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Global Analysis of Meningitis Disease With Optimal Control

Kwame Kyei Danquah1, Sampson Takyi Appiah1, Baaba A. Danquah1,Bernard Asamoah Afful2,∗ , Godfred Agyemang Safo3
1Department of Mathematics and Statistics, University of Energy and Natural Resources, Ghana

afriyiedanquah1@gmail.com, sampson.appiah@uenr.edu.gh, baaba.ghansah@uenr.edu.gh
2Department of Mathematics and Statistics, Utah State University, Logan, UT, USA

bernard.afful@usu.edu
3English International School of Bratislava, Radničné námestie 4, Bratislava, Slovakia

wisegas98@gmail.com
∗Correspondence: bernard.afful@usu.edu

Abstract. The meningitis epidemic has impacted lives negatively, especially in Sub-Sahara Africa,dubbed the ‘Meningitis Belt’. The epidemic has been a public health concern due to an improperunderstanding of the disease’s dynamics. To implement a control measure that will help minimize theepidemic, we introduce a non-linear Meningitis model that describes the dynamic behaviour of thedisease and explains the transmission trend. The model explores the condition that leads to local orglobal asymptomatic stability of the equilibria. The model is subjected to a sensitivity analysis to findthe parameters that influence the R0. The model is modified into an optimal control by adding time-dependent controls. The control model is solved qualitatively using Pontryagin’s maximum principleand numerically using MATLAB and the fourth-order Runge-Kutta method. We provide a controlstrategy that can be relied on for management decision-making based on the results.

1. Introduction
Meningitis, a deadly bacterial infection, is primarily attributed to Meningococcal meningitis.Meningitis kills over 100,000 individuals each year and affects 1.2 million people from all over theworld. In Africa, especially the Sub-Saharan Africa meningitis belt, which extends from Senegal toEthiopia, 10,000 people are expected to die each year [2]. This disease is widespread across sub-Saharan Africa, stretching from the meningitis belt in Senegal to Ethiopia. The illness reappears atthe start of each dry season and disappears at the start of the rainy season in Africa, a fascinatingpattern that warrants further study. Moreover, from the year 2003 to 2007, about 4100 casesof cerebrospinal Spinal Meningitis(CSM) were confirmed in the United States (CDC, 2017) [3].It is approximated that during most significant epidemics, over 1000 cases of the disease are
Received: 10 Jun 2024.
Key words and phrases. Meningitis Disease; Optimal control; Global stability; Local stability; Sensitivity analysis.1

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Eur. J. Math. Anal. 10.28924/ada/ma.4.21 2reported, typically happening every 5 to 12 years [4]. The bacteria is transmitted from infectedindividuals to susceptible ones through contact with respiratory and throat secretions like salivaand mucus. However, unlike flu and common cold viruses, the bacteria are not highly contagious,and it takes time before transmission occurs. Those at risk of contracting the disease are typicallyindividuals near the infected person, such as household members and roommates [5]. The mostcommon signs of the disease include fever, headache and stiffness of the neck. Diagnosing thedisease is sometimes difficult since the symptoms are often similar to other diseases [6]. Currently,a vaccine for meningitis exists, with the available vaccine primarily for bacteria such as meningitis.Bacterial meningitis is fatal when not diagnosed early [17]. Generally, infected individuals recoverwith permanent disabilities such as hearing loss and brain damage. These disabilities and thedisease itself are worsened when symptoms are not detected on time.Meningitis, a bacterial illness, has been a global concern, affecting numerous parts of the world.The disease has been endemic in several areas, with Sub-Saharan Africa being the hardest hit.Since its emergence, numerous models have been developed to describe the disease’s transmissionpatterns, yet there remains a need for further understanding of intervention strategies to curb thedisease in the Meningitis belt. In [15], and [16], the dynamical behaviour of the Meningitis diseasewas studied; however, the study failed to provide enough intervention and treatment strategies tominimize the disease. Against this background, we propose a non-linear mathematical Meningitismodel that would analyse the model’s stability and characterize a range of feasible control strategiesthat would aid management decision-making to curb the disease. In [31], the transmission behaviourof a meningitis disease is disclosed using an age-structured model that impacts the carriers’ input tothe model dynamics. The transmission behaviour of a Meningitis disease is revealed by building anage-structured model that affects the carriers’ contribution to the model dynamics. [32] formulateda compartmental meningitis model that predicted the behavioural pattern of individuals and thepopulation evolution by studying the dynamic trend of disease transmission. In their study, [33]studied the risk factor of Meningitis in adults by employing fuzzy cognitive maps and multi-criteriatechniques to determine the ranks of the various scenarios. [34], determined the numerical solutionof the Meningitis disease by considering the methods of Euler, Heun, and the fourth-order Runge-Kutta. In [35], the authors created a mathematical model to investigate the impact of sharedinformation on the dynamics of Meningitis disease. The authors in [36] modelled a co-infectionmathematical model of Listeriosis and Meningitis to unveil the parameters that impact the dynamicsof the co-infection model. In [37], the authors formulated a mathematical model of influenza-meningitis co-infection that analysed the infected’s outcome on the model’s dynamics. In the paperby [39], the authors looked at a mathematical model of meningitis that attempted to explain theinfection dynamics of the disease in Jirapa District, Ghana. To better understand the disease’stransmission mechanisms, the researchers in [38] developed a mathematical meningitis model. The

https://doi.org/10.28924/ada/ma.4.21


Eur. J. Math. Anal. 10.28924/ada/ma.4.21 3model was subjected to a thorough stability study, with disease-free equilibrium indicating stabilitywhen R0 ≤ 1 and endemic stability when R0 ≥ 1.Optimal controls are extensively used in dynamical systems, specifically those related to non-linear ordinary differential equations, and perceived as an intervention technique within controltheory [40–43]. Mathematical models that involve optimal control analysis are essential for un-derstanding disease spread and play a vital role in the policy-making process concerning diseasecontrol. The authors in [44] suggested a nonlinear mathematical model to see if public awarenesscampaigns affect the spread of infectious diseases. The model evaluated the population’s responseto media awareness because diseases spread through interaction between infected and suscepti-ble people. Thus, the ability of the susceptible individuals to avoid contact with the infectives.Their analysis showed that infectious disease spread can be controlled by employing an aware-ness program. However, due to human immigration, diseases will always remain endemic. In [44],the authors explored the impact of media coverage on controlling disease spread by formulating amathematical model incorporating media coverage. The analysis indicates that, even though theexistence of media was not the sole factor in the attempt to eradicate the disease, its presence,to some extent, can minimize the number of infections. In [45], the authors attempted to reduceEbola infection in the susceptible by constructing an optimal control theory from ordinary differ-ential equation modelling of the Ebola virus. Two control functions, education and treatments,were considered in modelling the control problem. The control system is solved by applying thetool of Pontryagin’s maximum principle. The analysis of the numerical results showed the controls’overall effect in reducing the disease. Also, the authors in [16] constructed a mathematical modelof syphilis transmission dynamics to aid in selecting the most effective syphilis screening choicesThe model created was an agent-based dynamic model that simulated a critical population of 2,000people. According to the model’s results, increasing the frequency of syphilis screening to everythree months was very effective in reducing syphilis infection cases. In [46], the authors devel-oped a mathematical model of COVID-19. To characterize a range of feasible controls that mightbe effective in minimizing the disease, the model was changed to an optimal control problem. Anumerical simulation of the problem was performed using a forward-backwards sweep and fourth-order Range-Kutta method. In [47], the authors created a mathematical model for the ongoingcoronavirus outbreak to determine intervention approaches to battle it. The model was turned intoan optimal control problem to provide a theoretical explanation for the disease, which was solvedqualitatively by utilizing Pontryagin’s maximal principle. MATLAB and an iterative technique wereused to solve the models numerically.The objective of this work is to design a mathematical model to investigate meningitis transmis-sion, to examine the equilibrium’s local and global stability, to conduct a sensitivity analysis of the

