^{1}

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In this work, we developed a compartmental bio-mathematical model to study the effect of treatment in the control of malaria in a population with infected immigrants. In particular, the vector-host population model consists of eleven variables, for which graphical profiles were provided to depict their individual variations with time. This was possible with the help of MathCAD software which implements the Runge-Kutta numerical algorithm to solve numerically the eleven differential equations representing the vector-host malaria population model. We computed the basic reproduction ratio
*R*
_{0} following the next generation matrix. This procedure converts a system of ordinary differential equations of a model of infectious disease dynamics to an operator that translates from one generation of infectious individuals to the next. We obtained
*R*
_{0} =
,
*i.e*., the square root of the product of the basic reproduction ratios for the mosquito and human populations respectively.
*R*
_{0m} explains the number of humans that one mosquito can infect through contact during the life time it survives as infectious.
*R*
_{0h} on the other hand describes the number of mosquitoes that are infected through contacts with the infectious human during infectious period. Sensitivity analysis was performed for the parameters of the model to help us know which parameters in particular have high impact on the disease transmission, in other words on the basic reproduction ratio
*R*
_{0}.

Malaria is a highly prevalent infectious disease especially in the tropical and subtropical areas.

In addition to being widespread, malaria is also a deadly disease. This is because statistics has shown that for Africa in particular, annually 145,000 million to 150,000 million infections are reported, among which, 800 to 850 cases result in deaths as shown in

It is interesting to do a quick statistical analysis of the data in

Year | 2000 | 2001 | 2002 | 2003 | 2004 | 2005 | 2006 | 2007 | 2008 | 2009 |
---|---|---|---|---|---|---|---|---|---|---|

Cases (×10^{3}) | 173,000 | 178,000 | 181,000 | 185,000 | 187,000 | 188,000 | 187,000 | 186,000 | 181,000 | 176,000 |

Deaths (×10^{3}) | 900 | 893 | 885 | 880 | 870 | 853 | 832 | 802 | 756 | 709 |

Model Summary and Parameter Estimates. Dependent Variable: C (Numbers of cases)

The independent variable is T.

Observation: It is quite clear from the WHO data, for the number of malaria cases reported over the 10 year period that the incidence of malaria infection follows a parabolic curve, rising sharply initially, to reach a maximum and then declining sharply thereafter (

C = 165283.3 + 7609.85 T − 647.73 T 2 with goodness of fit R 2 = 0.981 .

Model Summary and Parameter Estimates. Dependent Variable: D (Number of deaths)

The independent variable is T.

Observation: The number of malaria related deaths over the 10 year period as depicted in the above graph, follows a parabolic curve, rising from a high value initially, then reaching a maximum and then declining sharply thereafter. The equation of the parabola is given by:

D = 882.883 + 12.07 T − 2.89 T 2 with goodness of fit R 2 = 0.992 .

Malaria is a vector-borne disease [

Human infection starts from a blood meal of an infectious female mosquito. The parasites existing in the infectious mosquito’s saliva, called sporozoites at this stage, enter the bloodstream of the human through mosquito bites and migrate to the liver. Within minutes after entering in the human body, sporozoites infect hepatocytes, and multiply asexually and asymptomatically in liver cells for a period of 5 - 30 days [

According to the transmission procedure of malaria, there are three conditions for the prevalence of the disease:

1) High density of Anopheles mosquitoes,

2) High density of human population,

3) Large rate of transmission of parasites between human beings and mosquitoes.

Obviously, not too much can be done in respect to (2). So, (1) and (3) are naturally targeted. That is, either controlling the population of Anopheles female mosquitoes at a lower level, or avoiding biting by mosquitoes can reduce the chance of malaria becoming endemic. In the middle of the last century, people in Africa have already knew how to remove or poison the breeding grounds of mosquitoes or the aquatic habitats of the larva stages, such as by filling or applying oil to places with standing water, to control the population of mosquitoes [

In order to control mosquito-borne infections one can adopt the following measures;

• Reduce vector population: Make environment less mosquito-friendly by draining stagnant water.

• Use insecticides; not without problems: for example some mosquitoes become insecticide resistant.

• Prevent mosquitoes biting people. Insecticide-laced bed nets, although this is ineffective against mosquitoes that mainly bite during the day (e.g. A. aegypti).

