An Epidemiological Model of Malaria Transmission in Ghana.pdf

The International Journal Of Science & Technoledge (ISSN 2321 – 919X) www.theijst.com
171 Vol 3 Issue 1 January, 2015
THE INTERNATIONAL JOURNAL OF
SCIENCE & TECHNOLEDGE
An Epidemiological Model of Malaria Transmission in Ghana
1. Introduction
The World Health Organization (WHO) estimates that malaria caused approximately 627,000 deaths in 2012 mostly those of
children under five years of age in Africa. This means 1300 young lives are lost to malaria every day - a strong reminder that
victory over this ancient foe is still a long way off. The fact that so many people are dying from mosquito bites is one of the
greatest tragedies of the 21st century. (WHO, 2013) In Africa alone, costs of illness, treatment, and premature death from malaria
are at least billion per year. There are strong social and economic ways to the burden of the disease, which in so many ways
affects fertility, population growth, saving and investment, worker productivity, absenteeism, premature mortality and medical
costs (Sachs, 2002). In areas where malaria is highly endemic, young children bears a larger burden in terms of the disease
morbidity and mortality and affects fetal development during early stage of pregnancy in women due to loss of immunity.
Currently, strategies of controlling the disease includes, the use of chemotherapy, intermittent preventive treatment for children
and pregnant women (IPT), and use of insecticides treated bed nets and insecticides against the vector. The challenge posed by
malaria calls for urgent need for a better understanding of important parameters in the disease transmission and development of
effective and optimal strategies for prevention and control of the spread of malaria disease.
Mathematical modelling has become an important instrument in understanding the dynamics of disease transmission and in
decision making processes regarding intervention programs for disease control. Concerning malaria disease, (Ross, 1911)
developed the models of malaria transmission. He focused his study on mosquito control and showed that for the disease to be
eliminated the mosquito population should be brought below a certain threshold. His work was later extended by (Macdonald,
Francis T. Oduro
Senior Lecturer, Department of Mathematics, Kwame Nkrumah University of Science and Technology,
University Post office (PMB) Kumasi, Ghana, West Africa
Prince Harvim
Teaching Assistants, Department of Mathematics and Statistics, Ho Polytechnic, Ghana, West Africa
Akuamoah Worlanyo Saviour
Teaching Assistants, Department of Mathematics, Kwame Nkrumah University of Science and
Technology, (PMB) Kumasi, Ghana, West Africa
Reindorf Borkor
Teaching Assistants, Department of Mathematics, Kwame Nkrumah University of Science and
Technology, (PMB) Kumasi, Ghana, West Africa
Mahama Francois
Lecturer, Department of Mathematics and Statistics, Ho Polytechnic, Ghana, West Africa
Abstract:
In this paper malaria was modelled as a 7-dimensional system of ordinary differential equations. Analysis of the model showed
that there exists a domain where the model is epidemiologically and mathematically well-posed. The basic reproduction
number,( ) was computed. The stability analysis of the model was investigated and proved that the disease-free equilibrium is
locally asymptotically stable if ( ) and unstable when ( ). The Centre Manifold theorem is used to show that the
model has a unique endemic equilibrium which is locally asymptotically stable when ( ). Sensitivity analysis was
applied to find which parameters impacts the basic reproduction number the most, it was observed that the most sensitive
parameters are mosquitoes biting rate ( ) and mosquitoes death rate ( ). Simulations using MATLAB and the fourth order
Range-Kutta numerical method was done to determine the effectiveness of all possible combinations of malaria control
measures and the numerical results shows that the combination of four (4) control measures have the highest impact on the
control of the disease. Also with individuals total adherence to effective use of treated bednets and spray of insecticide for at
least 70 days in the population, little treatment efforts will then be required by the Ghanaian population in the control of the
spread of Malaria.
Keywords: Malaria, stability, sensitivity analysis, basic reproduction number