https://doi.org/10.28924/ada/ma.4.21


Eur. J. Math. Anal. 10.28924/ada/ma.4.21 4model parameters to identify the parameters that significantly affect the R0, to formulate a controlmodel for the Meningitis disease, and to perform a numerical simulation for the model.The rest of the research is divided into the following sections: Section 2 focuses on formulating anonlinear model for Meningitis disease. Section 3 explores the qualitative properties of the model,like positivity, solutions’ boundedness, basic reproduction numbers, and the existence of equilibriumalong with the local and global stability of disease-free and endemic equilibria. Section 4 centres onexamining the sensitivity of the model’s parameters on R0 using the normalized forward sensitivityindex. In Section 5, the model is modified by adding time-dependent and solved with Pontryagin’sMaximum Principle. Section 6 tackles computational investigations of the optimal control modelbased on the three control strategies, and the results are illustrated. Then, finally, we provideconclusions and discussions of the work in Section 7.
2. Mathematical model

In the current section, a deterministic model for Meningitis disease that partitions the totalpopulation into SHh, susceptible, EHh, exposed, AHh, asymptomatic, IHh, symptomatic, and RHh,recovered is formulated. The population N is given as N = SHh + EHh + AHh + IHh + RHh. Themodel assumes that people are recruited into the population by birth at the rate Λ. The susceptiblebecome exposed through contact with the symptomatic at rate η1. The exposed leaves at a rate τ1and enters the symptomatic while a fraction k1 enters the asymptomatic. The asymptomatic andsymptomatic die at rates τ3 and ψ2, respectively. The asymptomatic can leave to recovery classdue to natural immunity at rate τ2. The symptomatic enters the recovered compartments at a rate
ψ1. The recovered individuals could return to the susceptible class due to loss of immunity at rate
ω. With all the compartments, natural death occurs at a rate µ.

Figure 1. Schematic of the Meningitis model

https://doi.org/10.28924/ada/ma.4.21


Eur. J. Math. Anal. 10.28924/ada/ma.4.21 5

d

dt
SHh = Λ− µSHh − η1IHhSHh + ωRHh,

d

dt
EHh = η1IHhSHh − k1τ1EHh − (1− k1)τ1EHh − µEHh,

d

dt
AHh = (1− k1)τ1EHh − (τ2 + τ3 + µ)AHh, (1)
d

dt
IHh = k1τ1EHh − (ψ1 + ψ2 + µ)IHh,

d

dt
RHh = τ2AHh + ψ1EHh − (ω + µ)RHh,

with:
SHh0 ≥ 0, EHh0 ≥ 0, AHh0 ≥ 0, IHh0 ≥ 0 and RHh0 ≥ 0. (2)

3. Qualitative properties
3.1. Positivity and Boundedness.

Theorem 3.1. The set {SHh, EHh, AHh, IHh, RHh} being the solution of the state system (1) with
parameters which are non-negatives is positive with the initial condition given by;

{SHh0 ≥ 0, EHh0 ≥ 0, AHh0 ≥ 0, IHh0 ≥ 0, RHh0 ≥ 0} .

Proof. By inspection, the third equation of model (1) can be structured into a first-order differentialequation standard form as:
d

dt
AHh + (τ2 + τ3 + µ)AHh = (1− k1)τ1EHh. (3)

When equation (3) is solved with the integrating factor method, we get
AHh(t) = e−(τ2+τ3+µ)t

[
AHh(0) + (1− k1)τ1

∫ t

0

E(s)e−(τ2+τ3+µ)sds

]
.

The same method, when applied to the fourth equation, gives
IHh(t) = e−(ψ1+ψ2+µ)t

[
IHh(0) + k1τ1

∫ t

0

E(s)e−(ψ1+ψ2+µ)sds

]
.

Hence, we observe that d

dt
AHh ≥ 0 at t0, d

dt
IHh ≥ 0 at t0. Thus, we can generalise that the otherstate variables remain positive at t = 0. Hence, the state model system 1 is positively invariant in

R5+. �

Theorem 3.2. The model equation (1) is bounded within the invariant region, ϑ ∈ R5+ given as;

ϑ =
{

(SHh, EHh, AHh, IHh, RHh) ∈ R5+, SHh + EHh + AHh + IHh + RHh ≤ Λ− µN
}
.

https://doi.org/10.28924/ada/ma.4.21


Eur. J. Math. Anal. 10.28924/ada/ma.4.21 6

Proof. We add the respective compartments to prove the boundedness of model system (1). Thus,we get
N(t) = Λ− µSHh − τ3AHh − µAHh − µEHh − ψ2IHh − µIHh − µRHh,

dN(t)

dt
= Λ− τ3AHh − ψ2IHh − µN. (4)

It follows that from equation (4), setting H to be a solution of (4), we have a unique initial valueproblem, 
d

dt
H1(t) = Λ− µH1(t) t ≥ 0

H1(0) = N(0).
(5)

The solution of (5) gives;
H1(t) = N(0)e−µt +

Λ

µ
(1− e(−µt)). (6)

What happens next is that, from the comparison Theorem in [1], we notice that,
N(t) = N(0)e−µt +

Λ

µ
(1− e(−µt)). (7)

Therefore, from equation (7) the state variables (SHh, EHh, AHh, IHh, RHh) has the possible solutionset which bounded and the model equation (1) is invariant ϑ ∈ R5+. As a result, model (1) ismathematically well-posed and epidemiologically feasible. �

3.2. Existence of Disease-free equilibrium (DFE) point. Model system (1) has a trivial point
(0, 0, 0, 0, 0), which is usually ignored in the model’s analysis. The right-hand side of (1) is set tozero and solved, the disease-free equilibrium becomes

E0 =

(
Λ

µ
, 0, 0, 0, 0

)
. (8)

3.3. Basic reproduction number. The basic reproduction number, R0, is one of the things thatmodellers look for when it comes to infectious disease modelling. The basic reproduction numberis sufficient for determining the condition of the disease. In a completely naive population, thebasic reproduction number is defined as the number of persons one infected person may infect. Itis denoted by R0, and when R0 > 1, it means the disease will spread unless preventive strategiesare cautiously enacted. However, when R0 < 1, the infection dies without strenuous effort. Thederivation of R0 is important in modelling and can be derived by the method of [19]. The (9) is theformulae guaranteeing R0 derivation.
R0 = ρ(FV−1). (9)

The ρ is considered as the largest entry in the derivation of the next generation matrix of R0 =

ρ(FV−1), where F is the coming infection into compartment i and v . Thus, the transfer of indi-viduals out of compartment i by death. Technically, the R0 becomes the largest eigenvalue of the

https://doi.org/10.28924/ada/ma.4.21


Eur. J. Math. Anal. 10.28924/ada/ma.4.21 7matrix resulting from the partial derivative of (9). What happens next is the infected compartmentsof model (1) are given by
d

dt
EHh = η1IHhSHh − k1τ1EHh − (1− k1)τ1EHh − µEHh,

d

dt
AHh = (1− k1)τ1EHh − (τ2 + τ3 + µ)AHh,

d

dt
IHh = k1τ1EHh − (ψ1 + ψ2 + µ)IHh.

We notice from the diseased compartment that,
F =

η1IHhSHh0

0

 , and V =

 k1τ1EHh + (1− k1)τ1EHh + µEHh

−(1− k1)τ1EHh + (τ2 + τ3 + µ)AHh

−k1τ1EHh + (ψ1 + ψ2 + µ)IHh

 . (10)
When F is evaluated at E0, then FE0 becomes;

FE0 =


0 0

η1λ

µ

0 0 0

0 0 0

 . (11)

Evaluate V at E0, which gives;
VE0 =


(
k1τ1 + µ+ (1− k1)τ1

)
0 0

−(1− k1)τ1 (τ2 + τ3 + µ) 0

−k1τ1 0 (ψ1 + ψ2 + µ)

 . (12)
The basic reproduction number of model system (1) is determined by using the method of [19],which gives;

R0 =
η1Λk1τ1

µ(k1τ1 + µ+ (1− k1)τ1)(ψ1 + ψ2 + µ)
. (13)

3.4. Existence of an endemic equilibrium point (EEP). Endemic equilibrium exists when there isa presence of infection. The model (1) has a unique endemic equilibrium given by;
E∗ = (S∗Hh, E

∗
Hh, A

∗
Hh, I

∗
Hh, R

∗
Hh), (14)

https://doi.org/10.28924/ada/ma.4.21


Eur. J. Math. Anal. 10.28924/ada/ma.4.21 8where
S∗Hh =

Λ + ωR∗Hh
µ+ η1I

∗
Hh

,

E∗Hh =
η1I
∗
HhS

∗
Hh

(µ+ τ1)
,

A∗Hh =
(1− k1)τ1E∗Hh
(τ2 + τ3 + µ)

,

I∗Hh =
k1τ1E

∗
Hh

(ψ1 + ψ2 + µ)
,

R∗Hh =
τ2A

∗
Hh + ψ1E

∗
Hh

(ω + µ)
.

3.5. Stability of the disease-free equilibrium point. Here, the global and local stability analysesof the Meningitis model (1) at the disease-free equilibrium are studied. The geometrical approachof Lyapunov function theory by [20] would be used to prove that model (1) is globally asymptoticallystable at the disease-free equilibrium. The results are provided as follows;

J =


−µ− η1IHh 0 0 −η1SHh ω

η1IHh −
(
k1τ1 + µ+ (1− k1)τ1

)
0 η1SHh 0

0 (1− k1)τ1 −(τ2 + τ3 + µ) 0 0

0 k1τ1 0 −(ψ1 + ψ2 + µ) 0

0 0 τ2 ψ1 −(ω + µ)

 .
(15)Evaluating the Jacobian in (15) at the E0 gives;

J =



−µ 0 0 −η1
λ

µ
ω

0 −
(
k1τ1 + µ+ (1− k1)τ1

)
0 η1

Λ

µ
0

0 (1− k1)τ1 −(τ2 + τ3 + µ) 0 0

0 k1τ1 0 −(ψ1 + ψ2 + µ) 0

0 0 τ2 ψ1 −(ω + µ)


.

clearly, λ1 = −µ, λ2 = −(ω + µ), λ3 = −(τ2 + τ3 + µ). The remaining matrix becomes;
Ĵ =

−(k1τ1 + µ+ (1− k1)τ1
)

η1
Λ

µ

k1τ1 −(ψ1 + ψ2 + µ)

 .
The characteristic equation is given by

λ2 + b1λ+ b2 = 0 (16)

https://doi.org/10.28924/ada/ma.4.21


Eur. J. Math. Anal. 10.28924/ada/ma.4.21 9where
b1 = (k1τ1 + µ+ (1− k1)τ1 + (ψ1 + ψ2 + µ),

b2 = (k1τ1 + µ+ (1− k1)τ1(ψ1 + ψ2 + µ)−
Λ

µ
η1k1τ1,

then
λ3,4 =

−(b1)±
√
T1

2
,

where T1 = b21−4b2. If λ3 ≤ 0 and λ4 ≤ 0, then the disease free equilibrium is stable. Otherwise,it is unstable.
Theorem 3.3. When R0 < 1, the disease-free equilibrium E0 for the Meningitis model (1) is
globally asymptotically stable in R5+.

Proof. We construct a Lyapunov function
L = k1

(
τ1
d1d2

)
EHh +

1

d2
IHh,

where d1 = (k1τ1 + µ + (1 − k1)τ1) and d2 = (ψ1 + ψ2 + µ). Taking the derivative of L withrespect to EHh and IHh gives;
dL
dt

= k1

(
τ1
d1d2

)
d

dt
EHh +

1

d2

d

dt
IHh,

dL
dt

= k1

(
τ1
d1d2

)
(η1IHhSHh − k1τ1EHh − (1− k1)τ1EHh − µEHh) +

1

d2
(k1τ1EHh − (ψ1 + ψ2 + µ)IHh) ,

dL
dt

= k1

(
τ1
d1d2

)
(η1IHhSHh − d1EHh) +

1

d2
(kτ1EHh − d2IHh) .