• Vaccines and drug treatments. Not always available, there are problems with drugs and drug resistance.

The first and simplest model of malaria was developed by [

In the Ross-Macdonald model of malaria transmission, the flow of human from a susceptible class to an infected class and through recovery from infection, the reverse is shown in the upper part of the

The development of the means intended to reduce the spread of malaria infections and eradication necessitates decisive measures to curb the malaria epidemic. In particular, sustained minimization of the number of humans with incidence of malaria as a result of adequate control, can be attained by developing a suitable mathematical model which can enable us to understand better the dynamics and control of the vector-host endemic.

In developing the model, the human population is compartmentalized into seven classes including the susceptible, infected, exposed, treated, non-treated, recovered, and protected classes. For the mosquito population, we have four classes, namely; class of mosquito larva, susceptible mosquitoes, infected mosquitoes and exposed mosquitoes. We assume free interaction between the vector and host populations. The mathematical analysis of the compartmental models leads us to eleven coupled systems of nonlinear ordinary differential equations.

In this section we develop a compartmental bio-mathematical model (

to study the effect of treatment in the control of malaria in a population with infected immigrants.

From the above compartmental model we obtain the following equations for the dynamics of the human-mosquito interaction.

d S H d t = ( 1 − q ) Λ H + α 2 A + ρ R H − β H I M S H N H − α 1 S H − δ H S H (1)

d E H d t = β H I M S H N H − g E H − δ H E H (2)

d I H d t = g E H + q Λ H − k 1 I H − k 2 I H − δ H I H (3)

d I H N d t = k 2 I H − ( ω H + δ H ) I H N (4)

d T H d t = k 1 I H − γ T H − δ H T H (5)

d R H d t = γ T H − ( μ + ρ + δ H ) R H (6)

d A d t = α 1 S H + μ R H − α 2 A − δ H A (7)

d L M d t = Λ M − m L M − δ M L M (8)

d S M d t = m L M − β M I H S M N H − δ M S M (9)

d E M d t = β M I H S M N H − ϕ E M − δ M E M (10)

d I M d t = ϕ E M − δ M I M (11)

The state variables and parameters are defined in

Symbol | Description |
---|---|

S H ( t ) | Susceptible human population at time t |

E H ( t ) | Exposed human population at time t |

I H ( t ) | Infected human population at time t |

I H N ( t ) | Non-treated infected human population at time t |

T H ( t ) | Treated human population at time t |

R H ( t ) | Recovered human population at time t |

A ( t ) | Protected human population at time t |

L M ( t ) | Population of mosquito larva at time t |

S M ( t ) | Population of susceptible mosquitoes at time t |

E M ( t ) | Population of exposed mosquitoes at time t |

I M ( t ) | Population of infected mosquitoes at time t |

N H | Total population size of humans |

N M | Total population size of mosquitoes |

Symbol | Description |
---|---|

Λ H | Birth and immigrant rate of humans |

Λ M | Birth rate of mosquitoes |

ρ | Rate of loss of immunity |

β H | Transmission rate of infection from infected mosquitoes to susceptible human |

α 2 | Loss of immunity of protected class |

q | Fraction of infective immigrants |

α 1 | Progression rate of susceptible human to protected class |

k 1 | Treatment rate of human from infected state to treated class |

k 2 | Transmission rate of human from infected state to infectious none treated class |

g | Progression rate of human from exposed to infected compartments |

γ | Recovery rate of human from treated class |

δ H | Natural death rate of human from exposed to infected |

μ | Progression rate of human from recovery class to protected class |

M | Progression rate of mosquitoes from larva to susceptible |

β M | Transmission rate of infection from infected human to susceptible mosquitoes |

δ M | Natural death rate of mosquitoes |

ϕ | Progression rate of exposed mosquitoes to infected mosquitoes |

ω H | Disease-induced death rate of human |

The total population sizes N H and N M can be determined by N H = S H + E H + I H + I H N + T H + A + R H and N M = L M + S M + E M + I M . Thus

d N H d t = Λ H − δ H N H − ω H I H N (12)

Without loss of generality, we can write

d N H d t ≤ Λ H − δ H N H , d N M d t ≤ Λ M − δ M N M (13)

The model system has solution which are contained in the feasible Ω = Ω H × Ω M .

