmc . So that Nh = Sh + Im + Ic + Gmc + Rm + Rc + Rmc . The total vector population, denoted by Nv , is sub-divided into susceptible mosquitoes Sv , mosquitoes infected with malaria Iv . Thus, Nv = Sv + I v . The model is given by the following system of ordinary differential equations:

52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83

87 88 89 90 91 92 93 94 95 96 97

98

⎧d S = h + κ Rm + ωRc + ψ Rmc − βh Iv Sh − λSh − μh Sh ⎪ dt h ⎪ ⎪ ⎪ d ⎪ I = βh Iv Sh − λIm − (α + μh + φ)Im ⎪ ⎪ ⎪ dt m ⎪ ⎪ d ⎪ ⎪ ⎪ dt Ic = λSh − βh Iv Ic − (δ + μh + m)Ic ⎪ ⎪ d ⎪ ⎪ ⎪ dt Gmc = βh Iv Ic + λIm − (σ + μh + η + q)Gmc ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ d Rm = α Im − (κ + μ )Rm + (1 − σ )Gmc h dt d ⎪ R = δ I − (ω + μ ) Rc + (1 − )(1 − σ )Gmc ⎪ c c h ⎪ dt ⎪ ⎪ ⎪ d R = σ G − (ψ + μ )R ⎪ mc h mc ⎪ dt mc ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ d ⎪ B = ρ(Ic + θ Gmc ) − μb Bc ⎪ dt c ⎪ ⎪ ⎪ d ⎪ S = v − βv (Im + Gmc )Sv − μv Sv ⎪ ⎪ dt v ⎪ ⎩d I = βv Sv (Im + Gmc ) − μv Iv dt v

(1)

Bc ν , K + Bc

where Bc is the bacteria population, the ingestion rate is ν and K is the bacteria concentration. Also, m,η are cholera related death rates respectively, and φ , q are malaria related death rates respectively, while ρ is the average contribution of each cholera infected individual to the aquatic population of V. cholerae. The immunity waning rates are κ ,ω,ψ respectively and α ,δ ,σ respectively are the recovery rates. The term (1 − σ ) are the dually infected individuals who recovered from malaria only. And (1 − )(1 − σ ) are the dually infected individuals who recovered from cholera only. That is, σ + (1 − σ ) + (1 − )(1 − σ ) = 1. While μh and μv are respectively the humans and mosquitoes mortality rates and θ is the modification parameter. Our assumptions in the model are as follows: • •

Mosquitoes do not suffer mosquito-induced death. Individuals infected with both malaria and cholera can only infect mosquitoes with malaria parasites.

3. Cholera only model Here, we consider the cholera only model.

101 102 103 104 105 106 107 108 109 110 111 112 113 114

116

(3)

3.1. Stability of the disease-free equilibrium (DFE)

117

The cholera only model (3) has a DFE, obtained by setting the right-hand sides of the equations in the model to zero, given by

119

E 0c = (Sh∗ , Ic∗ , R∗c , B∗c ) =

The linear stability of E0c can be established using the next generation operator method in Driessche and Watmough [9] on the system (3). It follows that the reproduction number of the cholera only model (3), denoted by R0c , is given by R0c =

118

v , 0, 0, 0 . μh

νρ v , μb μh K (m + δ + μh )

120 121 122 123

(4)

Further, using Theorem 2 in Driessche and Watmough [9], the following result is established. The DFE is locally asymptotically stable if R0c < 1 and unstable if R0c > 1.

125

3.1.1. Existence of endemic equilibrium Lemma 1. The cholera only model has a unique endemic equilibrium if and only if R0c > 1.

128

Proof. Calculating the endemic equilibrium point, we obtain,

130

⎧ h + ωR∗c ⎪ ∗ ⎪ ⎪Sh = ⎪ μh + λ∗ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ λ∗ Sh∗ ⎪ ⎪ ⎪Ic∗ = ⎪ ⎨ m + δ + μh

P (λ

∗

124 126 127 129

(5)

δ Ic∗ ω + μh ρ Ic∗ μb

Hence, the cholera force of infection (see (2)), λ∗ , satisfies the following polynomial

(2)

99 100

115

⎧d ⎪ ⎪ dt Sh = h + ωRc − λSh − μh Sh ⎪ ⎪ ⎨ d Ic = λS − (δ + μ + m)Ic h h dt ⎪ d Rc = δ Ic − (ω + μh )Rc ⎪ ⎪ dt ⎪ ⎩d B = ρ I c − μb Bc dt c

⎪ ⎪ ⎪ ⎪ R∗c = ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩B∗c =

Here,

λ=

[m5G;September 23, 2014;14:20]

) = A(λ ) + B(λ ) = 0 ∗ 2

∗

(6)


131 132

ARTICLE IN PRESS

JID: MBS

[m5G;September 23, 2014;14:20]

K.O. Okosun, O.D. Makinde / Mathematical Biosciences 000 (2014) 1–14

Im + Gmc

Iv

μv Iv

βv Im

Sv μ v Sv

μb Bc

Bc

Sh

ρθIc ρIc

μh Ic

Λh

λSh

μh Im

Im βh Sh

μh Sh

3

αIm λIm (1 − σ)Gmc

(η + μh )Gmc

Gmc

Rm

μh Rm

σGmc

Ic

gGmc

Rmc

δIc Rc

μh Rmc

μh Rc

Fig. 1. Flow diagram for co-infection model. Please note that g = (1 − )(1 − σ ) in the flow chart.

133

where,

A = h ρ(μh + ω) + μb K (m(ω + μh ) + μh (δ + μh + ω)), B= 134 135 136

(7)

Clearly, A > 0 and B ≥ 0 whenever R0c < 1 so that λ∗ = −B/A 0. Therefore the model has no endemic equilibrium whenever R0c < 1.

139

The above result suggests the impossibility of backward bifurcation in the cholera model, since no endemic equilibrium exists when R0c < 1 (fig. 1).

140

4. Malaria only model

137 138 Q3

(ω + μh )(1 − R0c ),

141

144 145 146

R0m = 147 148 149

v h βh βv μh μ2v (α + φ + μh )

150 152

(10)

⎪ ⎪ v ⎪ ⎪ Sv∗ = ⎪ ⎪ μ + βv Im∗ ⎪ v ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ β S∗ I ∗ ⎪ ⎪ ⎩Iv∗ = v v m

154

(11)

where,

155

βv (v βh (κφ + μh (α + κ + φ + μh )) + μh (κ + μh )(α + φ + μh )μv ),

Bb = (κ + μh ) 1 − R20m ,

Aa = (8)

(12)

∗ = −B /A Clearly, Aa > 0 and Bb ≥ 0 whenever R0m < 1 so that Im a b 0. Therefore the model has no endemic equilibrium whenever R0m < 1.

v v , 0, 0, ,0 . μh μv

The linear stability of E0m can be established using the next generation operator method in Driessche and Watmough [9] on the system (8). It follows that the reproduction number of the malaria only model (8), denoted by R0m , is given by

153

⎧ h + κ R∗m ⎪ ⎪ Sh∗ = ⎪ ⎪ μh + βh Iv∗ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∗ ⎪ ⎪ ⎪R∗ = α Im ⎪ m ⎪ ⎨ κ + μh

∗ Pm (Im ) = Aa (Im∗ )2 + Bb (Im∗ ) = 0

The malaria only model (8) has a DFE, obtained by setting the right-hand sides of the equations in the model to zero, given by

143

Proof. Calculating the endemic equilibrium point, we obtain,

The endemic equilibrium satisfies the following polynomial

4.1. Stability of the disease-free equilibrium (DFE)

∗ , R∗m , Sv∗ , Iv∗ ) = E0m = (Sh∗ , Im

151

μv

In this section, we consider the malaria only model.