It follows that SHh =
Λ

µ
at t0. Hence

dL
dt

= k1

(
τ1Λ

d1d2µ

)
η1IHh −

k1τ1
d2

EHh +
k1τ1
d2

EHh − IHh,

= (R0 − 1)IHh.

From the model equation (1), the system variables and parameters are all non-negative, implyingthat dL
dt < 0 when R0 < 1, with dL

dt = 0 in the disease-free equilibrium. Hence, L is a Lyapunovfunction in ψ. Hence, from [20] principle, (EHh(t), IHh(t))→ (0, 0) as t →∞. �

3.6. Stability of the endemic equilibrium point. Here, we study the global and local stabilityof the Meningitis model (1) at the endemic equilibrium. The Lyapunov function method by [21]is employed to prove the globally asymptotic stability of model (1) at endemic equilibrium. The

https://doi.org/10.28924/ada/ma.4.21


Eur. J. Math. Anal. 10.28924/ada/ma.4.21 10underlying steps are, therefore, followed. The Jacobian evaluated at E∗ gives;

J =


−µ− η1I∗Hh 0 0 −η1S∗Hh ω

η1I
∗
Hh −

(
k1τ1 + µ+ (1− k1)τ1

)
0 η1S

∗
Hh 0

0 (1− k1)τ1 −(τ2 + τ3 + µ) 0 0

0 k1τ1 0 −(ψ1 + ψ2 + µ) 0

0 0 τ2 ψ1 −(ω + µ)


(17)We denote a = −µ − η1I∗Hh, b = −η1S∗Hh, c = ω, d = η1I

∗
Hh, e = −

(
k1τ1 + µ + (1 − k1)τ1

),
f = η1S

∗
Hh, g = (1 − k1)τ1, h = −(τ2 + τ3 + µ), i = k1τ1, j = −(ψ1 + ψ2 + µ), k = τ2, l = ψ1,and m = −(ω + µ). Then, the characteristics equation of model (1) is given by

Y 5 + A0Y
4 + A1Y

3 + A2Y
2 + A3Y + A4 = 0 (18)

with
A0 = (a + e + h + j +m),

A1 = ae + ah + aj + eh + am + ej +−f i + em + hj + hm + jm,

A2 = aeh + bdi + aej − af i + aem + ahj + ahm + ehj − f hi + ajm + ehm + ejm − f im + hjm,

A3 = ehjm − f him + bdhi + aehj − af hi − cdgk + aehm + bdim − cdi l + aejm − af im + ahjm,

A4 = −cdgjk + bdhim − cdhi l + aehjm − af him.

Based on the Routh-Hurwitz stability by [22], the condition for the characteristics equation (18) isgiven by

Yi =



Y1 Y3 Y5

Y0 Y2 Y4

0 Y1 Y3

0 Y0 Y2

0 0 Y1

0 0 Y0

0 0 0



> 0.

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Eur. J. Math. Anal. 10.28924/ada/ma.4.21 11The condition requires all the coefficients of the characteristics equation (18) to be positive, implyingthat all eigenvalues have negative real parts. If the condition Yi is satisfied, we conclude that theMeningitis model at the endemic equilibrium is stable and otherwise unstable.
Theorem 3.4. When R0 ≥ 1, the endemic equilibrium E∗ of model 1 is stable when SHh = S∗Hh,
EHh = E∗Hh, AHh = A∗Hh, IHh = I∗Hh, and RHh = R∗Hh, otherwise unstable.

Proof. We construct a Lyapunov function
Lp =

(
SHh − S∗Hh − S∗Hh ln

(
SHh
S∗Hh

))
+

(
EHh − E∗Hh − E∗Hh ln

(
EHh
E∗Hh

))
+

(
AHh − A∗Hh − A∗Hh ln

(
AHh
A∗Hh

))
+

(
IHh − I∗Hh − I∗Hh ln

(
IHh
I∗Hh

))
+

(
RHh − R∗Hh − R∗Hh ln

(
RHh
R∗Hh

))
.

The derivative of Lp with respect to t gives;
dLp
dt

=

(
SHh − S∗Hh

SHh

)
dSHh
dt

+

(
EHh − E∗Hh

EHh

)
dEHh
dt

+

(
AHh − A∗Hh

AHh

)
dAHh
dt

+

(
IHh − I∗Hh
IHh

)
dIHh
dt

+

(
RHh − R∗Hh

RHh

)
dRHh
dt

. (19)
Hence, substituting dSHh

dt , dEHhdt , dAHhdt , dIHhdt and dRHh
dt into equation (19) gives;

dLp
dt

=

(
SHh − S∗Hh

SHh

)
(Λ− µSHh − η1IHhSHh + ωRHh)

+

(
EHh − E∗Hh

EHh

)
(η1IHhSHh − kτ1EHh − (1− k1)τ1EHh − µEHh)

+

(
AHh − A∗Hh

AHh

)
((1− k1)τ1EHh − (τ2 + τ3 + µ)AHh)

+

(
IHh − I∗Hh
IHh

)
(kτ1EHh − (ψ1 + ψ2 + µ)IHh)

+

(
RHh − R∗Hh

RHh

)
(τ2AHh + ψ1EHh − (ω + µ)RHh) .

Hence, for SHh = S∗Hh, EHh = E∗Hh, AHh = A∗Hh, IHh = I∗Hh, and RHh = R∗Hh. We have that,
dLp
dt

= Λ− Λ

(
S∗Hh
SHh

)
− µ

(
(SHh − S∗Hh)2

SHh

)
− η1(IHh − I∗Hh)

(
(SHh − S∗Hh)2

SHh

)
+ ω(RHh − R∗Hh)

(
SHh − S∗Hh

SHh

)
+ η1

(
EHh − E∗Hh

EHh

)
(IHh − I∗Hh)(SHh − S∗Hh)

− kτ1
(

(EHh − E∗Hh)2

EHh

)
− (1− k1)τ1

(
(EHh − E∗Hh)2

EHh

)
− µ

(
(EHh − E∗Hh)2

EHh

)

https://doi.org/10.28924/ada/ma.4.21


Eur. J. Math. Anal. 10.28924/ada/ma.4.21 12

+ (1− k1)τ1(EHh − E∗Hh)

(
AHh − A∗Hh

AHh

)
− (τ2 + τ3 + µ)

(
(AHh − A∗Hh)2

AHh

)
+ kτ1(EHh − E∗Hh)

(
IHh − I∗Hh
IHh

)
− (ψ1 + ψ2 + µ)

(
(IHh − I∗Hh)2

IHh

)
+ τ2(AHh − A∗Hh)

(
RHh − R∗Hh

RHh

)
+ ψ1(EHh − E∗Hh)

(
RHh − R∗Hh

RHh

)
− (ω + µ)

(
(RHh − R∗Hh)2

RHh

)
.