1950) to account for super-infection. These two works were further extended by Ngwa and Shu (2000) with the popular
generalized SEIR malaria model, which includes both the human and mosquito interactions. Anderson and May (1991) derived a
malaria model with the assumption that acquired immunity in malaria is independent of exposure duration. Different control
measures and role of transmission rate on the disease frequency were further examined. (Hyun, 2000) studied a malaria
transmission model for different levels of acquired immunity and temperature dependent parameters, relating also to global
warming and local socioeconomic conditions in (Isao,2004). Kawaguchi et al. (2006) examined the combined use of insecticide
spray and zooprophylaxis as a strategy for malaria control. Dietz et al. (1974) proposed a model that accounts for acquired
immunity. (Chiyaka, 2008) formulated a deterministic model with two latent periods in the host and vector populations to assess
the impact of personal protection, treatment and possible vaccination strategies on the transmission dynamics of malaria. They
also considered treatment and spread of drug resistance in an endemic population (Chiyaka,2009). Jia (2008) formulated and
examined a compartmental mathematical model for malaria transmission that includes incubation periods for both infected
humans and mosquitoes. Mukandavire et al. (2009) proposed and investigated a deterministic model for the co-infection of HIV
and malaria in a community. Mwasa and Tchuenche (2011) examined a mathematical model that covers the dynamics of Cholera
transmission to study the influence of public health educational campaigns, vaccination and treatment in controlling the disease.
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 and the fourth order Range-Kutta numerical method. Some of the assumptions in this
paper on mathematical modelling of malaria are based on studies by Nakul Chitnis (2006), Okosun et al (2011) and Oduro et al
(2012).
2. Methodology
An (SEIR-SEI) epidemiological model of malaria transmission with four control measures is formulated as a system of ordinary
differential equations where the human population is made up of four compartments (Susceptible, Exposed, Infectious and
Recovered) and the vector mosquito population is made up of three compartments (Susceptible, Exposed and Infectious).
Sensitivity analysis is intended to determine which parameters impacts the basic reproduction number the most. Clinical malaria
data for the thesis is obtain from the National malaria control programme (NMCP) in Ghana and other parameter values are obtain
from the World Malaria Report 2013 by the World Health Organisation through the internet. Simulation using MATLAB and the
fourth order Range-Kutta numerical method will be conducted to establish the most effective control measures and the most
effective control combinations.
The model that we consider here is a slight modification of models for malaria transmission considered in Oduro et al.(2013),
Okosun et al. (2013), Ngwa and Shu (2000), Ngwa (2006) and Rafikov et al. (2009) it is not a generalization of these ones; nor is
it a special case of them. It is a standard model of SEIRS type for humans and SEI for mosquitoes in which we incorporated four
time dependent control measures simultaneously: (i) the use of treated bednets, (ii) treatment of infective humans, (iii) treatment
to protect pregnant women and their new born children: intermittent preventive treatment (IPTp) for pregnant women and (iv)
spray of insecticides. It is common knowledge that malaria treatment reduces the risk of disease but has only a low or negligible
transmission blocking effect. Here we consider a possible treatment that blocks transmission from infective humans to
mosquitoes. The model sub-divides the total human population, denoted by , into the following sub-classes of: individuals who
are susceptible to infection with malaria , those exposed to malaria parasite , individuals with malaria symptoms
and recovered individuals . So that, = + + + . The total vector (mosquito) population, denoted by , is
sub-divided into susceptible mosquitoes , mosquitoes exposed to the malaria parasite and infectious mosquitoes .
That is, = + + . Susceptible individuals are recruited at a rat . They either die from natural causes (at a rate )
or move to the exposed class by acquiring malaria through contact with infectious mosquitoes at a rate (1- ) , where is
the transmission probability per bite, is the per capita biting rate of mosquitoes, is the contact rate of vector per human per
unit time and 1- is the control on the use of treated bednets. Exposed individuals move to the infectious class at a rate
. Infectious individuals are assumed to recover at a rate + , where b is the rate of spontaneous recovery, is the
control on treatment of infected individuals, is treatment to protect pregnant women and their new born children: intermittent
preventive treatment (IPTp) for pregnant women and is the efficacy of treatment. Infectious individuals who do not
recover die at a rate . Susceptible mosquitoes are generated at a rate . They either die from natural causes (at a rate
or move to the exposed class by acquiring malaria through contacts with infected humans at a rate , where
is the probability for a vector to get infected by an infectious human. Exposed mosquitoes are assumed to die at a rate or move
to the infected class at a rate . Infected mosquitoes die at a rate . The mosquito population is reduced, due to the use of
insecticides spray, at a rate , where and p represent, respectively, the control and the efficacy of insecticides spray.
2.1. Model Formulation
Putting the above formulations and assumptions together gives the following vector-host model