⎧ d ⎪ ⎪ Sh = h + κ Rm − βh Iv Sh − μh Sh ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ d ⎪ ⎪ Im = βh Iv Sh − (α + μh + φ)Im ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎨ d Rm = α Im − (κ + μh )Rm ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪d ⎪ ⎪ Sv = v − βv Im Sv − μv Sv ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ d Iv = βv Sv Im − μv Iv dt 142

4.1.1. Existence of endemic equilibrium Lemma 2. The malaria only model has a unique endemic equilibrium if and only if R0m > 1.

(9)

Further, using Theorem 2 in Driessche and Watmough [9], the following result is established. The DFE is locally asymptotically stable if R0m < 1 and unstable if R0m > 1.

156 157 158

The above result suggests the impossibility of backward bifurcation in the model, since no endemic equilibrium exists when R0m < 1.

159

5. Malaria–cholera co-infection model

162

The malaria--cholera model (1) has a DFE, obtained by setting the right-hand sides of the equations in the model to zero, given by

164

E0 =

=

(

∗ Sh∗ , Im , Ic∗ , G∗mc , R∗m , R∗c , R∗mc , B∗c , Sv∗ , Iv∗

160 161

163 165

)

h v , 0, 0, 0, 0, 0, 0, 0, ,0 . μv μv

The linear stability of E0 can be established using the next generation operator method in Driessche and Watmough [9] on the system (1). It follows that the reproduction number of the malaria--cholera model (1), denoted by Rmc , is given by Rmc = max{Rm , Rc }


166 167 168 169

ARTICLE IN PRESS

JID: MBS 4

170


where

now differentiating Rm with respect to Rc , we get

v h βh βv , μh μ2v (α + φ + μh ) νρ h (η + q + σ + μh ) + θ (m + δ + μh ) Rc = . μb μh K (m + δ + μh )(η + q + σ + μh )

Rm =

171 172 173 174 175

176

178

5.1. Impact of cholera on malaria

2 μv R m

where,

and letting,

D21 R2m + 4D2 = D3 Rm + D4 , hence, Rm (D3 − D1 ) + D4 = 2 μv R m

βh = βh∗ = (14)

Rc =

Then we make the following change of variables Sh = x1 , Im = x2 , Ic = x3 , Gmc = x4 , Rm = x5 , Rc = x6 , Rmc = x7 , Bc = x8 , Sv = x9 , Iv = x10 , and N = x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 + x10 . In addition, using vector notation x = (x1 , x2 , x3 , x4 , x5 , x6 , x7 , x8 , x9 , x10 )T , the malaria--cholera model can then be written in the form dx/dt = F (x), with F = (f1 , f2 , f3 , f4 , f5 , f6 , f7 , f8 , f9 , f10 )T , as shown below:

185 186 187

Whenever m + δ ≥ η + q + σ is strictly positive, it implies that malaria enhances cholera infection. In other words whenever Eq. (16) is greater than zero, an increase in malaria cases results in increase of cholera cases in the community. If Eq. (16) is equal to zero, implying that malaria cases have no significant effect on the transmission dynamics of cholera. Similarly expressing μh in terms of Rc , we get

μh = 188

F1 − F2 Rc +

F3 R2c + F4 Rc + F5 2R c

(17)

where

F1 = (1 + θ )R0c ,

F2 = m + η + q + δ + σ ,

η+q+σ −m−δ F4 = 2(θ − 1)(m + δ − η − q − σ )R0c , F5 = (1 + θ )2 R20c F3 =

189

also let

μh =

F3 R2c + F4 Rc + F5 = F6 Rc + F7 .

(F6 − F2 )Rc + F7 + F1 2R c

R2m =

4v h βv βh R2c [F1 + F7 + (F6 − F2 )Rc ][F1 + F7 + 2(α + φ)Rc + (F6 − F2 )Rc ]μv (18)

195 196 197 198 199 200 201 202

203

μ μ K (m + δ + μh )(η + q + σ + μh ) ν = ν∗ = b h ρ h (η + q + σ + μh + θ (m + δ + μh ))

⎧ d ⎪ ⎪ x1 ⎪ ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎪ d ⎪ ⎪ x2 ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ d ⎪ ⎪ x3 ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ d ⎪ ⎪ x4 ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ d ⎪ ⎪ ⎪ x ⎪ ⎨ dt 5

194

μh μ (α + φ + μh ) v h βv

and

(16)

193

2 v

substituting (14) into the expression for Rc , we obtain

Differentiating Rc partially with respect to Rm leads to

191 192

(20)

5.1.1. Existence of backward bifurcation The phenomenon of backward bifurcation can be proved by using centre manifold theory on system (1). Adopting the Center Manifold theorem [7], we carry out bifurcation analysis. First, we consider the transmission rate βh and ν as bifurcation parameters so that Rm = 1 and Rc = 1 if and only if

D2 = v h βv βh ,

2D4 θ (m + δ − (η + q + σ ))μv R0c ∂ Rc = ∂ Rm [D4 + Rm (D3 − D1 + 2(η + q + σ )μv ]2

184

Whenever (19) is greater than zero, it means that an increase in cholera cases results in an increase of malaria cases in the community. The impact of malaria treatment on cholera is evaluated by partially differentiating Rm with respect to α . We have

Since Rm is a decreasing function of α , the treatment of cholera will have a positive impact on the dynamics of malaria.

180

183

(19)

μh =

−D1 Rm + D21 R2m + 4D2

(15)

182

∂ Rm = ∂ Rc 4(F1 + F7 )v h Rc [F1 + F7 + (α + φ + F6 − F2 )Rc ]βv βh [F1 + F7 + (F6 − F2 )Rc ]2 [F1 + F7 + (2(α + φ) + F6 − F2 )Rc ]2 μv

∂ Rm α =− ∂α α + φ + μh

R0c (D4 + (D3 − D1 )Rm + 2(η + q + σ )μv Rm + θ(D4 + (D3 − D1 )Rm + 2(m + δ)μv Rm )) . D4 + (D3 − D1 )Rm + 2(η + q + σ )μv Rm

181

190

To analyze the effects of cholera on malaria and vice versa, we begin by expressing Rc in terms of Rm . We solve for μh and we get

μh 179

(13)

Theorem 1. The disease free equilibrium E0 is locally asymptotically stable whenever Rmc < 1 and unstable otherwise.