Proof: let Ω = { S H , E H , I H , I H N , T H , A , R H , L M , S M , E M , I M } ∈ ℝ + 11 be any solution of the system with non-negative initial conditions. From Equation (13)

d N H d t ≤ Λ H − δ H N H (14)

Adopting Birhoff and Rotta [

0 ≤ N H ≤ Λ H δ H , Λ H − δ H N H ≥ C e − δ H t (15)

where C is a constant.

Therefore, all feasible solutions of the human population only of the model system are in the region.

Ω H = { ( S H , E H , I H , I H N , T H , A , R H ) ∈ ℝ + 7 : N H ≤ Λ H δ H }

Similarly the feasible set for model of the mosquitoes population only are in the region

Ω M = { ( L M , S M , E M , I M ) ∈ ℝ + 4 : N M ≤ Λ M δ M }

Therefore the feasible set for the model system is given by

Ω = { ( S H , E H , I H , I H N , T H , A , R H , L M , S M , E M , I M ) ∈ ℝ + 11 : N H ≤ Λ H δ H = N H * , N M ≤ Λ M δ M = N M * } (16)

The nonlinear system (1)-(11) will be qualitatively analyzed so as to find the conditions for existence and stability of disease free equilibrium points. Analysis of the model allows us to determine the impact of treatment on the transmission of malaria infection in a population. Also on finding the reproductive number R 0 , one can determine if the disease become endemic in a population or not [

d N H d t = Λ H − δ H N H , hence N H ( t ) → Λ H δ H as t → ∞ .

Thus Λ H δ H is the upper bound of N H ( t ) provided that N H ( 0 ) ≤ Λ H δ H . Similarly,

d N M d t = Λ M − δ M N M ⇒ N M ( t ) → Λ M δ M as t → ∞ .

Thus Λ M δ M is the upper bound of N M ( t ) provided that N M ( 0 ) ≤ Λ M δ M . Hence the invariant region is

Ω = { ( S H , E H , I H , I H N , T H , A , R H , L M , S M , E M , I M ) ∈ ℝ + 11 : N H ≤ Λ H δ H = N H * , N M ≤ Λ M δ M = N M * }

is positively invariant. Hence no solution path leaves through and boundary of Ω . Since path cannot leave Ω , solution remains non-negative for non negative initial conditions. This means that the solution exists for all positive time t. Therefore the model (1)-(11) is mathematically and epidemiological well-posed [

For convenience and to simplify the analysis of our model, we rewrite the model system (1)-(11) in terms of the proportions of individual in each class. Let

s h = S H N H , e h = E H N H , i h = I H N H , t h = T H N H , r h = R H N H , i h n = I H N N H , z = A N H , l m = L M N H , s m = S M N H , e m = E M N H , i m = I M N H .

Let π = N M N H be the female mosquito?human ratio, that is, the number of female mosquito per human host. The ratio π = N M N H is constant because a

mosquito takes a fixed number of blood meals per unit independent of the population density of the host [

Λ H = Λ h , Λ M = Λ m , β H = β h , δ H = δ h , β M = β m , δ M = δ m , ω H = ω h .

The simplified model now becomes modified human and mosquito population models.

d s h d t = ( 1 − q ) Λ h + α 2 z + ρ r h − β h i m s h − α 1 s h − δ h s h (17)

d e h d t = β h i m s h − g e h − δ h e h (18)

d i h d t = g e h + q Λ h − k 1 i h − k 2 i h − δ h i h (19)

d i h n d t = k 2 i h − ( ω h + δ h ) i h n (20)

d t h d t = k 1 i h − γ t h − δ h t h (21)

d r h d t = γ t h − ( μ + ρ + δ h ) r h (22)

d z d t = α 1 s h + μ r h − α 2 z − δ h z (23)

d l m d t = Λ m − m l m − δ m l m (24)

d s m d t = m l m − β m i h s m − δ m s m (25)

d e m d t = β m i h s m − ϕ e m − δ m e m (26)

d i m d t = ϕ e m − δ m i m (27)

It is necessary to prove that all solutions of system (17)-(27) with positive initial data will remain positive for all times t > 0 . This will be established by the following theorem.