Figure 1: The flow diagram for malaria disease transmission.
















































v
v
v
v
v
v
v
h
v
h
v
v
v
h
v
h
v
v
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
v
h
h
h
v
h
h
h
h
h
I
pu
E
dt
dI
E
pu
E
N
S
I
u
dt
dE
S
pu
N
S
I
u
dt
dS
R
R
I
u
u
b
dt
dR
I
u
u
b
I
E
dt
dI
E
E
N
S
I
u
dt
dE
N
S
I
u
S
R
dt
dS
)
(
=
)
(
)
(1
=
)
(
)
(1
=
)
(
=
)
(
)
(
=
)
(1
=
)
(1
)
(1
=
3
2
3
2
1
3
1
4
2
4
2
1
1
1
1























(1)
Where and . Here, and represent the force of infection of humans and mosquitoes, respectively.
In the model denote the rate the susceptible humans , become infected by infectious female anopheles mosquitoes and
refers to the rate at which the susceptible mosquitoes are infected by infectious humans . By Mwamtobe, 2010. It is
important to note that the rate of infection of susceptible human by infected mosquito is dependent on the total number of
Humans available per vector
2.1.1. Analysis of the Model
The model is analyse to check if the control measures have any impact on the malaria disease, that is, whether the malaria disease
can be control (eliminated) or not. The model parameters which determine persistence or elimination of malaria will be
determined and studied. Therefore, we start by determining the invariant region to check whether the SEIR-SEI malaria model is
in a biologically feasible region for both human and mosquito populations and showing that all solutions of equation (1) are
positive at all t and are attracted in that region. that is to show that the model equations are mathematically well posed.
The solutions of the system (1) are feasible for all if they enter the invariant region
. Therefore, the feasible solutions set for the model (1 )is given by
. Therefore, the
region is positively-invariant ( i.e. solution remain positive for all times, t)

Positivity of solutions
Lemma 1 Let the initial data be
.
Then the solution set of the system (1) is positive for all .
 Proof
From the first equation in the model (1), we have
Using separation of variables and integrating both sides gives
using the initial conditions: ,
Therefore
.
similarly, it can be shown that the remaining equations of the system (1) are positive for all , because for all
.
Steady states and Stability
In this section, we assume that the control parameters are constant and determine the basic reproductive number, the steady states
and their stability of system (1).
The disease free equilibrium (DFE) of the malaria model (1) exists and is given by
,0,0)
)
(
,0,0,0,
)
(1
(
=
)
,
,
,
,
,
,
(
=
3
*
*
*
*
*
*
*
0
pu
I
E
S
R
I
E
S
E
v
v
h
h
v
v
v
h
h
h
h







(2)
which represents the state in which there is no infection (in the absence of malaria) in the society.
The basic reproduction number, is calculated by using the next generation matrix (Van den and Watmough, 2002). It is given
by
)
)(
)(
(
)
(
)
(1
=
4
2
2
3
1
2
3
2
2
1
2
1
0

























u
b
pu
U
pu
u
R
h
v
h
v
h
h
v
Calculating the endemic equilibrium point, we obtain
0
*
*

h
I
)
(
)
(1
)
(
)
(1
=
3
2
1
2
3
1
*
*
pu
u
R
pu
u
S
v
o
h
v
h
h
h












1
4
2
*
* )
(
=




 



 u
u
b
E h
h








h
h
u
u
b
R
)
(
= 4
2
*
*
h
v
h
v
v
N
pu
u
N
S
)
(
)
(1
=
3
1
*
*







)
)
(
)
))((1
(
(
)
(1
=
3
1
3
2
*
*
1
*
*
h
v
v
h
h
h
v
v
N
pu
u
pu
N
I
N
u
E













)
(
)
(1
)
)(
(1
=
3
1
3
1
*
*
pu
N
u
R
pu
u
I
v
h
ov
v
v











Where C
BI
I
A h
h 
 *
*
2
*
*
)
( (3)
Using the quadratic formula
a
ac
b
b
x
2
4
=
2