D1 = μv (α + φ), 177

[m5G;September 23, 2014;14:20]

204 205 206 207 208 209 210

= h + κ x5 + ωx6 + ψ x7 − βh x10 x1 − λx1 − μh x1 = βh x10 x1 − λx2 − (α + μh + φ)x2 = λx1 − βh x10 x3 − (δ + μh + m)x3 = βh x10 x3 + λx2 − (σ + μh + η + φ r)x4 = α x2 − (κ + μh )x5 + (1 − σ )x4

⎪ d ⎪ ⎪ ⎪ x6 = δ x3 − (ω + μh )x6 + (1 − )(1 − σ )x4 ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ d ⎪ ⎪ ⎪ x = σ x4 − (ψ + μh )x7 ⎪ ⎪ dt 7 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ d ⎪ ⎪ x8 = ρ(x3 + θ x4 ) − μb x8 ⎪ ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ d ⎪ ⎪ x = v − βv (x2 + x4 )x9 − μv x9 ⎪ ⎪ ⎪ dt 9 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ d x10 = βv x9 (x2 + x4 ) − μv x10 dt

(21)

This method involves evaluation of the Jacobian of the system (21) at the disease free equilibrium (DFE) E0 , denoted by J(E0 ). This becomes


211 212

ARTICLE IN PRESS

JID: MBS

[m5G;September 23, 2014;14:20]


⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ J(E0 ) = ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 213

J3 J5 J8 J11

215 216 217 218 219

0

J1

κ

ω

ψ

0

J3

0

−J4

0

0

0

0

0

0

0

J3

0

0

−J5

J1

0

0

0

J2

0

0

0

0

0

−J6

0

0

0

0

0

0

0

α

0

J7

−J8

0

0

0

0

0

0

0

δ

J9

0

−J10

0

0

0

0

0

0

0

σ

0

0

−J11

0

0

0

0

0

ρc θ ρ c

0

0

0

−μb

0

0

0

−J12

0

−J12

0

0

0

0

−μv

0

0

J12

0

J12

0

0

0

0

0

−μv

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

J(E0 ) has a simple zero eigenvalue, with other eigenvalues having negative real parts. Hence, the Center Manifold theorem ([7]) can be applied. For this we need to calculate a and b. We first start by calculating the right and the left eigenvector of J(E0 ) denoted respectively by w = [w1 , w2 , w3 , w4 , w5 , w6 , w7 , w8 , w9 , w10 ]T , and v = [v1 , v2 , v3 , v4 , v5 , v6 , v7 , v8 , v9 , v10 ]. We obtain

v10 =

−μv (η + q + σ + μh )

v βv

v2 =

v10 βv v , μv (α + φ + μh )

.

After rigorous computations, it can be shown that

τ w10 μ (η + q + σ + μh ) μ (η + q + σ + μh ) − 2w10 βv μh v βv 2v βv2 2 α K μv (η + q + σ + μh ) + , v βv μh (κ + μh )(α + φ + μh ) h b = v2 w10 > 0. μh a=

223 224 225

mc

h

⎪ d ⎪ ⎪ Rm = u3 α Im − (κ + μh )Rm + (1 − u5 σ )Gmc ⎪ ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎪ d ⎪ ⎪ Rc = u4 δ Ic − (ω + μh )Rc + (1 − )(1 − u5 σ )Gmc ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ d ⎪ ⎪ ⎪ Rmc = u5 σ Gmc − (ψ + μh )Rmc ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎪ d ⎪ ⎪ Bc = (1 − u2 )ρ(Ic + θ Gmc ) − μb Bc ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎪ d ⎪ ⎪ ⎪ Sv = v − (1 − u1 )βv (Im + Gmc )Sv − μv Sv ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ d Iv = (1 − u1 )βv Sv (Im + Gmc ) − μv Iv dt (22)

The control functions, u1 (t), u2 (t), u3 (t), u4 (t) and u5 (t) are bounded, Lebesgue integrable functions. The control u1 (t) and u2 (t) represents the efforts on preventing malaria and cholera infections respectively. The control on treatment of malaria infected individuals u3 (t) satisfies 0 u3 g2 , where g2 is the drug efficacy use for treatment of malaria infected individuals. The control on treatment of cholera infected individuals u4 (t) satisfies 0 u4 g3 , where g3 is the drug efficacy use for treatment of cholera infected individuals and the control on treatment of co-infected individuals u5 (t) satisfies 0 u5 g4 , where g4 is the drug efficacy use for treatment of co-infected individuals. Our control problem involves a situation in which the number of malaria infected individuals, cholera infected individuals, co-infected individuals and the cost of applying preventions and treatments controls u1 (t), u2 (t), u3 (t), u4 (t) and u5 (t) are minimized subject to the system (22). tf is the final time and the coefficients, z1 , z2 , z3 , z4 , A, B, C, D, E are the balancing cost factors due to scales and importance of the ten parts of the objective function. We seek to find an optimal control, u∗1 , u∗2 , u∗3 , u∗4 and u∗5 , such that

w10 = 1

and

v4 = v4 ,

222

5

w9 = −w10 ,

+

2 v

3 v

2 v

Since the coefficient b is always positive, it follows from Theorem [7] that the system (1) will undergo backward bifurcation if the coefficient a is positive. This is implying that the disease free is not globally stable.

230 231

⎧ d ⎪ ⎪ Sh = h + κ Rm + ωRc + ψ Rmc − (1 − u1 )βh Iv Sh ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎪ Bc ν ⎪ ⎪ ⎪ − (1 − u2 ) Sh − μ h Sh ⎪ ⎪ K + Bc ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪d Bc ν ⎪ ⎪ Im = (1 − u1 )βh Iv Sh − (1 − u2 ) Im − (u3 α + μh + φ)Im ⎪ ⎪ dt K + Bc ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ d Bc ν ⎪ ⎪ Sh − (1 − u1 )βh Iv Ic − (u4 δ + μh + m)Ic ⎪ Ic = (1 − u2 ) ⎪ ⎪ dt K + Bc ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Bc ν ⎪d ⎪ ⎪ Gmc = (1 − u1 )βh Iv Ic + (1 − u2 ) Im ⎪ ⎪ dt K + Bc ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ − (u σ + μ + η + q)G

For this, we consider the objective functional

μh

w2 μ

the model (1) to determine the optimal strategy for controlling the disease. Hence we have,

(α + φ + μh ) μ , w2 = , μh v βv αμ2v w3 = w4 = w6 = w7 = w8 = 0, w5 = , v βv (κ + μh ) Kw5

2 v

v1 = v3 = v5 = v6 = v7 = v8 = v9 = 0,

221

J2

ρ h μb (δ + m + μh )(η + q + σ + μh ) = , J2 = , μh ρ(η + q + σ + μh + θ (m + δ + μh )) μ3 (α + φ + μh ) = v , J4 = α + φ + μh , v βv = m + δ + μh , J6 = η + q + σ + μh , J7 = (1 − σ ), = κ + μh , J9 = (1 − )(1 − σ ), J10 = ω + μh v βv = ψ + μh , J12 = μv

w1 =

220

0

where

J1

214

−μh

5

J(u1 , u2 , u3 , u4 , u5 ) =

tf 0

232

[z1 Im + z2 Ic + z3 Gmc + z4 Iv + Au21

+ Bu22 + Cu23 + Du24 + Eu25 ]dt

(23) 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251

J(u∗1 , u∗2 , u∗3 , u∗4 , u∗5 ) = min{J(u1 , u2 , u3 , u4 , u5 )|u1 , u2 , u3 , u4 , u5 ∈ U } (24)

