Transmission Dynamics of Malaria in Ghana

In this paper, a deterministic mathematical model to investigate the transmission dynamics of malaria in Ghana is formulated taking into account human and mosquito populations. The model consists of seven non-linear differential equations which describe the dynamics of malaria with 4 variables for humans and 3 variables for mosquitoes. The state vector for the model is (S h, Eh, Ih,R, S m, Em, Im, ) where S h, Eh, Ih, R, S m, Em and Im respectively represent populations of susceptible humans, exposed humans, infectious humans, recovered humans, susceptible mosquitoes, exposed mosquitoes and infectious mosquitoes. Stability analysis of the model is performed and we make use of the next generation method to derive the basic reproduction number R0. A mathematical analysis of the dynamic behaviour indicates that the estimated model has a unique endemic equilibrium point and malaria will persist in Ghana. The basic reproduction number for Ghana is found to be R0 = 0.8939. Further, both the disease-free and endemic equilibria are locally asymptotically stable. Numerical simulations indicate that reducing current biting rate of female Anopheles mosquitoes by 1/16 could assist Ghana to achieve malaria free status by the year 2037. If, in addition, the number of days it takes to recover from malaria infection were reduced to three 3 days malaria free status could be achieved by the year 2029.


Introduction
Malaria is a life-threatening disease caused by a protozoan parasite called Plasmodium, which lives part of its life in humans and part in Anopheles mosquitoes.The disease is endemic in tropical and subtropical regions, including Africa, Asia, Latin America, the Middle East and some parts of Europe.According to the Anti-Malaria Drug Policy for Ghana document in 2009, Malaria remains hyper endemic in Ghana and is the single most important cause of mortality and morbidity especially among children under five years, pregnant women and the poor (UNICEF -At a glance: Ghana, www.unicef.org).Chitnis (2005) proposed a model similar to the malaria model in this paper.The main differences of our model, from that of Chitnis (2005) is that we have excluded the infection of female Anopheles mosquito by recovered humans, because we assume that these humans do not have sufficient plasmodium parasites in their bodies to transmit the infection to mosquitoes.Also, in our model, the infectious humans recover with clinical treatment and the death of the female Anopheles mosquito is caused by natural death rate and insecticides.Further this paper fills in an information gap pertaining to the dynamics of transmission of malaria in Ghana.To the best of our knowledge enough work has not been done in this area to assist health workers and policy makers to make informed decisions.In this paper we use clinical malaria data from Ghana Health Service at WHO Website to develop a mathematical model to investigate and understand the disease transmission dynamics in Ghana taking both host and vector populations into account.Our model will be non-linear ordinary differential equations.Stability analysis and numerical simulations are performed.The simulations are conducted using MATLAB's ode45.Some of the assumptions in this paper on mathematical modeling of malaria are based on studies by Chitnis (2005), Chitnis et al. (2006), Danso-Addo (2009) and Mwamtobe (2010).

Method
The model is based on important intervention strategies currently relevant in Ghana such as clinical treatment and the mortality of the female Anopheles mosquito which is caused not only by natural death rate, but by indoor residual spraying (IRS) and insecticide treated bed nets (ITNs).We divide the human population into 4 classes: the Susceptible, denoted, S h , representing the fraction of host population that is susceptible to infection; the Exposed, denoted E h , being the fraction of population who are infected but not infectious in that they cannot transmit the infection.There is also the Infectious class, denoted I h , corresponding to persons who have the malaria infection and can transmit it to persons in a susceptible class.Finally, the recovered, denoted by R, being people who recover from the infection through clinical treatment and are endowed with temporary immunity.These latter humans can not transmit the infection to mosquitoes because we assume that they have no plasmodium parasites in their bodies.People enter the susceptible class, either through birth or through immigration (assume to be at a constant rate).There is a finite probability of movement to the exposed class when a susceptible person is bitten by an infectious anopheles mosquito.The parasite is then passed on to humans in the form of sporozoites.The clinical onset of the disease is characterized by the parasite entering the blood stream as merozoites.Later there is a movement of infectious humans to recovery class where they acquire some immunity to malaria.After some further time, this immunity is lost and they revert to the susceptible class.Humans leave the population through natural death and those in the infectious class have additional disease-induced death rate.
We do not include the immigration of infectious humans because we assume that most people who are sick will not travel.The movement of Exposed humans are excluded because, given the short time of the exposed stage, the number of exposed people is small.We do make a simplifying assumption that there is no immigration of recovered humans.The female Anopheles mosquito population is divided into 3 classes: Susceptible S m , Exposed E m and Infectious I m .Anopheles male mosquitoes are not included in the model because only female mosquitoes bite humans for blood meals.Female mosquitoes enter the susceptible class through birth.The parasite (in the form of gametocytes) enters the mosquito, with some probability, when the mosquito bites an infectious human and the mosquito moves from the Susceptible to the Exposed class.After some period of time, dependent on the ambient temperature and humidity, the parasite develops into sporozoites and enters the mosquito's salivary glands; and the mosquito moves from the exposed class to the infectious class.The mosquito remains infectious for life.Mosquitoes leave the population through natural death rate and death caused by insecticides.We assume that longevity of the female Anopheles mosquitoes is unaffected by the parasite infection and they do not die from the infection.There is no super infection of the disease.Mosquitoes cannot survive without human host as they need human blood to feed their developing eggs.
The parameters in Table 1 and the state variables in Table 2 are used in Figure 1 to formulate the malaria model.Total human population at time t N m (t) Total mosquito population at time t