we have
A
AC
B
B
Ih
2
4
=
2
* 


A
AC
B
B
2
4
=
2



or
A
AC
B
B
2
4
=
2









=
2
4
=
2
4
=
2
2
A
B
AC
B
A
AC
B
B
Where
)
)
(1
)(
)
(
)(
(
)
(
)
(1
)
(
)
)(1
(
)
)(
)(1
)
)(
(1
)(
(
=
1
2
0
3
3
2
0
1
2
0
3
2
1
2
0
3
1
2
0
3
1
4
2




















u
R
pu
N
pu
R
u
R
pu
u
R
pu
u
R
pu
u
N
u
u
b
A
h
h
v
h
h
h
v
v
h
v
h
v
h
h
















)
)
(1
)(
)
(
)(
(
)
(
)
)
(
)
(1
)(
(
)
)
(
(
)
(
)
)(1
(
)
)
(
(
)
(
)
)(1
(
)
)
(
)
(1
(
)
(
)
)(1
(
)
(1
=
1
2
0
2
3
2
3
2
0
3
1
2
3
2
0
3
2
1
2
0
2
3
2
2
0
3
2
1
4
2
2
0
3
1
2
0
3
2
1
































u
R
pu
N
pu
R
pu
u
N
R
pu
R
pu
u
R
pu
N
R
pu
u
u
u
b
R
pu
u
N
R
pu
u
B
h
h
h
v
h
h
v
h
h
v
h
h
h
ov
v
h
v
h
h
h
v
h
v
h
v
h
h
v
h
h





























)
(
)
)
(
)(
(
)
)
(
(
)
(
)
)(1
(
)
(1
=
3
2
0
2
3
2
2
0
2
3
2
2
0
3
2
1
pu
R
pu
N
R
pu
N
R
pu
u
C
v
h
h
h
v
h
h
h
h
v
h
v
h
h
























From the quadratic equation (3) we analyze the possibility of multiple equilibria. It is important to note that the coefficient A is
always positive with B and C having different signs. We realize that C is positive if is less than unity, and negative if is
greater than unity. Hence, we have established the following results:
 Proposition 1.
(1) Precisely one unique endemic equilibrium if and or ,
(2) Precisely one unique endemic equilibrium if
(3) Precisely two endemic equilibrium if , and ,
(4) No endemic otherwise.
The stability of the endemic equilibrium of the model (1) can be analysed using the Centre Manifold Theory described by (Gumel
and Song, 2008; Castillo-Chavez and Song, 2004) we carry out a bifurcation analysis of system (1) at . For this, we
consider the transmission rate as a bifurcation parameter so that if and only if
Let be the bifurcation parameter, the system is linearised at disease free equilibrium point when with
Thus can be solved when with . Thus can be solved from when
The Jacobian matrix of (1) calculated at is given by

A right eigenvector associated with the eigenvalue zero is
Solving the system we have the following right eigenvector
From the left eigenvector we have the following results.
therefore free
we now compute the sign of a and b as indicated in the theorem
similarly partial derivatives that are not zero when calculating b are
therefore
Hence and . Therefore the following theorem holds.

 Theorem 2
The model (1) has a unique endemic equilibrium which is locally asymptotically stable when and unstable when
.
2.1.2 Sensitivity Analysis
We would like to know different factors that are responsible for the disease transmission and prevalence. In this way we can try to
reduce human mortality and morbidity due to disease. Initial disease transmission depends upon the reproductive number whereas
disease prevalence is directly related to the endemic equilibrium point. The class of infectious humans is the most important class
because it represents the persons who may be clinically ill and is directly related to the disease induced deaths. We will calculate
the sensitivity indices of the reproductive number, , and the endemic equilibrium point with respect to the parameters given in
Table 1 for the model. By the analysis of these indices we could determine which parameter is more crucial for disease
transmission and prevalence.
 Definition
The normalized forward sensitivity index of a variable, h , that depends on a parameter, l , is defined as :
l
h
h
l
l



=

Therefore the sensitivity index of 0
R that depends on 1
 is given as
1
0
0
1
0
1
=