226

6. Analysis of optimal control

227

In this section, we apply Pontryagin’s Maximum Principle to determine the necessary conditions for the optimal control of the malaria-cholera co-infection. We incorporate time dependent controls into

228 229

where U = {(u1 , u2 , u3 , u4 , u5 ) such that u1 , u2 , u3 , u4 , u5 are measurable with 0 ≤ u1 ≤ 1, 0 ≤ u2 ≤ 1, 0 ≤ u3 ≤ g2 , 0 ≤ u4 ≤ g3 and 0 ≤ u5 ≤ g4 , for t ∈ [0, tf ]} is the control set. The necessary conditions that an optimal solution must satisfy come from the Pontryagin et al. [30] Maximum Principle. This


252 253 254 255 256

ARTICLE IN PRESS

JID: MBS 6

257 258


principle converts (22) and (23) into a problem of minimizing pointwise a Hamiltonian H, with respect to u1 , u2 , u3 , u4 and u5

H = z1 Im + z2 Ic + z3 Gmc + z4 Iv + Au21 + Bu22 + Cu23 + Du24 + Eu25 + MSh h + κ Rm + ωRc + ψ Rmc − (1 − u1 )βh Iv Sh Bc ν − (1 − u2 ) Sh − μ h Sh K + Bc Bc ν + MIm (1 − u1 )βh Iv Sh − (1 − u2 ) Im K + Bc

and

u∗5 = min 1, max 0,

−

−

−

+ MBc {(1 − u2 )ρ(Ic + θ Gmc ) − μb Bc }

+ MSv {v − (1 − u1 )βv (Im + Gmc )Sv − μv Sv }

262 263 264 265 266 267

268 269

(25)

where MSh , MIm , MIc , MGmc , MRm , MRc , MRmc , MBc , MSv and MIv are the adjoint variables or co-state variables. The system of equations is found by taking the appropriate partial derivatives of the Hamiltonian (25) with respect to the associated state variable. Theorem 2. Given optimal controls u∗1 , u∗2 , u∗3 , u∗4 , u∗5 and solutions Sh , Im , Ic , Gmc , Rm , Rc , Rmc , Bc , Sv , Iv of the corresponding state system (22) and (23) that minimize J(u1 , u2 , u3 , u4 , u5 ) over U. Then there exists adjoint variables MSh , MIm , MIc , MGmc , MRm , MRc , MRmc , MBc , MSv , MIv satisfying

−dMi = dt

∂H ∂i

(26)

where i = Sh , Im , Ic , Gmc , Rm , Rc , Rmc , Bc , Sv , Iv and with transversality conditions

270

(27)

and

u∗1 = min 1, max 0,

βh Iv Sh (MIm − MSh ) + βh Iv Ic (MGma − MIc ) + Yy 2A

, (28)

271

u∗2 = min ⎧ ⎛ ⎨ 1, max ⎝0, ⎩

Bc ν(MIc −MSh )Sh K+Bc

+

Bc ν(MGmc −MIm )Im K+Bc

2B

+ ρ(Ic + θ Gmc )MBc

dMSh = dt

−

dMIm dt

u∗3 273

= min 1, max 0,

u∗4 = min 1, max 0,

α(MIm − MRm )Im

⎞⎫ ⎬ ⎠ , ⎭

.

δ(MIc − MRc )Ic 2D

278 279 280 281 282 283

μh MSh + (1 − u1 )βh Iv (MSh − MIm )

Bc ν (MSh − MIc ) K + Bc Bc ν = −z1 − u3 α MRm + (1 − u2 ) (MIm − MGmc ) K + Bc + (u3 α + μh + φ)MIm + (1 − u1 )βv Sv (MSv − MIv )

+ (1 − u2 )βv Sv (MSv − MIv ) dMRm − dt dMRc − dt dMRmc − dt dMBc − dt

= −κ MSh + (κ + μh )MRm = −ωMSh + (ω + μh )MRc = −ψ MSh + (ψ + μh )MRmc = (1 − u2 )Sh

K

(K + Bc )2

+ (1 − u2 )Im

(MSh − MIc )

K

(K + Bc )2

(MIm − MGmc ) + μb MBc

dMSv = (1 − u1 )βv (Im + Gmc )(MSv − MIv ) + μv MSv dt dMIv = −z4 + (1 − u1 )βh Sh (MSh − MIm ) − dt + (1 − u1 )βh Ic (MIc − MGmc ) + μv MIv Solving for u∗1 , u∗2 , u∗3 , u∗4

(30)

(31)

(33)

and u∗5

subject to the constraints, the characterization (28)--(32) can be derived and we have

0=

0=

0=

.

276 277

dMGmc = −z3 + (u5 σ + μh + η + q)MGmc − (1 − u5 σ )MRm dt − (1 − )(1 − u5 σ )MRc − u5 σ MRmc − (1 − u2 )ρθ MBc

0=

2C

275

dMIc = −z2 + (1 − u1 )βv Iv (MIc − MGmc ) − u4 δ MRc dt − (1 − u2 )ρ MBc + (u4 δ + μh + m)MIc

(29) 272

.

−

MSh (tf ) = MIm (tf ) = MIc (tf ) = MGmc (tf ) = MRm (tf ) = MRc (tf ) = MRmc (tf ) = MBc (tf ) = MSv (tf ) = MIv (tf ) = 0

2E

+ (1 − u2 )

+ MRmc {u5 σ Gmc − (ψ + μh )Rmc }

261

Proof. Corollary 4.1 of Fleming and Rishel [10] gives the existence of an optimal control due to the convexity of the integrand of J with respect to u1 , u2 , u3 , u4 and u5 , a priori boundedness of the state solutions, and the Lipschitz property of the state system with respect to the state variables. The differential equations governing the adjoint variables are obtained by differentiation of the Hamiltonian function, evaluated at the optimal control. Then the adjoint equations can be written as

+ MRc {u4 δ Ic − (ω + μh )Rc + (1 − )(1 − u5 σ )Gmc }

260

σ (MGmc + MRm + (1 − )MRc − MRmc )Gmc

where Yy = βv (Im + Gmc )Sv (MIv − MSv ).

+ MRm {u3 α Im − (κ + μh )Rm + (1 − u5 σ )Gmc }

+ MIv {(1 − u1 )βv Sv (Im + Gmc ) − μv Iv .}

274

(32)

− (u3 α + μh + φ)Im } Bc ν + MIc (1 − u2 ) Sh − (1 − u1 )βh Iv Ic K + Bc − (u4 δ + μh + m)Ic Bc ν + MGmc (1 − u1 )βh Iv Ic + (1 − u2 ) Im K + Bc − (u5 σ + μh + η + q)Gmc

259

[m5G;September 23, 2014;14:20]

0=

∂H = 2Au1 + βh Iv Sh (MSh − MIm ) + βh Iv Ic (MIc − MGmc ) ∂ u1 + βv (Im + Gmc )Sv (MSv − MIv ) B c ν Sh B νI ∂H = 2Bu2 + (MSh − MIc ) + c m (MIm − MGmc ) ∂ u2 K + Bc K + Bc − ρ(Ic + θ Gmc )MBc ∂H = 2Cu3 + α(MRm − MIm )Im ∂ u3 ∂H = 2Du4 + δ(MRc − MIc )Ic ∂ u4 ∂H = 2Cu5 + σ (MRmc − MGmc − MRm − (1 − )MRc )Gmc (34) ∂ u5