We generate the below equation after thorough algebraic manipulations;
dLp
dt

= G1 − G2, (20)
where

G1 = Λ + ω(RHh − R∗Hh)

(
SHh − S∗Hh

SHh

)
+ η1

(
EHh − E∗Hh

EHh

)
(IHh − I∗Hh)(SHh − S∗Hh)

+ (1− k1)τ1(EHh − E∗Hh)

(
AHh − A∗Hh

AHh

)
+ kτ1(EHh − E∗Hh)

(
IHh − I∗Hh
IHh

)
+ τ2(AHh − A∗Hh)

(
RHh − R∗Hh

RHh

)
,

and
G2 = Λ

(
S∗Hh
SHh

)
+ µ

(
(SHh − S∗Hh)2

SHh

)
+ η1(IHh − I∗Hh)

(
(SHh − S∗Hh)2

SHh

)
+ kτ1

(
(EHh − E∗Hh)2

EHh

)
µ

(
(EHh − E∗Hh)2

EHh

)
+ (τ2 + τ3 + µ)

(
(AHh − A∗Hh)2

AHh

)
+ (ψ1 + ψ2 + µ)

(
(IHh − I∗Hh)2

IHh

)
+ (ω + µ)

(
(RHh − R∗Hh)2

RHh

)
.

Hence, dLpdt = 0 when SHh = S∗Hh, EHh = E∗Hh, AHh = A∗Hh, IHh = I∗Hh, and RHh = R∗Hh. It canbe shown that the inequality G1 ≤ G2. Evidently, it can be verified that dLp
dt ≤ 0 when G1 ≤ G2.Hence dLp

dt = 0, when SHh = S∗Hh, EHh = E∗Hh, AHh = A∗Hh,IHh = I∗Hh and RHh = R∗Hh. Thisindicates that the largest compact invariant set is a Singleton. Hence, from [20], E∗ is globallystable. �

4. Sensitivity analysis of R0
Getting the correct estimation of the R0 in infection disease modelling is crucial because ithelps us in the decisions concerning the management of the infection. However, the possibilityof the parameters linked to the R0 to change makes sensitivity analysis an important subject inepidemiology.

https://doi.org/10.28924/ada/ma.4.21


Eur. J. Math. Anal. 10.28924/ada/ma.4.21 13

Definition 4.1. The normalized forward sensitivity index of R0 computed using the formula used
by [23] for a given parameter α1 is

ϑR0α1 =
∂R0
∂α1

α1
R0

. (21)
The parameters with positive indices contribute to the epidemic spreading since they enhancethe R0. The parameters with a negative index, on the other hand, aid in disease control by lowering

R0. From Table 1, Λ, τ1, ψ1, η1, k1, µ, and ψ2 are the parameters which are most sensitive on R0.
Table 1. Model Parameter Sensitivity Indices for the Reproduction Number

Parameter Sensitivity Index
Λ 1.000

τ1 −2.333

ψ1 −0.526

ψ2 −0.473

η1 1.000

µ −1.000

k1 1.000

This is because, any increment in the parameter values of Λ, η1, and k1 will lead to a 100% increasein R0. Also, an increase in µ, τ1, ψ1, and ψ2, will decrease R0 by 100%, 233.3%, 52.6% and 47.3%respectively. Therefore, effective measures must be put in place to decrease Λ, η1, and k1 and toincrease µ, τ1ψ1, and ψ2. Although intervention measures are geared towards increasing and/orincreasing the most significant parameters, it is paramount that the control of the other parametersnot be completely ignored.
5. Optimal control analysis

In this section, model system (1) is modified by putting in three time-dependent controls, viz. per-sonal protection, vaccination and treatment controls, to examine the impact of the control schemeson the Meningitis disease. In model system (1), the associated infection force is lowered by afactor of (1 − u1), where u1 is the personal protection control that ensures the attempt to reduceroom heat and avoid close contact with the infected. The rate of vaccinating susceptible individualsagainst Meningitis is represented by the control function u2. As a result, the model assumes thatvaccinated individuals shift from the susceptible compartment to the removed compartment at anytime. Furthermore, we assume that the control function u3 reflects the rate at which sick patients

https://doi.org/10.28924/ada/ma.4.21


Eur. J. Math. Anal. 10.28924/ada/ma.4.21 14are treated. Hence, the modified nonlinear control system becomes;
dSHh
dt

= Λ− µSHh − (1− u1)η1IHhSHh + ωRHh − u2SHh,

dEHh
dt

= (1− u1)η1IHhSHh − kτ1EHh − (1− k1)τ1EHh − µEHh,

dAHh
dt

= (1− k1)τ1EHh − (τ2 + τ3 + µ)AHh, (22)
dIHh
dt

= kτ1EHh − (ψ1 + ψ2 + u3 + µ)IHh,

dRHh
dt

= τ2AHh + ψ1EHh + u2SHh + u3IHh − (ω + µ)RHh.

To examine the efforts needed to control the disease, we define an optimal functional J thatminimizes the exposed, asymptomatic and symptomatic individuals and maximizes the recoverythrough personal protection, vaccination and treatment controls of u1, u2 and u3. Hence, theobjective functional J is given by;
J (u1, u2, u3) =

∫ tf

0

[
B1EHh + B2AHh + B3IHh +

1

2
(u21d1 + u22d2 + u23d3)

]
dt. (23)

Referring to (23), the quantities B1, B2, and B3 are the weight coefficients of the exposed, asymp-tomatic and symptomatic individuals. In addition, the terms u21d1
2 , u22d22 and u23d3

2 represents the costrelated to minimizing the exposed, asymptomatic and symptomatic individual. The control modelconsiders a quadratic cost on the controls as in other works. We target optimal control u∗1, u∗2, u∗3such that
J (u∗1, u

∗
2, u
∗
3) = min{J (u1, u2, u3) : (u1, u2, u3) ∈ U}, (24)

where
U = {(u1, u2, u3)|0 ≤ ui ≤ 1, i = 1, 2, 3 Lebesgue measurable} (25)

With the method of Pontryagin’s maximum principle [24], system (22) and (23) are transformed intoa problem of Hamiltonian minimization H with respect to the controls u1, u2 and u3 where;
H =

[
B1EHh + B2AHh + B3IHh +

1

2
(u21d1 + u22d2 + u23d3)

]
,

+ λ1 {Λ− µSHh − (1− u1)η1IHhSHh + ωRHh − u2SHh} ,

+ λ2 {(1− u1)η1IHhSHh − kτ1EHh − (1− k1)τ1EHh − µEHh} , (26)
+ λ3 {(1− k1)τ1EHh − (τ2 + τ3 + µ)AHh} ,

+ λ4 {kτ1EHh − (ψ1 + ψ2 + u3 + µ)IHh} ,

+ λ5 {τ2AHh + ψ1EHh + u2SHh + u3IHh − (ω + µ)RHh} .

https://doi.org/10.28924/ada/ma.4.21


Eur. J. Math. Anal. 10.28924/ada/ma.4.21 15

Theorem 5.1. There exists an optimal control U∗ = (u∗1, u
∗
2, u
∗
3) ∈ U such that

J (u∗1, u
∗
2, u
∗
3) = min

U∈U
J (u1, u2, u3), (27)

subject to the control system 22 with the initial conditions.