R
R
R
2
1
1
2
1
1
1
0
1 )
(
=




 
h
R
and
)
(
2
1
)
(
2
1
)
(
=
1
2
1
1
2
1
1
2
1
1
2
1
1
1
0








 








h
h
h
R
)
)
(
2
1
)
(
2
1
)
(
(
)
)
(
(
=
1
2
1
1
2
1
1
2
1
1
2
1
1
2
1
1
2
1
1
1
0
1 





















h
h
h
h
R
therefore
)
2(
=
1
0
1
h
h
R





similarly the sensitivity index of other parameters are as follows

)
)(
2(
=
)
2(
=
)
2(
=
)
2(
=
)
)(
2(
))
3(
(2
=
)
2(
=
1
4
2
4
1
2
1
1
1
2
0
4
2
0
4
2
0
3
2
3
0
2
3
3
2
3
2
0
3
2
3
0
1














































































h
h
h
R
h
h
R
b
h
R
v
v
R
v
v
v
v
R
v
v
v
R
u
u
b
u
u
b
u
u
b
b
u
u
b
pu
pu
pu
pu
pu
pu
pu
Sensitivity indices for the control parameters
1
1
0
1 1
=
u
u
R
u



)
2(
=
2
2
0
2 u
b
u
h
R
u









)
)(
2(
)
2
3
(3
=
2
3
3
3
2
3
0
3 











v
v
v
R
u
pu
pu
u
pu
p
)
2(
=
4
4
0
4 u
b
u
h
R
u









Parameter Description Estimated values
Mosquito contact rate with human
Mosquito biting rate +1
Death rate of mosquitoes
Probability of human getting infected
Probability of a mosquito getting infected
Human birth rate
Mosquitoes birth rate
b Spontaneous recovery
Human natural death rate
Humans progression rate from exposed to infected
Table 1: Sensitivity Indices of R0
3. Analysis and Results
In this section, we study numerically an epidemiological model of malaria transmission with four control parameters. We carry
out numerical simulations using a fourth order Rung-Kutta scheme in Matlab. The main aim is to verify some of the analytical
results on the stability of the system (1). The parameter values were obtained from literature. We simulate the basic malaria model

with different combinations of the control measures to find out the effects of varying each control parameter. The figures are
plotted using the parameter values in table (2) and the initial conditions. Using various combinations of the four controls, one
control at a time and two controls at a time, we investigate and compare numerical results from simulations with the following
scenarios:
• Strategy A. using personal protection ( 1
u ) without insecticide spraying ( 0
=
3
u ) and no treatment of the symptomatic humans
( 0
=
2
u )
• Strategy B. treating the symptomatic humans ( 2
u ) without using insecticide spraying ( 0
=
3
u ) and no personal protection
( 0
=
1
u ),
• Strategy C. using insecticide spraying ( 3
u ) without personal protection ( 0
=
1
u ) and no treatment of the symptomatic humans
( 0
=
2
u ),
• Strategy D. treating the symptomatic humans ( 2
u ) and using insecticide spraying ( 3
u ) with no personal protection ( 0
=
1
u ),
• Strategy E. using personal protection ( 1
u ) and insecticide spraying ( 3
u ) with no treatment of the symptomatic humans
( 0
=
2
u ),
• Strategy F using treatment ( 2
u ) and personal protection ( 1
u ) with no insecticide spraying ( 0
=
3
u ), finally
• Strategy G. using all four control measures ( 1
u , 2
u , 3
u , and 4
u ).
Parameter Description Estimated values References
Mosquito contact rate with human Kbenesh et al.(2009)
Mosquito biting rate 0.2 Kbenesh et al.(2009)
Probability of human getting infected 0.8333 Nakul et al.(2006)
Probability of a mosquito getting infected 0.09 Kbenesh et al.(2009)
Natural death rate in humans Hyun(2001)
Natural death rate in mosquitoes Nakul et al.(2006)
Recovered individuals' loss of immunity NMC Ghana (2013)
Humans' progression rate from exposed to infected Kbenesh et al.(2009)
Mosquitoes' progression rate from exposed to infected Kbenesh et al.(2009)
Human birth rate NMC Ghana.(2013)
Mosquitoes birth rate Oduro et al.(2012)
Proportion of effectively treated individuals Assumed
Disease induced death NMC Ghana.(2013)
b Spontaneous recovery Chiyaka et al. (2008)
p Insecticide's efficacy Assumed
Table 2: Description of variables and parameters of the malaria model
4. Numerical Simulations
Numerical simulation using the fourth order Range-Kutta method in matlab is use to solve the malaria model(1).
)
2
2
(
6
1
= 4
3
2
1
1 k
k
k
k
y
y i
i 