284 285

ARTICLE IN PRESS

JID: MBS

[m5G;September 23, 2014;14:20]


286

Hence, we obtain (see Lenhart and Workman (2007))

7. Numerical methods

u∗1 =

βh Iv Sh (MIm − MSh ) + βh Iv Ic (MGma − MIc ) + βv (Im + Gmc )Sv (MIv − MSv )

u∗2 = u∗3 u∗4

= =

Bc ν(MIc − MSh )Sh (K + Bc )

2A +

Bc ν(MGmc − MIm )Im (K + Bc )


2B

α(MIm − MRm )Im 2C

δ(MIc − MRc )Ic

2D σ (MGmc + MRm + (1 − )MRc − MRmc )Gmc ∗ u5 = 2E

287 288

289

(35)

By standard control arguments involving the bounds on the controls, we conclude

⎧ 0 If ξi∗ ≤ 0 ⎪ ⎪ ⎨ u∗i = ξi∗ If 0 < ξi∗ < 1 ⎪ ⎪ ⎩ 1 If ξi∗ ≥ 1

for i ∈ 1, 2, 3, 4, 5 and where

ξ1∗ = βh Iv Sh (MIm − MSh ) + βh Iv Ic (MGma − MIc ) + βv (Im + Gmc )Sv (MIv − MSv ) ξ = ∗ 2

ξ = ∗ 3

ξ = ∗ 4

ξ5∗ 290 291 292 293

Bc ν(MIc − MSh )Sh K + Bc

7

294

The numerical solutions are illustrated using MATLAB program with computation times of 1.39 s on a Windows Vista Operating System. The optimality system, which consists of the state system and the adjoint system, is solved to obtain the optimal control solution. An iterative scheme, fourth order Runge--Kutta, is used to solve the optimality system. The adjoint equations are solved by the backward fourth order Runge--Kutta scheme using the current iterations solutions of the state equations because of the transversality conditions (27). Then the controls are updated by using a convex combination of the previous controls and the value from the characterizations (28)– (33). This process is repeated and the iterations are stopped if the values of the unknowns at the previous iterations are very close to the ones at the present iteration [3,4,15,17]. Find below in Table 1 the parameter descriptions and values used in the numerical simulation of the co-infection model.

295 296 297 298 299 300 301 302 303 304 305 306 307 308 309

8. Numerical results

310

We discuss the numerical results as listed below to show the effect of various optimal control strategies on the spread of malaria--cholera co-infection in a population.

312

8.1. Prevention (u1 ) and treatment (u3 ) of malaria

314

311 313

2A +

α(MIm − MRm )Im

Bc ν(MGmc − MIm )Im (K + Bc )


2B

2C

δ(MIc − MRc )Ic

2D σ (MGmc + MRm + (1 − )MRc − MRmc )Gmc = 2E

Next, we discuss the numerical solutions of the optimality system and the corresponding results of varying the optimal controls u1 , u2 , u3 , u4 and u5 , the parameter choices, and the interpretations from various cases.

The malaria prevention (treated bednet) control u1 and the malaria treatment control u3 are used to optimize the objective function J while we set the other controls (u2 , u4 and u5 ) relating to cholera to zero. We observed in Fig. 2(a) that the number of malaria infected humans Im was initially controlled significantly but start rising again at the final time. This may not be unconnected to the fact (from our previous analysis) that an uncontrolled cholera epidemic enhances malaria infection. This same trend is also observed in Fig. 2(e) in the control of number of malaria infected mosquitoes Iv . While in Fig. 2(b) the impact of this strategy in controlling the cholera infected individuals Ic yielded no positive results because there was no intervention put in place against cholera. Also, the effect of not controlling the cholera infected population is clearly depicted in Fig. 2(d), this made

Table 1 Description of variables and parameters of the co-infection model (1). Parameter

Value

Reference

Malaria parameters φ Malaria related death q Malaria related death among co-infected βh Probability of human getting infected βv Probability of mosquitoes getting infected λ Bacteria contact rate with humans μh Natural death rate in humans μv Natural death rate in mosquitoes κ Malaria immunity waning rate ω Cholera immunity waning rate ψ Malaria--cholera immunity waning rate h Human birth rate v Mosquitoes birth rate σ Recovery rate of co-infected individual δ Recovery rate of cholera infected individual α Recovery rate of malaria infected individual

Co-infected who recovered from malaria only

Description

0.05--0.1 day−1 0.01 0.034 day−1 0.09 0.05 0.00004 day−1 1/15 to 0.143 1/(60 × 365) day−1 0.001 0.0001--0.02 100 day−1 1000 day−1 0.5 0.07 1/(2 × 365) day−1 0.1

[31] Assumed Assumed [6] Assumed [12] [6] [6] [33] Assumed [6] [6] Assumed [33] [6] Assumed

Bacteria and cholera parameters ν Ingestion rate K Bacteria concentration in water η Cholera related death among co-infected m Cholera related death ρ Cholera infected contribution to the aquatic θ Modification parameter μb Bacteria mortality rate

0.5 1000 0.05 0.02407 0.7 1.2 0.123

[25] [33] Assumed [35] Assumed [23] [25]


315 316 317 318 319 320 321 322 323 324 325 326 327

ARTICLE IN PRESS

JID: MBS 8

[m5G;September 23, 2014;14:20]


0.3

u = u = u =u =u =0 1

2

3

4

u = u = u =u =u =0

5

1

0.35

u1 ≠ 0, u3 ≠ 0

2

3

4

5

u1 ≠ 0, u3 ≠ 0

Cholera Infected Individuals

Malaria Infected Individuals

0.25

0.2

0.15

0.1

0.05

0.3

0.25

0.2

0.15

0.1

0

20

40

60 Time (days)

80

100

120

0

20

40

(a) 0.2

100

120

u = u = u =u =u =0

1

1

0.45

u ≠ 0, u ≠ 0

2

3

4

5

u1 ≠ 0, u3 ≠ 0

3

0.4

0.16 0.14 Bacteria Population

Malaria-Cholera Co-Infected Individuals

80

(b) u1 = u2 = u3 =u4 =u5=0

0.18

60 Time (days)

0.12 0.1 0.08 0.06

0.35 0.3 0.25 0.2

0.04 0.15 0.02 0

20

40

60 Time (days)

80

100

120

(c) 0.22

0.1

0

20

40

60 Time (days)

80

100

120

(d)

u1 = u2 = u3 =u4 =u5=0 u ≠ 0, u ≠ 0

0.2

1

3

Infected Mosquitoes

0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0

20

40

60 Time (days)

80

100

120

(e) Fig. 2. Simulations of the malaria--cholera model showing the effect of malaria prevention and treatment only on transmission.


JID: MBS

ARTICLE IN PRESS

[m5G;September 23, 2014;14:20]


331

this strategy of no effect in effectively putting the bacteria population Bc under control. While the population of the malaria--cholera co-infected humans Gmc shown in Fig. 2(c) show little significant difference between the cases without control and the controlled cases.