Proof. The work of [25] would be considered the grounds for proving the existence of optimal control.In minimizing the control problem, the necessary and convexity of the objective functional in u1, u2and u3 are satisfied. The control space U is also convex and closed by definition.The optimal control system is bounded, which verifies the compactness necessary for the optimalcontrol. Also, the integrand in the functional (23) is convex on U . Therefore, we notice that thereexist a constant q > 1, positive numbers u1, u2 and u3 such that,
J(u1, u2, u3) ≥ u1

(
|u1|2 + |u2|2 + |u3|2

) q
2 − u2

. �

Hence, there exists an optimal control. It follows that determining the optimal solution, thePontryagins’s maximum principle by [26] is applied to the Hamiltonian (26) such that given (y , u)is an optimal solution of the optimal control problem, then there exist a non-trivial vector function
λ = (λ1, · · · , λ5) satisfying the below equation;

dy

dt
= −

∂H(t, y , u, λ)

∂λ
,

0 =
∂H(t, y , u, λ)

∂u
, (28)

dλ

dt
=
∂H(t, y , u, λ)

∂y
.

Hence, the necessary condition related to the Hamiltonian (26) is applied.
Theorem 5.2. Given that S∗Hh, E∗Hh, AHh, I∗Hh and R∗Hh are optimal state solutions with associated
optimal control variables (u∗1, u

∗
2, u
∗
3) for the optimal control problem (22) and (23), then there exist

adjoint variables λi for i = 1, . . . , 5, satisfying;
dλ1
dt

= (λ1 − λ2)(1− u1)η1IHh + (λ2 − λ5)u2 + µλ1,

dλ2
dt

= −B1 + (λ2 − λ3)(1− k1)τ1 + (λ2 − λ4)kτ1 + µλ2,

dλ3
dt

= −B2 + (τ3 + µ)λ3 + (λ3 − λ5)τ2, (29)
dλ4
dt

= −B3 + (λ1 − λ2)(1− u1)η1SHh + (λ4 − λ5)ψ1 + (ψ2 + µ)λ4 + (λ4 − λ5)u3,

dλ5
dt

= (λ5 − λ1)ω + µλ5,

with boundary condition;
λi(tf ) = 0, i = 1, 2, . . . , 5, (30)

https://doi.org/10.28924/ada/ma.4.21


Eur. J. Math. Anal. 10.28924/ada/ma.4.21 16

and the optimal control u∗1 , u∗2 and u∗3 are given by
u∗1 = min

{
1,max

{
0,

(
(λ2 − λ1)

η1IHhSHh
d1

)}}
u∗2 = min

{
1,max

{
0, (λ1 − λ5)

SHh
d2

}}
u∗3 = min

{
1,max

{
0, (λ4 − λ5)

IHh
d3

}} (31)
Proof. The adjoint and transversality conditions are derived using the Hamiltonian (26). Thus weequate SHh = S∗Hh, EHh = E∗Hh, AHh = A∗Hh, IHh = I∗Hh and RHh = R∗Hh and differentiating theHamiltonian with respect to SHh, EHh, AHh, IHh and RHh to obtain (29). Further, the equations

∂H
∂u1

= 0,
∂H
∂u2

= 0,
∂H
∂u3

= 0. (32)
are determined on the interior of the control set, and using the optimality conditions and theproperty of the control space u1 and u2, we can determine (22). From (22), we can characterize thecontrol found by solving the optimality system. In solving the optimality system, the transversalityand the characterization of the optimal control (u1, u2, u3) are used. The controls u∗1, u∗2 and u∗3when substituted into the control system (22) gives;

dSHh
dt

= Λ− µSHh −
(

1−min

{
1,max

{
0,

(
(λ2 − λ1)

η1IHhSHh
d1

)}})
η1IHhSHh

+ωRHh − (λ1 − λ5)
SHh
d2

SHh,

dEHh
dt

=

(
1−min

{
1,max

{
0,

(
(λ2 − λ1)

η1IHhSHh
d1

)}})
η1IHhSHh

−kτ1EHh − (1− k1)τ1EHh − µEHh,
dAHh
dt

= (1− k1)τ1EHh − (τ2 + τ3 + µ)AHh,

d

dt
IHh = kτ1EHh − (ψ1 + ψ2 + min

{
1,max

{
0, (λ1 − λ5)

SHh
d2

}}
+ µ)IHh,

d

dt
RHh = τ2AHh + ψ1EHh + min

{
1,max

{
0, (λ1 − λ5)

SHh
d2

}}
SHh

+ min

{
1,max

{
0, (λ4 − λ5)

IHh
d3

}}
IHh − (ω + µ)RHh.

(33)

�

6. Numerical simulation and discussion
The present section focuses on obtaining a numerical solution for the model. Besides the qual-itative analysis that has been carried out, it becomes imperative to find a numerical solution forthe model. Hence, our task here is to derive a numerical solution that solves the without and withcontrol models and evaluates the effectiveness of the considered control strategies. A numericalalgorithm uses a 4th-order Runge-Kutta method and MATLAB to solve the optimality system. Thus,a numerical solution of the control optimality system involves running the adjoint system backwards

https://doi.org/10.28924/ada/ma.4.21


Eur. J. Math. Anal. 10.28924/ada/ma.4.21 17and the state forward in time, with the associated boundary conditions and the controls. The pro-cess involves continuously upgrading the controls and the characterization value until the earlierresults become close to the currently obtained value; then, the algorithm stops, and a solution isobtained. The MATLAB simulation was done with values taken from the published work. Beloware the parameters considered for the simulation.
Table 2. Meningitis Model Parameters

Parameter Description Range Estimated
Value

Reference

Λ Recruitment rate (100− 100000) 1000 [3]
τ1 Modification parameter (0.001− 0.8) 0.3 [3]
τ2 Rate at individuals leaves theasymptomatic class (0.002− 0.3) 0.03 [3]
τ3 Disease induced death rate (0.002− 0.2) 0.2 [3]
ψ1 Rate at which individuals leavethe infected class to the recov-ered class