)
,
(
=
1 i
i y
x
hf
k
)
2
1
,
2
(
= 1
2 k
y
h
x
hf
k i
i 

)
2
1
,
2
(
= 2
3 k
y
h
x
hf
k i
i 

)
2
1
,
(
= 3
4 k
y
h
x
hf
k i
i 

The function )
,
( i
i y
x
f is the malaria model(1) with h as the step size.
For the numerical simulation we used the following initial state variables and the parameter values from table (1) 700
=
(0)
h
S ,
100
=
(0)
h
E , 0
=
(0)
h
I , 0
=
(0)
h
R , 5000
=
(0)
v
S , 500
=
(0)
v
E , 30
=
(0)
v
I and parameter values 0.00011
=
h
 ,
0.071
=
v
 , 0.030
=
 , 0.01
=
 , 0.05
=
 , 0.0000457
=
h
 , 0.0667
=
v
 , 0.0014
=
 , 0.058
=
1
 ,

0.0556
=
2
 , 0.5
=
b , 0.502
=
 , 0.02
=
 , 0.5
=
 , 0.85
=
p to illustrate the effect of different control strategies on
the spread of malaria in the Ghana population. Thus, we have considered the spread of malaria in an endemic population.
4.1. Simulation of treated bednets
Only the control ( 1
u ) on treated bednets is used while the control on treatment ( 2
u ) and the control on insecticide spray ( 3
u ) are
set to zero. In Figure 2, the results show a significant difference in the h
I and v
I with control strategy compared to h
I and v
I
without control. Specifically, we observed in Figure 2(a) that the control strategies lead to a decrease in the number of
symptomatic human ( h
I ) as against an increases in the uncontrolled case. Similarly, in Figure 2(b), the uncontrolled case resulted
in increased number of infected mosquitoes ( v
I ), while the control strategy lead to a decrease in the number infected. The control
profile shows that the personal protection control 1
u is at the upper bound till the time 100
=
t days, before dropping to the
lower bound.
Figure 2: Simulations showing the effect of treated bednets only on infected human and mosquitoes populations
4.1.1. Simulation of Treatment
With this strategy, only the control ( 2
u ) on treatment is used while the control on treated bednets ( 1
u ) and the control on
insecticide spray ( 3
u ) are set to zero. In Figure 3, the results show a significant difference in the h
I and v
I with control strategy
compared to h
I and v
I without control. But this strategy shows that effective treatment only has a significant impact in reducing
the disease incidence among human population. The control profile shows that the treatment control 2
u rises to and stabilizes at
the upper bound for 70
=
t days, before dropping to the lower bound.
Figure 3: Simulations showing the effect of treatment only on infected human and mosquitoes populations

4.1.2. Simulation of Insecticide Spraying
With this strategy, only the control on insecticide spraying ( 3
u ) is used while the control on treatment ( 2
u ) and the control on
treated bednets ( 1
u ) are set to zero. The results in Figure 4 show a significant difference in the h
I and v
compared to h
I and v
I without control. We see in Figure 4(a) that the control strategies resulted in a decrease in the number of
symptomatic human ( h
I ) as against an increase in the uncontrolled case. Also in Figure 4(b), the uncontrolled case resulted in
increased number of infected mosquitoes ( v
I ), while the control strategy lead to a drastic decrease in the number of infected
mosquitoes. The control profile shows that the insecticide spray control 3
u is at the upper bound till the time 90
=
t days, it then
reduces gradually to the lower bound.
Figure 4: Simulations showing the effect of insecticide spraying only on infected human and mosquitoes populations
4.1.3. Simulation of Treatment and Insecticide Spray
With this strategy, the control ( 2
u ) on treatment and the control on ( 3
u ) insecticide spraying are both used while the control on
treated bednets ( 1
u ) is set to zero. In Figure 5, the result shows a significant difference in the h
I and v
compared to h
I and v
I without control. We observed in Figure 5(a) that the control strategies resulted in a decrease in the
number of symptomatic humans ( h
I ) as against increases in the uncontrolled case. Similarly in Figure 5(b), the uncontrolled case
resulted in increased number of infected mosquitoes ( v
I ), while the control strategy lead to a decrease in the number of infected
mosquitoes. The control profile shows that the treatment control 2
u is at the upper bound till time 50
=
t , while the insecticide
spray 3
u is at the upper bound for 90 days before reducing gradually to the lower bound.
Figure 5: Simulations showing the effect of treatment and insecticide spray only on infected human and mosquitoes populations