332

8.2. Prevention (u2 ) and treatment (u4 ) of cholera

333

347

The cholera prevention control u2 and the cholera treatment control u4 are used to optimize the objective function J while we set the other controls (u1 , u3 and u5 ) relating to malaria to zero. We observed in Fig. 3(a) a continuous rise in the number of malaria infected humans Im cases. This may be connected to the absence of interventions against malaria. The negative impact of this strategy is also shown in Fig. 3(e), where the number of malaria infected mosquitoes Iv cases is seen to be on the increase at the end of the intervention period. The result in the depicted Fig. 3(b) clearly suggest that this strategy is very efficient and effective in the control of the number of cholera infected humans Ic and similarly, the positive impact of the cholera controlled cases resulted in the significant control of the bacteria population Bc as shown in Fig. 3(d). While the population of the malaria--cholera co-infected humans Gmc shown in Fig. 3(c) show little significant difference between the cases without control and the controlled cases.

348

8.3. Malaria and cholera prevention (u1 and u2 ) only

328 329 330

334 335 336 337 338 339 340 341 342 343 344 345 346

370

The malaria and cholera prevention controls u1 and the control u2 are used to optimize the objective function J while we set the other controls (u3 , u4 and u5 ) to zero. That is, only the prevention mechanisms are optimized without treatments. We observed in Fig. 4(a) that the number of malaria infected humans Im was controlled significantly before it start rising again at the final time. This may be connected to the fact that treatment of infected individuals is neglected and as a result the disease persists in the community. This effect is also observed in Fig. 4(e) in the control of number of malaria infected mosquitoes Iv . While in Fig. 4(b) the impact of this strategy in controlling the cholera infected individuals Ic yielded no positive results because there was no treatment of cholera infected individuals in place. The effect of not treating the cholera infected population is shown clearly in Fig. 4(d), making this strategy of no effect in effectively putting the bacteria population Bc under control. The population of the malaria--cholera co-infected humans Gmc shown in Fig. 4(c) show little significant difference between the cases without control and the controlled cases. This strategy suggest that optimal preventive strategies against malaria and cholera in a community while adequate treatment regime is not put in place at the same time would not be an effective approach to controlling either of the disease at the final time.

371

8.4. Malaria and cholera treatment (u3 and u4 ) only

349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369

372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387

The malaria and cholera treatment controls u3 and u4 are used to optimize the objective function J while we set the other interventions, that is the preventive measures (u1 , u2 and u5 ) to zero. That is, only the treatment mechanisms are optimized without preventions. We observed in Fig. 5(a) a continuous rise in the number of malaria infected humans Im cases at the end. This may be due to the absence of prevention strategy against malaria and cholera. The impact of this strategy is also shown in Fig. 5(e), where the number of malaria infected mosquitoes Iv cases is seen to rise again at the end of the intervention period after a significant decrease. The result in the depicted Fig. 5(b) clearly suggest that this strategy is very efficient and effective in controlling the number of cholera infected humans Ic , this again reaffirm our assertion that cholera infection is not enhanced by increased risk of malaria infection. Also, the significant effect of this strategy on cholera controlled cases leads to the significant control of the Bacteria population Bc as shown in Fig. 5(d). The population of

9

the malaria--cholera co-infected humans Gmc shown in Fig. 5(c) show little significant difference between the cases without control and the controlled cases. This strategy suggest that optimal treatment regime against malaria and cholera in a community while preventative measures are neglected would only be an effective approach to control cholera only while malaria persists at the final time. 8.5. Malaria and cholera preventions with treatment (u1 , u2 , u3 , u4 , u5 ) In this strategy all the control mechanism (u1 , u2 , u3 , u4 , u5 ) are used to optimize the objective function J. That is, both the preventions and treatments of malaria and cholera are optimized. We observed in Fig. 6(a) that the number of malaria infected humans Im is effectively controlled. The impact of this strategy is also shown in Fig. 6(e), where the number of malaria infected mosquitoes Iv is significantly reduced to zero at the end of the intervention period. The result shown in Fig. 6(b) clearly suggest that this strategy is also very efficient and effective in controlling the number of cholera infected humans Ic and leading also to effective control of the bacteria population Bc as shown in Fig. 6(d). The population of the malaria--cholera co-infected humans Gmc shown in Fig. 6(c) show significant difference between the cases without control and the controlled cases. This strategy suggests that optimal prevention and treatment regime against both malaria and cholera in a community would be a very effective approach to effectively control both diseases malaria--cholera at the final intervention time. 9. Concluding remarks In this paper, we formulated and analyzed a deterministic model for the transmission of malaria--cholera co-infection that includes use of preventions, treatments of infectives and also performed optimal control analysis of the model. The model was rigorously analyzed to gain insights into its qualitative dynamics. We obtained the following results: 1. The cholera only model has a locally-stable disease free equilibrium whenever the associated reproduction number is less than unity. Also, the model has a unique endemic equilibrium whenever R0c > 1. 2. The malaria only model also has a locally-stable disease free equilibrium whenever the associated reproduction number is less than unity. It also has a unique endemic equilibrium whenever R0m > 1. 3. The malaria--cholera co-infection model has a locally-stable disease free equilibrium whenever the associated reproduction number is less than unity and exhibits the phenomenon of backward bifurcation, which suggests a case where stable disease-free equilibrium co-exists with a stable endemic equilibrium whenever the basic reproductive number is less than unity. 4. We found from the analysis of the impact of cholera on malaria that malaria infection may be associated with an increased risk of cholera, while cholera infection is not associated with an increased risk for malaria. 5. Focusing only on malaria intervention strategies (optimal preventions and treatments) while cholera is not under control would not lead to effective control of either malaria or cholera at the end of the intervention. As clearly shown in Fig. 2(a) and (e), where the number of malaria infected individuals and mosquitoes are seen to be increasing at the final time. This is an indication that malaria infection may be associated with an increased risk of cholera. 6. That optimal efforts on cholera intervention strategies (optimal preventions and treatments) while malaria is not under control would only result in effective control of cholera only


388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448

ARTICLE IN PRESS

JID: MBS 10


0.3

u = u = u =u =u =0 1

2

3

4

u = u = u =u =u =0

5

1

0.35

u2 ≠ 0, u4 ≠ 0 0.25

2

3

4

5

u2 ≠ 0, u4 ≠ 0

0.3 Cholera Infected Individuals


[m5G;September 23, 2014;14:20]

0.2

0.15

0.1

0.25 0.2 0.15 0.1 0.05

0.05 0

20

40

60 Time (days)

80

100

120

0

20

40

(a) 0.2

80

100

120

(b) u1 = u2 = u3 =u4 =u5=0

u1 = u2 = u3 =u4 =u5=0

0.45

u ≠ 0, u ≠ 0

0.18

2

u ≠ 0, u ≠ 0

4

2

0.4

4

0.16 0.35 0.14 Bacteria Population


60 Time (days)

0.12 0.1 0.08

0.3 0.25 0.2

0.06

0.15

0.04

0.1

0.02

0.05

0

20

40

60 Time (days)

80

100

120

(c)

0

20

40

60 Time (days)

80

100

120

(d)

u1 = u2 = u3 =u4 =u5=0

0.22

u ≠ 0, u ≠ 0 2

4

Infected Mosquitoes

0.2

0.18

0.16

0.14

0.12

0.1 0

20

40

60 Time (days)

80

100

120

(e) Fig. 3. Simulations of the malaria--cholera model showing the effect of cholera prevention and treatment only on transmission.