Let the initial data be

{ s h ( 0 ) ≥ 0 , i h ( 0 ) ≥ 0 , i h n ( 0 ) ≥ 0 , t h ( 0 ) ≥ 0 , z ( 0 ) ≥ 0 , r h ( 0 ) ≥ 0 , e h ( 0 ) ≥ 0 , s m ( 0 ) ≥ 0 , l m ( 0 ) ≥ 0 , e m ( 0 ) ≥ 0 , i m ( 0 ) ≥ 0 } ∈ Ω

Then the solution set ( s h , e h , i h , i h n , t h , z , r h , l m , s m , e m , i m ) ( t ) of the model system (4) is positive for all t > 0 .

Proof: From first equation of (17)

d s h d t = ( 1 − q ) Λ h + α 2 z + ρ r h − β h i m s h − α 1 s h − δ h s h ≥ − ( β h i m + α 1 + δ h ) s h

⇒ ∫ 1 s h d ( s h ) ≥ − ∫ ( β h i m + α 1 + δ h ) d t

∴ s h ( t ) ≥ s h ( 0 ) e − ( β h i m + α 1 + δ h ) t ≥ 0

Following the above procedure, from equations (18)-(23), we obtain respectively the positivity conditions;

e h ( t ) ≥ e h ( 0 ) e − ( g + δ h ) t ≥ 0 , i h ( t ) ≥ i h ( 0 ) e − ( k 1 + k 2 + δ h ) t ≥ 0 , i h n ( t ) ≥ i h n ( 0 ) e − ( ω h + δ h ) t ≥ 0 , t h ( t ) ≥ t h ( 0 ) e − ( γ + δ h ) t ≥ 0 , r h ( t ) ≥ r h ( 0 ) e − ( μ + ρ + δ h ) t ≥ 0 , z ( t ) ≥ z ( 0 ) e − ( δ h + α 2 ) t ≥ 0.

Similarly for the modified mosquito population, equations (20)-(27) gives the positivity conditions;

l m ( t ) ≥ l m ( 0 ) e − ( m + δ m ) t ≥ 0 , s m ( t ) ≥ s m ( 0 ) e − ( β m i h + δ m ) t ≥ 0 , e m ( t ) ≥ e m ( 0 ) e − ( ϕ + δ m ) t ≥ 0 , i m ( t ) ≥ i m ( 0 ) e − δ m t ≥ 0.

Let E 0 = ( s h 0 , e h 0 , i h 0 , i h n 0 , t h 0 , z 0 , r h 0 , l m 0 , s m 0 , e m 0 , i m 0 ) be the steady-state of the system (17)-(27) which can be calculated by setting the right hand side of the model (17)-(27) to zero, giving us the following;

( 1 − q ) Λ h + α 2 z + ρ r h − β h i m s h − α 1 s h − δ h s h = 0 (28)

β h i m s h − g e h − δ h e h = 0 (29)

g e h + q Λ h − k 1 i h − k 2 i h − δ h i h = 0 (30)

k 2 i h − ( ω h + δ h ) i h n = 0 (31)

k 1 i h − γ t h − δ h t h = 0 (32)

γ t h − ( μ + ρ + δ h ) r h = 0 (33)

α 1 s h + μ r h − α 2 z − δ h z = 0 (34)

Λ m − m l m − δ m l m = 0 (35)

m l m − β m i h s m − δ m s m = 0 (36)

β m i h s m − ϕ e m − δ m e m = 0 (37)

ϕ e m − δ m i m = 0 (38)

Disease-free equilibrium points (DFE) are steady-state solutions where there is no disease (malaria). The disease free equilibrium of the normalized model (17)-(27) is obtained by setting

d s h d t = d e ˙ h d t = d i h d t = d i h n d t = d t h d t = d r h d t = d z d t = d l m d t = d s m d t = d e m d t = d i m d t = 0

At disease free equilibrium we have,

s h = Λ h δ h , s m = m Λ m δ m ( m + δ m ) , e h = i h = i h n = t h = r h = l m = e m = i m = z = q = 0.

Therefore the disease free equilibrium (DFE) denoted by E 0 of the system (28)-(38) is given by

E 0 = ( s h 0 , e h 0 , i h 0 , i h n 0 , t h 0 , z 0 , r h 0 , l m 0 , s m 0 , e m 0 , i m 0 ) = ( Λ h δ h , 0 , 0 , 0 , 0 , 0 , 0 , 0 , m Λ m δ m ( m + δ m ) , 0 , 0 )

that represents the state in which there is no infection in the society and is known as the disease-free equilibrium point (DFE). This implies that at the disease-free equilibrium, the susceptible human population is equal to the total human population and the susceptible mosquito population is equal to the total mosquito population.