4.1.4. Simulation of Treated Bednets and Insecticide Spray
Here, the control on treated bednets ( 1
u ) and the spray of insecticide ( 3
u ) are used while setting the control on treatment
0
=
2
u . For this strategy, shown in Figure 6, we observed that the number of symptomatic human ( h
I ) and mosquitoes ( v
I )
differs considerably from the uncontrolled case. Figure 6(a), reveals that symptomatic humans ( h
I ) is lower in comparison with
the case without control. While Figure 6(b), reveals a similar result of decreased number of infected mosquitoes ( v
I ) for the
controlled strategy as compared with the strategy without control. The control profile shows that the control on treated bednets
( 1
u ) is at upper bound for 60 days, while insecticide spray ( 3
u ) is at upper bound for 100
=
t days before reducing to the lower
bound.
Figure 6: Simulations showing the effect of treated bednets and insecticide spray only on infected human and mosquitoes
populations
4.1.5. Simulation of Treated Bednets and Treatment
With this strategy, the control on treated bednets ( 1
u ) and the treatment ( 2
u ) are used while setting the control on spray of
insecticide 3
u to zero. For this strategy, shown in Figure 7, there is a significant difference in the h
I and v
compared to h
I and v
I without control. We observed in Figure 7(a) that due to the control strategies, the number of symptomatic
humans ( h
I ) decreases as against the increase in the uncontrolled case. A similar decrease is observed in Figure 7(b) for infected
mosquitoes ( v
I ) in the control strategy, while an increased number is observed for the uncontrolled case resulted. In the control
profile, the control 1
u is at the upper bound for 118 (days) and drops gradually until reaching the lower bound, while control on
treatment 2
u starts and remain at upper bound for 12 days before dropping gradually to the lower bound. The result here shows
that with a treated bednets coverage of 100% for 118 days and treatment coverage of 100% for 12 (days), the disease incidence
will be greatly reduced.

Figure 7: Simulations showing the effect of treated bednets and treatment only on infected human and mosquitoes populations
4.1.6. Simulation of Treated Bednets, Treatment and Insecticide Spray
Here, all three controls ( 1
u , 2
u and 3
u ) are used. For this strategy in Figure 8, we observed in Figure 8(a) and 8(b) that the
control strategies resulted in a decrease in the number of symptomatic humans ( h
I ) and infected mosquitoes ( v
I ) as against the
increased number of symptomatic humans ( h
I ) and infected mosquitoes in the uncontrolled case. The control profile shows that
the control 1
u is at upper bound for 60
=
t days, while control 2
u , starts high at about 77% and reduces to the lower bound
gradually over time. The control 3
u on the other hand is at upper bound for about 100 days before reducing to the lower bound.
Figure 8: Simulations showing the effect of treated bednets, treatment and insecticide spray only on infected human and
mosquitoes populations
4.1.7. A Comparison of All Four Control Strategies
A comparison of all four control strategies in Figures 9(a) and 9(b) shows that while all four strategies lead to a decrease in the
number of infected, both in human and in mosquitoes. The control strategy without treatment resulted in a higher number of
infected humans, followed by the strategy without treated bednets. The strategy without the spray of insecticide even though, it
gave a better result in reducing the infection in human, gave a poorer result in reducing the mosquitoes population. This result
shows that with individuals total adherence to effective use of treated bednets and spray of insecticide in the population, little
treatment efforts will then be required by the nation Ghana in the control of the spread of the disease.