ARTICLE IN PRESS

JID: MBS

[m5G;September 23, 2014;14:20]


0.3

11

u = u = u =u =u =0 1

2

3

4

u = u = u =u =u =0

5

1

0.35

u1 ≠ 0, u2 ≠ 0

2

3

4

5

u1 ≠ 0, u2 ≠ 0

Cholera Infected Individuals


0.25

0.2

0.15

0.1

0.05

0.3

0.25

0.2

0.15

0.1

0

20

40

60 Time (days)

80

100

120

0

20

40

(a) 0.2

100

120

1

u1 = u2 = u3 =u4 =u5=0

0.45

u ≠ 0, u ≠ 0

u ≠ 0, u ≠ 0

2

1

2

0.4

0.16 0.14 Bacteria Population


80

(b) u1 = u2 = u3 =u4 =u5=0

0.18

60 Time (days)

0.12 0.1 0.08 0.06

0.35 0.3 0.25 0.2

0.04 0.15 0.02 0

20

40

60 Time (days)

80

100

120

(c) 0.22

0.1

0

20

40

60 Time (days)

80

100

120

(d)

u1 = u2 = u3 =u4 =u5=0 u ≠ 0, u ≠ 0

0.2

1

2

Infected Mosquitoes

0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0

20

40

60 Time (days)

80

100

120

(e) Fig. 4. Simulations of the malaria--cholera model showing the effect of malaria and cholera prevention only on transmission.


ARTICLE IN PRESS

JID: MBS 12


0.3

u = u = u =u =u =0 1

2

3

4

u = u = u =u =u =0

5

1

0.35

u3 ≠ 0, u4 ≠ 0 0.25

0.2

0.15

0.1

3

4

5

0.25 0.2 0.15 0.1 0.05

0.05

0

20

40

60 Time (days)

80

100

120

0

20

40

(a) 0.2

60 Time (days)

80

100

120

(b) u1 = u2 = u3 =u4 =u5=0

u1 = u2 = u3 =u4 =u5=0

0.45

u ≠ 0, u ≠ 0

0.18

3

u ≠ 0, u ≠ 0

4

3

0.4

4



2

u3 ≠ 0, u4 ≠ 0



[m5G;September 23, 2014;14:20]

0.12 0.1 0.08

0.3 0.25 0.2

0.06

0.15

0.04

0.1

0.02

0.05

0

20

40

60 Time (days)

80

100

120

(c) 0.22

0

20

40

60 Time (days)

80

100

120

(d)

u1 = u2 = u3 =u4 =u5=0 u ≠ 0, u ≠ 0

0.2

3

4

Infected Mosquitoes

0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0

20

40

60 Time (days)

80

100

120

(e) Fig. 5. Simulations of the malaria--cholera model showing the effect of malaria and cholera treatment only on transmission.


ARTICLE IN PRESS

JID: MBS

[m5G;September 23, 2014;14:20]


0.3

u = u = u =u =u =0 1

2

3

4

u = u = u =u =u =0

5

0.25

2

3

4

5

u1 ≠ 0, u2 ≠ 0, u3 ≠ 0, u4 ≠ 0, u5 ≠ 0



1

0.35

u1 ≠ 0, u2 ≠ 0, u3 ≠ 0, u4 ≠ 0, u5 ≠ 0

0.2

0.15

0.1

0.05

0.25 0.2 0.15 0.1 0.05

0

20

40

60 Time (days)

80

100

120

0

20

40

(a) 0.2

80

100

120

u1 = u2 = u3 =u4 =u5=0

0.45

u ≠ 0, u ≠ 0, u ≠ 0, u ≠ 0, u ≠ 0 1

60 Time (days)

(b)

u1 = u2 = u3 =u4 =u5=0

0.18

2

3

4

u ≠ 0, u ≠ 0, u ≠ 0, u ≠ 0, u ≠ 0

5

1

0.4

2

3

4

5



13

0.12 0.1 0.08

0.3 0.25 0.2

0.06

0.15

0.04

0.1

0.02

0.05

0

20

40

60 Time (days)

80

100

120

(c) 0.22

0

20

40

60 Time (days)

80

100

120

(d)

u = u = u =u =u =0 1

2

3

4

5

u ≠ 0, u ≠ 0, u ≠ 0, u ≠ 0, u ≠ 0

0.2

1

2

3

4

5

0.18 Infected Mosquitoes

0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0

20

40

60 Time (days)

80

100

120

(e) Fig. 6. Simulations of the malaria--cholera model showing the effect of malaria and cholera prevention with treatments on transmission.


JID: MBS 14

449 450 451 452 453 454 455 456 457 458

459 460

ARTICLE IN PRESS K.O. Okosun, O.D. Makinde / Mathematical Biosciences 000 (2014) 1–14

while malaria still persist, this is shown in Fig. 3(b) and (d), where the number of cholera infected individuals are seen to be effectively controlled at the final time. This suggests that cholera infection is not associated with an increased risk for malaria, that is, malaria infection does not enhance cholera infection. 7. Whenever there is co-existence of malaria and cholera in the community, our model suggests the incorporation of cholera control measures with the malaria intervention strategies for effective malaria control.

Uncited References [1], [22], [14], [18], [21], [28], [29], [37], [38], [39].

461

Supplementary Materials

462 464

Supplementary material associated with this article can be found, in the online version, at doi:http://dx.doi.org/10.1016/j.mbs.2014.09. 008.

465

References

463

466 467 468 469 470 471 472 473 474 475 476 477 Q4 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494

[m5G;September 23, 2014;14:20]

[1] B.M. Adams, H.T. Banks, H. Kwon, H.T. Tran, Dynamic multidrug therapies for HIV: optimal and STI control approaches. Math. Biosci. Eng. vols. 1 and 2 (2004) 223–241. [2] M. Agarwal, V. Verma, Modelling and analysis of the spread of an infectious disease cholera with environmental fluctuations. Appl. Appl. Math. 7(1) (2012) 406–425. [3] F.B. Agusto, Optimal chemoprophylaxis and treatment control strategies of a tuberculosis transmission model. World J. Model. Simul. 5(3) (2009) 163–173. [4] F.B. Agusto, K.O. Okosun, Optimal seasonal biocontrol for Eichhornia crassipes, Int. J. Biomath. 3(3) (2010) 383–397. 10.1142/S1793524510001021. [5] R.M. Anderson, R.M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, Oxford, 1991 [6] K.W. Blayneh, Y. Cao, H.D. Kwon, Optimal control of vector-borne diseases: treatment and Prevention, Discrete Cont. Dyn. Syst. B 11 (3) (2009) 587–611. [7] C. Castillo-Chavez, B. Song, Dynamical model of tuberculosis and their applications, Math. Biosci. Eng. 1 (2004) 361–404. [8] M.K. Christopher, X.V. Jorge, A simple vaccination model with multiple endemic states, Math. Biosci. 164 (2000) 183–201. [9] P.V.D. Driessche, J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci. 180 (2002) 29–48. [10] W.H. Fleming, R.W. Rishel, Deterministic and Stochastic Optimal Control, Springer Verlag, New York, 1975. [11] L. Guihua, J. Zhen, Global stability of a SEIR epidemic model with infectious force in latent, infected and immune period, Chaos Solut. Fractals 25 (2005) 1177– 1184. [12] M.Y. Hyun, A mathematical model for malaria transmission relating global warming and local socioeconomic conditions, Rev. Saude Publica 35 (3) (2001) 224–231 www.fsp.usp.br/rsp.