(0.001− 0.1) 0.02 [27]
ψ2 Disease induced death rate (0.002− 0.1) 0.018 [3]
ω Loss of immunity (0.01− 0.1) 0.084 [48]
η1 Contact rate (0.1− 0.9) 0.5 [28]
µ Rate at which individuals nat-urally leaves the compartment (0.00001− 0.2) 0.0000391 [29]
k1 Rate at which individualsleaves the exposed class (0.01− 0.5) 0.3 [30]

6.1. Strategy A : u3 = 0. Strategy A uses the controls u1 and u2, with u3 set to zero. The graphsof 2a, 2b, 2c and 2d Indicate the exposed, asymptomatic, and symptomatic people, as well as thecontrols. The without-control graph of 2a showed a swift increase of an estimated 5000 in the first
20 days. Moreover, it remained at this level throughout the remaining simulated time. With theasymptomatic non-control graph of 2b, we notice a gentle rise of the graph in the first 20 days to
4300 of the asymptomatic population. The asymptomatic control graph progressed steadily to a new
5000 in 140 days and retained it till the end of the simulation. The symptomatic non-control graphof 2c increased smoothly and moved to the maximum height of 430 of the symptomatic populationin 130 days, which remained until the end of the simulation. Control figures of the exposed,asymptomatic and symptomatic produced results with substantially minimized graphs. From theexposed graph of 2a, the graph increases similarly but could not rise to the level of the without

https://doi.org/10.28924/ada/ma.4.21


Eur. J. Math. Anal. 10.28924/ada/ma.4.21 18control plot. We see that the control graph has been dramatically minimized. The control figures2b of the asymptomatic lie far below the non-control figure. The symptomatic control figure of2c lay slightly above zero and maintained the level throughout the simulated time. Figure 2d isthe control profile plot of strategy A. The plot shows that the personal protection procedure u1remained at the Upper bound throughout the simulation, while the vaccine control u2 remained atthe upper bound until 178 days before dropping to the lower bound.
6.2. Strategy B : u2 = 0. Strategy B sets u2 = 0 and generates the exposed, asymptomatic, andsymptomatic graphs. From the exposed graph of a, we observed that the without control curve swiftlyraised to 5000 when t = 20. It moved gently to about 5100 for the next 20 days and remainedat that level for the remaining time. The asymptomatic without control curve steadily increasedin the early days of 20 to 4500, increased gently for the next 120 days to 5000, and maintainedthe level for the remaining time. The symptomatic without control graph increased smoothly andprogressed to a height of 430 at the final time. The exposed, asymptomatic and symptomatic controlgraphs produced graphs with desired results. Thus, we noticed a completely minimized exposure,asymptomatic and symptomatic, with the control simulations. Figure 3d is the control profile graph.The graph shows that the personal protection and treatment controls remained at the upper bounduntil 100 and 98 when they dropped to the lower bound. The simulated plots of the exposed,asymptomatic and symptomatic confirmed that strategy B is effective.
6.3. Strategy C : u1 = 0. Strategy C uses the control u1 = 0 and the remaining controls forthe simulated. The graphs of 4a, 4b, 4c, and 4d are the exposed, asymptomatic, symptomatic andcontrol profile plots of strategy C. In the early days, the exposure surged swiftly for the withoutcontrol graphs but maintained a stable level after 40 days. The asymptomatic graph moved steeplyin the early days of 20 to 4500, increased further to 5000 for the next 120, and retained the levelfor the remaining days. The symptomatic graph gently increased throughout the simulation andmaintained a steady progression. The exposed control plot showed a swift increase in the graphin the early days and progressed gently for the entire simulated time. The asymptomatic controlgraph smoothly increased in the early days and progressed with the same momentum for the rest ofthe simulation. The symptomatic control curve was noticed to be minimized throughout the entiresimulation. The plot of figure 4d is the optimal control profile of strategy C. The vaccine (u2) andtreatment (u3) controls remained at the upper bound throughout the simulation until 180 days,when they decreased to the lower bound.
6.4. Strategy D : u1 6= 0, u2 6= 0 and u3 6= 0. Strategy D considered the three controls in itssimulation and generated the exposed, asymptomatic and symptomatic plots. Without control, theexposed graph sparked to 2000 at t = 0. The graphs increased steadily to 5000 in 20 days andthen retained it for the rest of the time. The asymptomatic graph also increased early in the first

https://doi.org/10.28924/ada/ma.4.21


Eur. J. Math. Anal. 10.28924/ada/ma.4.21 19

0 20 40 60 80 100 120 140 160 180

Time (days)

0

1000

2000

3000

4000

5000

6000

E
xp

os
ed

u
1
=0, u

2
=0, u

3
=0

u
1

 0, u
2

 0, u
3
=0

(a)

0 20 40 60 80 100 120 140 160 180

Time (days)

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000
A

sy
m

pt
om

at
ic

u
1
=0, u

2
=0, u

3
=0

u
1

 0, u
2

 0, u
3
=0

(b)

0 20 40 60 80 100 120 140 160 180

Time (days)

0

50

100

150

200

250

300

350

400

450

S
ym

pt
om

at
ic

u
1
=0, u

2
=0, u

3
=0

u
1

 0, u
2

 0, u
3
=0

(c)

0 20 40 60 80 100 120 140 160 180

Time (days)

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

C
on

tr
ol

 p
ro

fil
e

u
1

u
2

(d)
Figure 2. plot of phase portraits with u3 = 0

https://doi.org/10.28924/ada/ma.4.21


Eur. J. Math. Anal. 10.28924/ada/ma.4.21 20

0 20 40 60 80 100 120 140 160 180

Time (days)

0

1000

2000

3000

4000

5000

6000

E
xp

os
ed

u
1
=0, u

2
=0, u

3
=0

u
1

 0, u
2
=0, u

3
 0

(a)

0 20 40 60 80 100 120 140 160 180

Time (days)

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

A
sy

m
pt

om
at

ic

u
1
=0, u

2
=0, u

3
=0

u
1

 0, u
2
=0, u

3
 0

(b)

0 20 40 60 80 100 120 140 160 180

Time (days)

0

50

100

150

200

250

300

350

400

450

S
ym

pt
om

at
ic

u
1
=0, u

2
=0, u

3
=0

u
1

 0, u
2
=0, u

3
 0

(c)

0 20 40 60 80 100 120 140 160 180

Time (days)

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

C
on

tr
ol

 p
ro

fil
e

u
1

u
3

(d)
Figure 3. plot of phase portraits with u2 = 0

https://doi.org/10.28924/ada/ma.4.21


Eur. J. Math. Anal. 10.28924/ada/ma.4.21 21

0 20 40 60 80 100 120 140 160 180

Time (days)

0

1000

2000

3000

4000

5000

6000

E
xp

os
ed

u
1
=0, u

2
=0, u

3
=0

u
1
=0, u

2
 0, u

3
 0

(a)