Figure 9: Simulations showing the comparison of the effect of all four different control strategies
4.1.8. Simulation of Insecticide Spraying
A scenario with reducing different fraction of vector population is simulated, the result shows that the value of 0.2
=
p gave the
lowest number of susceptible ( v
S ) vectors while 0.85
=
p gave the least value of infected ( v
I ) vectors, this is followed by
0.6
=
p , 0.85
=
p and lastly by 14
=
p (a case corresponding to no use or ineffective insecticide) as expected. This has the
resultant effect (not depicted here) on total number of vectors susceptible to malaria v
S . When 0.85
=
p , the total number of
vectors susceptible to malaria, v
S is 4900, when 0.6
=
p , 2000
=
v
S , and lastly when 0.2
=
p , the total number of
susceptible vectors to malaria, 1000
=
v
S .
4.1.9. Sensitivity Indices
By analysing the sensitivity indices we observe that the most sensitive parameters are mosquitoes biting rate ( ) and mosquitoes
death rate ( v
 ). The reproductive number 0
R is directly related to the mosquitoes biting rate and inversely related to the
mosquitoes death rate. It can realized that an increase (or decrease) in biting rate  by 10% increases (or decreases) 0
R by 10%.
Similarly increase (or decrease) in death rate v
 of mosquitoes by 10% decreases (or increases) 0
R by 10%. Therefore it suggest
that strategies that can be applied in controlling and eradicating the disease are to target the mosquito biting rate and the mosquito
death rate. These are; 1
u The use of insecticide-treated bed nets and 3
u Indoor residual spray.
5. Conclusion
The malaria model was formulated with four control parameters using a deterministic system of differential equations to
effectively investigate the transmission dynamics of malaria in Ghana and to find out the most effective control measure.
Mathematically, malaria was modelled as a 7-dimensional system of ordinary differential equations. It was first showed that there
exists a domain where the model is epidemiologically and mathematically well-posed. The basic reproduction number,( 0
R ) was
calculated and it was established that the model is locally asymptotically stable when the associated reproduction number is less
than unity and the model is unstable when the reproduction number is greater than unit. That is to say if ( 1
<
0
R ), the disease can
not persist in Ghana and when ( 1
>
0
R ) the disease can persist in Ghana. The stability analysis of the model was investigated and
it proved that the disease-free equilibrium is locally asymptotically stable if ( 1
<
R ) and unstable when ( 1
>
R ). The Centre
Manifold theorem was used to show that the model has a unique endemic equilibrium which is locally asymptotically stable when
( 1
<
R ). The sensitivity analysis was applied to find which parameters impacts the basic reproduction number the most and it was
observed that the most sensitive parameters are mosquitoes biting rate ( ) and mosquitoes death rate ( v
 ). The reproductive
number 0
R is directly related to the mosquitoes biting rate and inversely related to the mosquitoes death rate. It was realized that
an increase (or decrease) in biting rate  by 10% increases (or decreases) 0
R by 10%. Similarly increase (or decrease) in death
rate v
 of mosquitoes by 10% decreases (or increases) 0
R by 10%. Also numerical simulation using the fourth order Range-
Kutta method in matlab was done to determine the effectiveness of all possible combinations of the four malaria control measures.

In the control model considered, we use one control at a time and the combination of two controls at a time, while setting the
other(s) to zero to investigate and compare the effects of the control strategies on malaria eradication in Ghana. Clinical malaria
data from Ghana was used for the control analysis and our numerical results shows that the combination of the four (4) controls
these are; the use of treated bednets ( 1
u ), treatment of infective humans( 2
u ), spray of insecticides( 3
u ) and the intermittent
preventive treatment for pregnant women ( 4
u ) has the highest impact on the control of the disease. This is followed by the
combination of treatment of infective human ( 2
u ) and the use of treated bednets ( 1
u ) among the human population and lastly by
the combination involving the use of treated bednets ( 1
u ) and spray of insecticide ( 3
u ) . In communities where resources are
scarce, it was suggested that the combination of treatment of infective human ( 2
u ) and the use of treated bednets ( 1
u ) should be
adopted, having observed from the comparison of all four control strategies in Figure 9, that there is no significant difference
between this strategy and the combination of the three (3) controls. Our result however shows two possible control strategies, each
with two combinations of control measures that are sufficient to effectively achieve and maintain interruption of transmission of
malaria in Ghana. A result which addresses the WHO concern about the insufficiency of only one control measure to achieve and
maintain interruption of transmission of malaria.
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