[13] M.D. John Snow, On the Mode of Communication of Cholera, John Churchill, New Burlington Street, England, 1855. [14] Karrakchou, M. Rachik, S. Gourari, Optimal control and infectiology: application to an HIV/AIDS model, Appl. Math. Comput. 177 (2006) 807–818. [15] D. Kirschner, S. Lenhart, S. Serbin, Optimal control of the chemotherapy of HIV, J. Math. Biol. 35 (1997) 775–792. [16] J.C. Koella, R. Anita, Epidemiological models for the spread of anti-malaria resistance, Malaria J. 2 (2003) 3. [17] S. Lenhart, J.T. Workman, Optimal Control Applied to Biological Models, Chapman and Hall, 2007. [18] O.D. Makinde, K.O. Okosun, Impact of chemo-therapy on optimal control of malaria disease with infected immigrants, BioSystems 104 (2011) 32–41. [19] E. Mtisi, H. Rwezaura, J.M. Tchuenche, A mathematical analysis of malaria and tuberculosis co-dynamics, Discrete Cont. Dyn. Syst. B 12 (4) (2009) 827–864. [20] Z. Mukandavire, A.B. Gumel, W. Garira, J.M. Tchuenche, Mathematical analysis of a model for HIV-malaria co-infection, Math. Biosci. Eng. 6 (2009) 333– 362. [21] Z. Mukandavire, W. Garira, J.M. Tchuenche, Modelling effects of public health educational campaigns on HIV/AIDS transmission dynamics, Appl. Math. Model. 33 (2009) 2084–2095. [22] S. Mushayabasa, C.P. Bhunu, Modeling schistosomiasis and HIV/AIDS co-dynamics, Comput. Math. Methods Med. 2011 (2011) 15 pp., doi:10:1155/2011/846174. [23] S. Mushayabasa, C.P. Bhunu, Is HIV infection associated with an increased risk for cholera? Insights from mathematical model, BioSystems 109 (2012) 203–213. [24] A. Mwasa, J.M. Tchuenche, Mathematical analysis of a cholera model with public health interventions, BioSystems 105 (2011) 190–200. [25] R.L.M. Nielan, E. Schaefer, H. Gaff, K.R. Fister, S. Lenhart, Modeling optimal intervention strategies for cholera, Bull. Math. Biol. 72 (2010) 2004–2018. [26] I.S. Nikolaos, K. Dietz, D. Schenzle, Analysis of a model for the pathogenesis of AIDS, Math. Biosci. 145 (1997) 27–46. [27] K.O. Okosun, R. Ouifki, N. Marcus, Optimal control analysis of a malaria disease transmission model that includes treatment and vaccination with waning immunity, BioSystems 106 (2011) 136–145, doi:10.1016/j.biosystems.2011.07.006. [28] K.O. Okosun, R. Ouifki, N. Marcus, Optimal control strategies and costeffectiveness analysis of a malaria model, BioSystems 111 (2013) 83–101, doi:10.1016/j.biosystems.2012.09.008. [29] K.O. Okosun, O.D. Makinde, Optimal control analysis of malaria in the presence of non-linear incidence rate, Appl. Comput. Math. 12 (1) (2013) 20–32. [30] L.S. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze, E.F. Mishchenko, The mathematical Theory of Optimal Processes, Wiley, New York, 1962 [31] J.S. Robert, S.D. Hove-Musekwa. Determining effective spraying periods to control malaria via indoor residual spraying in sub-saharan Africa, J. Appl. Math. Decis. Sci. (2008) Article ID 745463, Hindawi Publishing Corporation, 19 pp. [32] R. Ross, The Prevention of Malaria, second ed., London, Murray, 1911. [33] R.P. Sanches, C.P. Ferreira, R.A. Kraenkel, The role of immunity and seasonality in cholera epidemics, Bull. Math. Biol. 73 (2011) 2916–2931. [34] Z. Shuai, P. van den Driessche, Global dynamics of cholera models with differential infectivity, Math. Biosci. 234 (2011) 118–126. [35] Z. Shuai, J.H. Tien, P. van den Driessche, Cholera models with hyperinfectivity and temporary immunity, Bull. Math. Biol. 74 (2012) 2423–2445. [36] J.P. Tian, J. Wang, Global stability for cholera epidemic models, Math. Biosci. 232 (2011) 31–41. [37] A. Tripathi, R. Naresh, D. Sharma, Modelling the effect of screening of unaware infectives on the spread of HIV infection, Appl. Math. Comput. 184 (2007) 1053– 1068. [38] G. Zaman, H.K. Yong, I.H. Jung. Stability analysis and optimal vaccination of an SIR epidemic model, BioSystems 93 (2008) 240–249. [39] http://www.ukessays.com/essays/health/malaria-and-cholera-control-insouth-africa-health-essay.php#ixzz2MHtCb4Ov.


495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 Q5 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554

Optimal Population-Level Infection Detection Strategies for Malaria Control and Elimination in a Spatial Model of Malaria Transmission.

Modelling Optimal Control of Cholera in Communities Linked by Migration.

Epidemiology and control of malaria and other arthropod-borne diseases.

A coinfection model for HIV and HCV.

Optimal control problems of epidemic systems with parameter uncertainties: application to a malaria two-age-classes transmission model with asymptomatic carriers.

Coinfection with malaria and intestinal parasites, and its association with anaemia in children in Cameroon.

A model of bi-mode transmission dynamics of hepatitis C with optimal control.

Coinfection Dynamics of Two Diseases in a Single Host Population.

The Blantyre Integrated Malaria Initiative: a model for effective malaria control.

Algorithm for optimal linear model-based control with application to pharmacokinetic model-driven drug delivery.

Treatment of rheumatic diseases and hepatitis B virus coinfection.

visceral leishmaniasis co-infection.

Robust model predictive control for optimal continuous drug administration.

Dendritic Immunotherapy Improvement for an Optimal Control Murine Model.

An experimental model to study tuberculosis-malaria coinfection upon natural transmission of Mycobacterium tuberculosis and Plasmodium berghei.

An optimal control model for maximum-height human jumping.

Delay differential model for tumour-immune response with chemoimmunotherapy and optimal control.

Framework for Optimal Global Vaccine Stockpile Design for Vaccine-Preventable Diseases: Application to Measles and Cholera Vaccines as Contrasting Examples.

Optimal tuberculosis prevention and control strategy from a mathematical model based on real data.

Ivermectin and malaria control.

An imported case of severe falciparum malaria with prolonged hemolytic anemia clinically mimicking a coinfection with babesiosis.

Predicting optimal transmission investment in malaria parasites.

AIDS co-infection model.

Epidemiology and Synergistic Hepatopathology of Malaria and Hepatitis C Virus Coinfection.