Equations of the Malaria Model
Applying the assumptions, definitions of state variables and parameters above, the system of non-linear differential equations which describe the dynamics of malaria are formulated below: with initial conditions and α m = θ hm φI h N h .For the purpose of this paper, we refer to Equation (1) as malaria model.The total population sizes are

Invariant Region
The invariant region can be obtained by the following theorem.
Theorem 1 The solutions set to the malaria model are feasible for all t > 0 if they enter the invariant region + be any solution of the malaria model with non-negative initial conditions.In absence of the malaria,that is and solving for N h we have, N h ≤ ψ μ + Ce −μt .Using the intial condtions at t = 0, N h (0) = N h0 , we have Applying the theorem of differential inequality (Birkhof & Rota, 1982), we obtain, Therefore, as t → ∞, the human population N h approaches K = ψ mu , the parameter K is usually called the carrying capacity (Namawejje, 2011).Hence all feasible solution set of the human population of the mmalaria model enters the region Similarly, the feasible solutions set of the mosquito population enters the region Therefore, the feasible solutions set for malaria model given by Ω = Ω h × Ω m is positiveinvariant and hence it is biologically meaningful and mathematically well-posed in the domain Ω.

Positivity of Solutions
Proof.From the first equation in the model (1), we have From the second equation of model malaria model, Similary, it can be shown that the remaining equations in system malaria model are also positive ∀t > 0, because e η > 0 η ∈ R.

Disease-Free Equilibrium
Disease-free equilibrium points (DFE) are steady state solutions where there is no malaria in the human population or Plasmodium parasite in the mosquito population.Let define the "diseased" classes as the human or mosquito populations that are either exposed or infectious; that is, E h , I h , E m and I m .In absence of the disease, this implies that ( E h = I h = E m = I m = 0) and when the right-hand side of a nonlinear system (1) is set to zero, we have, Therefore, the disease-free equilibrium point of the malaria model ( 6) is given by which represents the state in which there is no infection(in the absence of malaria) in the society.

Basic Reproduction Number
We use the next generation operator approach as described by Diekmann et al. (1990) to define the basic reproduction number, R 0 , as the number of secondary infections that one infectious individual would create over the duration of the infectious period, provided that everyone else is susceptible.When R 0 < 1, each infected individual produces on average less than one new infected individual, so we would expect the disease to die out.On the other hand, if R 0 > 1, each individual produces more than one new infected individual, so we would expect the disease to spread in the population.This means that the threshold quantity for eradicating the disease is to reduce the value of R 0 to be less than one.We determine R 0 using the next generation operator approach.The associated next generation matrices are and l = θ hm φμρ ψω(τ+ μ+ π) .We can now calculate the eigenvalues λ to determine the basic reproduction number R 0 by taking the spectral radius (dominant eigenvalue) of the matrix FV −1 .We have λ 2 (λ 2 − kp) = 0 ⇒ λ = 0 or λ = ± kp.From the four eigenvalues, the dominant eigenvalue of the matrix FV −1 is λ = kp.Therefore the basic reproduction number R 0 = kp.Hence The threshold parameter R 0 can be defined as square roots of the product of number of humans one mosquito infects during its infectious lifetime R 0h and number of mosquitoes one human infects during the duration of the infectious period R 0m , provided all humans and mosquitoes are susceptible.Therefore, The basic reproduction number can be used to determine the local stability of the disease free equilibrium point.

Local Stability of Disease-Free Equilibrium
The local stability of the disease-free equilibrium can be analyzed using the Jacobian matrix of the malaria model at the disease free equilibrium point.Referring to the results of Van den Driessche and Watmough (2002), the following theorem holds.
Theorem 2 The disease free equilibrium point for the malaria model is locally asymptotically stable if R 0 < 1 and unstable if R 0 > 1.
Proof.The Jacobian matrix J of the malaria model ( 1) with S h = N h − (E h + I h + R) and S m = N m − (E m + I m ) at the disease-free equilibrium point is given by The eigenvalues of the Jacobian matrix are the solutions of the characteristic equation Then where Using the Routh-Hurwitz Criteria on (6), we can prove that all roots of the polynomial (6) have negative real parts.
For the characteristic polynomial in (6), when n = 4, the Routh-Hurwitz criteria are B 1 > 0, B 2 > 0, B 3 > 0, B 4 > 0 the determinants of Hurwitz matrices are positive.Hence all the eigenvalues of the jacobian (5) have negative real part and R 0 < 1. Therefore disease-free equilibrium point is stable.
Conversrly, if R 0 > 1 it implies that B 4 < 0 and since the remaining coefficients (B 1 , B 2 and B 3 ) of the polynomial (6) are positive, then all the roots of this polynomial cannot have negative real parts.Therefore, the disease-free equilibrium point is unstable.