The disease free equilibrium of the model (17)-(27) was given by

E 0 = ( s h 0 , e h 0 , i h 0 , i h n 0 , t h 0 , z 0 , r h 0 , l m 0 , s m 0 , e m 0 , i m 0 ) = ( Λ h δ h , 0 , 0 , 0 , 0 , 0 , 0 , 0 , m Λ m δ m ( m + δ m ) , 0 , 0 )

R_{0} is often found through the study and computation of the eigenvalues of the Jacobian at the disease- or infectious-free equilibrium Diekmann [

The dynamics of the model is specified by the IVP;

d x i d t = f i ( x ) , x ( 0 ) ∈ ℝ + n (39)

We define Θ 0 as the set of all disease-free states as

Θ 0 = { x ∈ ℝ + n : x i = 0 , 1 ≤ i ≤ m } (40)

Next we recast the IVP (4.39) in the form;

d x i d t = F i ( x ) − V i ( x ) (41)

where F i ( x ) is the rate of new infections entering compartment i, and

V i = V i − ( x ) − V i + ( x ) (42)

where V i + ( x ) is the rate of transfer into compartment i by any other means, and V i − ( x ) is the rate of transfer out of compartment i. Given a disease-free equilibrium point x D F E of (39), with x D F E and f ( x ) satisfying certain important assumptions [

F i j = ∂ F i ( x ) ∂ x j | x D F E , V i j = ∂ V i ( x ) ∂ x j | x D F E for 1 ≤ i , j ≤ m (43)

It then follows that F V − 1 is the next generation matrix and the basic reproduction ratio R 0 is the spectral radius of F V − 1 ,

⇒ R 0 = ρ ( F V − 1 ) (44)

Rewriting the system (41) starting with the infected compartments for both populations; e h , i h , e m , i m , i h n , t h and then followed by uninfected classes; s h , z , r h , l m , s m also from the two populations, gives;

d e h d t = β h i m s h − g e h − δ h e h

d i h d t = g e h + q Λ h − k 1 i h − k 2 i h − δ h i h

d e m d t = β m i h s m − ϕ e m − δ m e m

d i m d t = ϕ e m − δ m i m

d i h n d t = k 2 i h − ( ω h + δ h ) i h n

d t h d t = k 1 i h − γ t h − δ h t h

d s h d t = ( 1 − q ) Λ h + α 2 z + ρ r h − β h i m s h − α 1 s h − δ h s h

d z d t = α 1 s h + μ r h − α 2 z − δ h z

d r h d t = γ t h − ( μ + ρ + δ h ) r h

d l m d t = Λ m − m l m − δ m l m

d s m d t = m l m − β m i h s m − δ m s m

The method of next generation matrix has been used to show the rate of appearance of new infection in compartments; e h and e m , from the system (12);

F = ( β h i m s h 0 β m i h s m 0 0 0 ) , V = ( ( g + δ h ) e h − g e h − q Λ h + ( k 1 + k 2 + δ h ) i h ( ϕ + δ m ) e m − ϕ e m + δ m i m − k 2 i h + ( ω h + δ h ) i h n − k 1 i h + ( γ + δ h ) t h )

By linearization approach, the associated matrix at disease free equilibrium is obtained as

F = ( 0 0 0 Λ h β h δ h 0 0 0 0 0 0 0 0 0 m Λ m β m δ m ( m + δ m ) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 )

F V − 1 = ( 0 0 Λ h β h ϕ δ h δ m ( ϕ + δ m ) Λ h β h δ h δ m 0 0 0 0 0 0 0 0 m Λ m β m g δ m ( m + δ m ) ( g + δ h ) ( k 1 + k 2 + δ h ) m Λ m β m δ m ( m + δ m ) k 1 + k 2 + δ h 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 )

∴ R 0 = ( Λ h β h ϕ δ h δ m ( ϕ + δ m ) ) ( m Λ m β m g δ m ( m + δ m ) ( g + δ h ) ( k 1 + k 2 + δ h ) )

Here the term Λ h β h ϕ δ h δ m ( ϕ + δ m ) explains the number of humans that one mosquito infect through contact during the life time it survives as infectious. On the other hand m Λ m β m g δ m ( m + δ m ) ( g + δ h ) ( k 1 + k 2 + δ h ) describes the number of mos-

quitoes that are infected through contacts with the infectious human during infectious period. Hence

R 0 = R 0 m × R 0 h

where R 0 m = Λ h β h ϕ δ h δ m ( ϕ + δ m ) and R 0 h = m Λ m β m g δ m ( m + δ m ) ( g + δ h ) ( k 1 + k 2 + δ h ) .