0 20 40 60 80 100 120 140 160 180

Time (days)

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000
A

sy
m

pt
om

at
ic

u
1
=0, u

2
=0, u

3
=0

u
1
=0, u

2
 0, u

3
 0

(b)

0 20 40 60 80 100 120 140 160 180

Time (days)

0

50

100

150

200

250

300

350

400

450

S
ym

pt
om

at
ic

u
1
=0, u

2
=0, u

3
=0

u
1
=0, u

2
 0, u

3
 0

(c)

0 20 40 60 80 100 120 140 160 180

Time (days)

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

C
on

tr
ol

 p
ro

fil
e

u
2

u
3

(d)
Figure 4. plot of phase portraits with u1 = 0

https://doi.org/10.28924/ada/ma.4.21


Eur. J. Math. Anal. 10.28924/ada/ma.4.21 22

20 days to 4500 and then smoothly progressed to 5000 for the next 100 days and maintained thelevel. The symptomatic level gradually rose from 10 in a day (1) to 430 in 180 days. With thecontrol application, witnessed the exposure raised 1000 to 1900 in 180 days. The asymptomaticslightly raised slightly below 500 for the 1180 days. The symptoms could barely move above 10.Plot 4d is the control profile plot of strategy D, We notice from the control plot that the personalprotection, vaccination and treatment controls remained at the upper bound until 180, 178, and177 respectively when they dropped to the lower bound.
7. Discussion and Conclusion

The study considered a non-linear compartmental model of meningitis disease to explain thetransmission dynamics. The compartmental Meningitis model was presented, with compartmentsfor Susceptible(S), Exposed(E), Asymptomatic(A), Symptomatic(I), and Recovered(R). The model’sequilibria and the basic reproduction number were determined. The model’s local stability at thetwo equilibrium points, the disease-free and endemic, was determined by using the linearisationapproach. The global stabilities of the equilibria were investigated by employing the geometricmethod of the Lyapunov function. In addition, a sensitivity analysis was carried out on the R0to determine the parameters that significantly affect the R0. It was seen that the most sensitiveparameters on R0 are Λ, τ1, ψ1, η1, k1, µ, and ψ2.An optimal control model was formulated by adding time-dependent optimal controls. By defin-ing time-dependent policies that might help decrease or eradicate the disease, the model waschanged into an optimal control problem. To discover the optimality conditions of the systems, thecontrol model was solved using Pontryagin’s maximum principle. Several works have been done onmeningitis transmission, but few have considered optimal control. As a result, we formulated theMeningitis model that was modified to optimal control problems to characterise a range of possiblestrategies to help control the disease. Therefore, we considered possible pairing of the controls toexamine their combined effect on the disease.With strategy A, the controls of personal protection and vaccination were considered. Thegraphs of 2a, 2b, 2c, and 2d denote the exposed, asymptomatic and symptomatic individuals andthe control profile. Without control, the graph of 2a showed a swift increase of an estimated 5000in the first 20 days. Moreover, it remained at this level throughout the remaining simulated time.With the asymptomatic non-control graph of 2b, we notice a gentle rise of the graph in the first
20 days to 4300 of the asymptomatic population. The asymptomatic control graph progressedsteadily to a new level of 5000 in 140 days and retained at that level till the end of the simulation.The symptomatic non-control graph of2c increased smoothly and moved to the maximum height of
430 of the symptomatic population in 130 days, which remained until the end of the simulation.Control figures of the exposed, asymptomatic and symptomatic produced results with substantiallyminimized graphs. From the exposed graph of 2c, the graph increases similarly but could not rise

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0 20 40 60 80 100 120 140 160 180

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Figure 5. plot of phase portraits with u1 6= 0,u2 6= 0 and u3 6= 0

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Eur. J. Math. Anal. 10.28924/ada/ma.4.21 24to the level of the without control plot. We see that the control graph has been dramaticallyminimized. The control figures 2b of the asymptomatic lies far below the non-control figure. Thesymptomatic control figure of 2d lay slightly above zero and maintained the level throughout thesimulated time.Strategy B considered personal protection and treatment control. From the exposed graph of2a, we observed that the without control curve swiftly raised to 5000 when t = 20. It movedgently to about 5100 for the next 20 days and remained at that level for the remaining time. Theasymptomatic without control curve steadily increased in the early days of 20 to 4500, increasedgently for the next 120 days to 5000, and maintained the level for the remaining time. Thesymptomatic without control graph increased smoothly and progressed to a height of 430 at the finaltime. The exposed, asymptomatic and symptomatic control graphs produced graphs with desiredresults. Thus, we noticed a completely minimized exposure, asymptomatic and symptomatic, withthe control simulations.Strategy C paired the vaccination and treatment controls. We noticed that the exposure surgedswiftly for the graphs without control in the early days but maintained a stable level after 40 days.The asymptomatic graph moved steeply in the early days of 20 to 4500, increased further to 5000for the next 120, and retained it for the remaining days. The symptomatic graph gently increasedthroughout the simulation and maintained a steady progression. The exposed control plot showeda swift increase in the graph in the early days and progressed gently for the entire simulated time.The asymptomatic control graph climbed gradually in the early days and continued throughout thesimulation. Throughout the simulation, the symptomatic control curve was found to be minimized.Strategy D paired all three controls. Without control, the exposed graph rose to 2000 at t = 0.The graphs increased steadily to 5000 in day 20 and then maintained that level for the rest of thetime. The asymptomatic graph also increased early in the first 20 days to 4500 and then smoothlyprogressed to 5000 for the next 100 days and retained the level. The symptomatic level graduallyrose from 10 in a day (1) to 430 in 180 days. With the control application, we witnessed theexposed raised 1000 to 1900 in 180 days. The asymptomatic slightly raised slightly below 500for the 1180 days. The symptomatic could not move above 10. The simulated results showed thatthe strategies significantly minimise the disease. It can be concluded that the combination of thethree controls or two controls can be employed when it comes to meningitis control. A cost-benefitanalysis of the combinations shown needs to be performed.
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	1. Introduction
	2. Mathematical model
	3. Qualitative properties
	3.1. Positivity and Boundedness
	3.2. Existence of Disease-free equilibrium (DFE) point
	3.3. Basic reproduction number
	3.4. Existence of an endemic equilibrium point (EEP)
	3.5. Stability of the disease-free equilibrium point
	3.6. Stability of the endemic equilibrium point

	4. Sensitivity analysis of 
	5. Optimal control analysis
	6. Numerical simulation and discussion
	6.1. Strategy A : 
	6.2. Strategy B : 
	6.3. Strategy C : 
	6.4. Strategy D : 

	7. Discussion and Conclusion
	References