The Endemic Equilibrium Point
Endemic equilibrium points are steady state solutions where the disease persists in the population (all state variables are positive).That is, malaria infection will persists in the population and the endemic equilibrium point (EEP) of the model is given by EEP = (S e 1 h , E e 1 h , I e 1 h , R e 1 , S e 1 m , E e 1 m , I e 1 m ) > 0. To derive the EEP, we have to solve After some algebraic manipulation, we get We now consider the possibility of multiple endemic equilibria for equation ( 6).It may also indicate three distinct situations which we have to consider depending on the signs of B and C since A is always positive.The C is negative if R 0 < 1 and positive if R 0 > 1. Hence the three situations will lend to the following theorem.

Local Stability of the Endemic Equilibrium
The stability of the endemic equilibrium of the malaria model can be analysed using the Centre Manifold Theory described by Castillo-Chavez and Song (2004).
To apply this theorem we make the following change of variables in the malaria model.
The malaria model is written in the form, where X i = (x 1 , x 2 , . . ., x 7 ) T and H = (h 1 , h 2 , . . ., h 7 ) T are transposed matrices.The malaria model becomes where Let Ψ * be the bifurcation parameter, the system ( 7) is linearized at disease free equilibrium point when Ψ = Ψ * with R 0 = 1.Thus Ψ * can be solved from (3) when R 0 = 1 as Then zero is a simple eigenvalue of the following Jacobian matrix, J bi f with the application of the bifurcation parameters.
A right eigenvector associated with the eigenvalue zero is w = (w 1 , w 2 , . . ., w 7 ).We have the following right eigenvector The left eigenvector is given as follows: Therefore Hence a < 0 and b > 0. Therefore the following theorem holds.
Theorem 1 The malaria model has a unique endemic equilibrium which is locally asymptotically stable when R 0 < 1 and unstable when R 0 > 1.

Results and Discussion
The parameters in the malaria model were estimated using clinical malaria data and demographic statistics of Ghana.Those that were not available were obtained from literature published by researchers in malaria endemic countries which have similar environmental conditions compared to Ghana.According to the Ghana Living Standards Survey Report of the Fifth Round (GLSS 5), 2008, the estimated number of households in Ghana is 5.5 million.Conservatively, it is assumed that there are 10 female Anopheles mosquitoes per household in Ghana.The female Anopheles mosquito population is then approximately given by: 5, 500, 000×10 = 55, 000, 000 mosquitoes.The Table 3 below shows the estimated parameters and their sources for the model (1).The rates are given per day.After substituting the estimated parameter values in Table 3 into the malaria model, we obtain the basic reproduction number from (3) as R 0 = 0.8939.Since R 0 = 0.8939 < 1, hence malaria disease can be eliminated or eradicated in the susceptible population in Ghana.

Local Stability of the Disease-free Equilibrium
Using (6), we have λ 4 + 0.3853λ 3 + 0.05209λ 2 + 0.002868λ + 0.00001075 = 0 Since the coeeficients of the polynomial are positive, it follows by the Routh-Hurwitz stability criteria that, the disease-free equilibrium point is asymptotically stable.This means that malaria free society can be achieved.

The Endemic Equilibrium Point
The quadratic equation for calculating the value of I e 1 h is given below: Since a < 0 and b > 0, by the Centre Manifold Theory described by Castillo-Chavez and Song (2004) the endemic equilibrium point is locally asymptotically stable.This means malaria will persist in Ghana.

Numerical Simulations
A numerical simulation of the estimated malaria model is conducted to explore scenarios of the dynamics of the disease in the human population.The time-axes in all phase portraits below start from the year 2000.We mainly consider the effects of varying key parameters responsible for controlling malaria 1) Reducing the biting rate of mosquitoes.
2) The treatment rate of infectious humans.
3) Combining the reduction in the biting rate of mosquitoes and the increase in the treatment rate of infectious humans.
The biting rate of mosquitoes can be reduced by using the Insecticide-treated bed nets (ITN) and Indoor residual spraying (IRS).The values of the biting rate of mosquitoes, transmission rate of infection from an infectious mosquito to a susceptible human, rate of loss of immunity for humans and the mosquito population are reduced by 1/16, while the values of the other parameters are maintained.This is illustrated in the Figure 2

Table 1 . Mode parameters and their interpretations for the malaria model (1)
hm Probability of transmission of infection from an infectious human to a susceptible mosquito provided there is a bite φ Biting rate of mosquitoes

Table 2 .
The state variables for the malaria model (1) h (t) Number of susceptible humans at time t E h (t) Number of exposed humans at time t I h (t) Number of infectious humans at time t R(t) Number of recovered (immune) humans at time t S m (t) Number of susceptible mosquitoes at time t E m (t) Number of exposed mosquitoes at time t I m (t) Number of infectious mosquitoes at time t N h (t)

Table 3 .
Mode parameters and their interpretations for the malaria model (1)