In this section, we carry out the sensitivity analysis of the model parameter to help us know the parameters that have high impact on the disease transmission, which is on the reproduction ratio R 0 .

We used the normalized forward sensitivity index of a variable to parameter approach used in Okosun [

We compute the sensitivity of R 0 to each of the parameters described in

γ n m = ∂ m ∂ n × n m

where n represents the variables of the model, and m the parameters.

Sensitivity index of ϕ given by − 1 2 ( ϕ ϕ + δ m )

Parameter | β H | k 1 | k 2 | g | δ H | M | β M | δ M | ϕ |
---|---|---|---|---|---|---|---|---|---|

Sensitivity Index | 0.5 | ?0.25 | ?0.156 | ?0.375 | ?0.719 | 0.125 | 0.5 | ?1.352 | ?0.272 |

Sensitivity index of g given by − 1 2 ( g g + δ h )

Sensitivity index of δ m given by 1 2 ( − 2 − δ m ϕ + δ m − δ m m + δ m )

Sensitivity index of δ h given by 1 2 ( − 1 − δ h g + δ h − δ h k 1 + k 2 + δ h )

Sensitivity index of k 1 given by − 1 2 ( k 1 k 1 + k 2 + δ h )

Sensitivity index of k 2 given by − 1 2 ( k 2 k 1 + k 2 + δ h )

Sensitivity index of m given by 1 2 ( 1 − m m + δ m )

Sensitivity index of Λ m = β m = β h = Λ h = 1 2

Remark: Sensitivity indices of R_{0} evaluated at the baseline parameter values are given in the

From _{0} by 5%.

Parameter values:

q : = 0.1 , Λ H : = 0.5 , α 1 : = 0.8 , α 2 : = 0.6 , ρ : = 0.02 , β H : = 0.5 , δ H : = 0.3 , g 1 : = 0.9 k 1 : = 0.8 k 2 : = 0.5 , ω H : = 0.5 , γ : = 0.7 , μ : = 0.4 , N H : = 100 , Λ M : = 0.2 , m 1 : = 0.3 , δ M : = 0.1 , β M : = 0.4 , ϕ : = 0.12

D ( t , Y ) : = [ ( 1 − q ) Λ H + α 2 Y 6 + ρ Y 5 − β H Y 10 Y 0 N H − α 1 Y 0 − δ H Y 0 β H Y 10 Y 0 N H − g 1 Y 1 − δ H Y 1 g 1 Y 1 + q 1 Λ H − k 1 Y 2 − k 2 Y 2 − δ H Y 2 k 2 Y 2 − ( ω H + δ H ) Y 3 k 1 Y 2 − γ Y 4 − δ H Y 4 γ Y 4 − ( μ + ρ + δ H ) Y 5 α 1 Y 0 + μ Y 5 − α 2 Y 6 − δ H Y 6 Λ M − m 1 Y 7 − δ M Y 7 m 1 Y 7 − β M Y 8 Y 2 N H − δ M Y 8 β M Y 8 Y 2 N H − ϕ Y 9 − δ M Y 9 ϕ Y 9 − δ M Y 10 ]

Vector of derivative values at any solution point (t, Y):

Define additional arguments for the ODE solver:

t 0 : = 0 : Initial value of independent variable

t 1 : = 0 : Initial value of independent variable

Y 0 : = [ 50 15 25 2 2 4 2 5 3 2 1 ] T : Vector of initial function values

n u m : = 1 × 10 3 : Number of solution values on [t0, t1]

S 1 : = Rkadapt ( Y 0 , t 0 , t 1 , n u m , D ) : Solution matrix

Human (

t : = S 1 〈 0 〉 : Independent variable values

S H : = S 1 〈 1 〉 : First solution function values

E H : = S 1 〈 2 〉 : Second solution function values

I H : = S 1 〈 3 〉 : Third solution function values

I H N : = S 1 〈 4 〉 : Fourth solution function values

T H : = S 1 〈 5 〉 : Fifth solution function values

R H : = S 1 〈 6 〉 : Sixth solution function values

A H : = S 1 〈 7 〉 : Seventh solution function values

Param | Λ H | Λ M | ρ | β H | α 2 | q | α 1 | k 1 | k 2 | g | γ | δ H | μ | M | β M | δ M | ϕ | ω H |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Value | 0.5 | 0.4 | 0.02 | 0.5 | 0.6 | 0.1 | 0.8 | 0.8 | 0.5 | 0.9 | 0.7 | 0.3 | 0.4 | 0.3 | 0.15 | 0.1 | 0,12 | 0.5 |

0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | |
---|---|---|---|---|---|---|---|---|

0 | 0 | 50 | 15 | 25 | 2 | 2 | 4 | 2 |

1 | 0.01 | 49.469 | 14.824 | 24.737 | 2.108 | 2.178 | 3.99 | 2.394 |

2 | 0.02 | 48.946 | 14.649 | 24.476 | 2.214 | 2.352 | 3.97 | 2.78 |

3 | 0.03 | 48.431 | 14.477 | 24.218 | 2.317 | 2.523 | 3.96 | 3.159 |

4 | 0.04 | 47.924 | 14.307 | 23.963 | 2.419 | 2.689 | 3.95 | 3.53 |

5 | 0.05 | 47.425 | 14.138 | 23.71 | 2.518 | 2.852 | 3.94 | 3.894 |

6 | 0.06 | 46.933 | 13.972 | 23.46 | 2.616 | 3.012 | 3.93 | 4.25 |

7 | 0.07 | 46.449 | 13.808 | 23.212 | 2.711 | 3.167 | 3.93 | 4.6 |

8 | 0.08 | 45.972 | 13.645 | 22.966 | 2.804 | 3.32 | 3.92 | 4.942 |

9 | 0.09 | 45.503 | 13.485 | 22.723 | 2.896 | 3.469 | 3.92 | 5.278 |

10 | 0.1 | 45.041 | 13,326 | 22.483 | 2.985 | 3.614 | 3.91 | 5.607 |

11 | 0.11 | 44.585 | 13.17 | 22.245 | 3.073 | 3.756 | 3.91 | 5.929 |

12 | 0.12 | 44.137 | 13.015 | 22.009 | 3.159 | 3.895 | 3.91 | 6.245 |

13 | 0.13 | 43.695 | 12.862 | 21.776 | 3.242 | 4.03 | 3.91 | 6.554 |

14 | 0.14 | 43.26 | 12.711 | 21.545 | 3.324 | 4.163 | 3.91 | 6.857 |

15 | 0.15 | 42.832 | 12.561 | 21.316 | 3.405 | 4.292 | 3.91 | … |

Mosquitoes

L M : = S 1 〈 8 〉 : Eighth solution function values

S M : = S 1 〈 9 〉 : Ninth solution function values

E M : = S 1 〈 10 〉 : Tenth solution function values

I M : = S 1 〈 11 〉 : Eleventh solution function values

The susceptible human population S H against time (

Despite the availability of drugs, the malaria disease is still endemic in many parts of the world including developed countries. Elimination of malaria requires maintaining the effective reproduction number R_{0} less than unity, as well as achieving low levels of susceptibility. In this research work, we developed a compartmental bio-mathematical model to study the effect of treatment in the control of malaria in a population with infected immigrants. We obtained the basic reproduction number, R_{0} and studied the stability of the disease-free equilibrium of the model. Sensitivity analysis of R_{0} with respect to the model parameters was carried out on the compartmental vector-host malaria model with

eleven compartments. From the literature on modelling of vector-host malaria models, we discovered that many researchers failed to consider protective measures in their models, though some discussed it theoretically. Our major contribution to the existing body of knowledge is incorporating the protective measure in our mathematical model.

The authors declare no conflicts of interest regarding the publication of this paper.

Maliki, O.S., Romanus, N. and Onyemegbulem, B.O. (2018) A Mathematical Modelling of the Effect of Treatment in the Control of Malaria in a Population with Infected Immigrants. Applied Mathematics, 9, 1238-1257. https://doi.org/10.4236/am.2018.911081