EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6129 ISSN 1307-5543 – ejpam.com Published by New York Business Global Optimal Control Strategies Applied to a Mathematical Model of Meningitis Sahar A. Mohamed1, Hasim A. Obaid2,3,∗, Eihab B. M. Bashier4, Kailash C. Patidar 5 1 Department of Physics, Faculty of Science and Technology, Al Neelain Uiversity, Khartoum, Sudan 2 Department of Applied Mathematics, Faculty of Mathematical Sciences and Informatics, University of Khartoum, Khartoum, Sudan 3 Ahmed Bin Mohammed Military College, Doha, Qatar 4Department of Mathematics, Faculty of Education and Arts, University of Sohar, Sohar, Oman 5Department of Mathematics and Applied Mathematics, Faculty of Natural Sciences, University of the Western Cape, Bellville, South Africa Abstract. In this paper, we deal with the problem of optimal control for the transmission dynam- ics of the meningococcal meningitis. The problem is a mathematical model described by a system of nonlinear differential equations. Based on this, two controls are formulated and the resulting system is solved as an optimal control problem. Aiming to minimize the number of illnesses or deaths in the population, we use a control representing a vaccination and another one representing a treatment strategy. We prove that these controls are capable of reducing the number of carriers and infectious individuals. Numerical simulations are carried out to show how to perform the two strategies. 2020 Mathematics Subject Classifications: 49J15, 49K15, 34A12, 34A45 Key Words and Phrases: Mathematical Biology, Meningitis, Optimal Control, Vaccination, Treatment 1. Introduction Meningitis is an inflammation of a membrane, called the meninges, that covers the brain and spinal cord. Meningitis is a severe acute infectious disease caused by several microorganisms, including viruses, bacteria, parasites, and fungi. Fatality rates associated with this disease can be as low as 2% in infants and children, and as high as 20–30% in ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6129 Email addresses: abdeensahar@yahoo.com (S. A. Mohamed), hassem.ab@abmmc.edu.qa (H. A. Obaid), ebashier@su.edu.om (E. B. M. Bashier), kpatidar@uwc.ac.za (K. C. Patidar) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) S. A. Mohamed et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6129 2 of 17 neonates and adults [1]. Damage to the meninges produces a variety of problems, ranging from a high fever with headaches to unconsciousness and death [2]. As mentioned in [3], meningococcal meningitis affects sub-Saharan Africa in a unique and distinctive way. In a region known as the meningitis belt, which spans the continent from Senegal to Ethiopia there is an increase in incidence of meningococcal meningitis every dry season, which dies out when the first rains arrive. At intervals ranging from 6 to 14 years, major epidemics occur, the largest of which killed more than 25 000 people in 10 countries in 1996 –1997. In some areas, this epidemic lasts for more than 1 year, dying out in the rainy season, only to return afterwards. In a single community, however, an epidemic may last only a few weeks. It is well known that mathematical models play a significant role in understanding the dynamics of biological systems. In most cases, these models are described by autonomous systems of nonlinear ordinary/or partial differential equations. Mathematical models for meningitis are explored by many researchers. Below, we mention a few of them. Mart́ınez et al. [4] introduced a novel mathematical model to study the spreading of meningococcal meningitis. The model is a discrete mathematical model based on cellu- lar automata where the population is divided in five classes: susceptible, asymptomatic infected, infected with symptoms, carriers, recovered and died. It catches the individual characteristics of people in order to give a prediction of both the individual behavior, and whole evolution of population. They found that both the global and the individual be- havior of the disease can be predicted and the simulations that they did give results that agree with empirical predictions regarding the basic role played by the carriers. Lawi et al. [5] presented and analyzed a deterministic model for the dynamics of malaria and meningitis co-infection. They established the basic reproduction number Rmm. The analysis showed that the disease free equilibrium of the model may not be globally asymptotically stable whenever Rmm < 1. The Centre Manifold theory was used to show that the model has a unique endemic equilibrium which is locally asymptotically stable when Rmm < 1 and unstable when Rmm > 1. It was deduced further that a reduc- tion in malaria infection cases either through protection or prompt effective treatment, which is dependent on the socio-economic status of a community, would reduce the number of new co-infection cases. In [6], Opoku et al. proposed an epidemiological model to accurately estimate menin- gitis occurrence and transmission in Ghana. They used optimal control theory to identify the ideal strategies for disease eradication and to study the the impact of vaccination and treatment plans. It was deduced from the numerical simulations that high coverage of sus- ceptible individuals receiving vaccinations, combined with the development of an efficient vaccine, is the best strategy for managing the bacteria. Tilahun [7] proposed a deterministic model of pneumonia-meningitis coinfection. Firstly, the qualitative behaviours of the model were studied, and then the basic model was ex- tended to optimal control by incorporating four control interventions, such as prevention of pneumonia as well as meningitis and also treatment of pneumonia and meningitis dis- eases. Their findings showed that each of the listed strategies is effective in minimizing the expansion of pneumonia-only, meningitis-only, and coinfectious population in the specified S. A. Mohamed et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6129 3 of 17 period of time. A mathematical model that governs the epidemiology of cerebrospinal meningitis in Jirapa, Ghana was developed by Wiah and Adetunde [8]. They studied the dynamics of the model where existence of a solution, the stability of equilibria, and bifurcations of the system were investigated. They showed that the epidemic of meningococcal disease was already on the decline when nationwide vaccination campaign was launched. They found that the introduction of the vaccine reduced the incidence of infection to a predicted low of five cases per year, compared with 40 per year with no vaccination. Yano et al. [9] formulated and analyzed a Meningococcal meningitis epidemic model that describes the spreading mechanisms of meningitis in a community with varying pop- ulation. Using Pontryagin’s approach to an optimal problem, their model was extended to include the four control intervention measures: effort to prevent a disease infection, efforts to treat sensitive and resistant strains and immunity control effort. Their optimal control study revealed that the use of these prevention techniques and treatment efforts led to a larger decrease of infections. The above are a few works regarding mathematically modeling the meningitis disease. One can find more studies dealing with different aspects of meningitis, for instance [10– 12]. It worth mentioning that, as can be seen in the above studies, optimal control strategies have not been applied widely in the models describing meningitis. To control the meningitis disease, some models include vaccination as strategies to tackle the evolution of the disease [4, 8, 13]. However, such models did not account for time dependent control strategies, rather fixed parameters values were used. Optimal control strategies have been applied for various mathematical models of epi- demic diseases, for example one can refer to [14–22]. Results obtained in these works as well as in other works are of high interest and can be used to propose and design programs for controlling epidemic diseases. In this paper, we consider two optimal control strategies applied to a meningitis trans- mission model developed in [3]. Aiming to reduce the number of individuals developing meningitis, the first control is associated with vaccination to represent the fraction of individuals who are vaccinated and considered as immune individuals, while the second control is associated with the treatment strategy. The rest of the paper is organized as follows. Section 2 is devoted to the problem statement where the mathematical model along with the set of controls provided. Analysis of the optimal control problem is presented in Section 3. We solve the resulting optimality system numerically in Section 4. A thorough discussion on the results obtained is presented in Section 5. S. A. Mohamed et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6129 4 of 17 2. A model for optimal control of meningitis In this section, we consider the deterministic model presented in [3] where the state system is described by the following nonlinear system of ordinary differential equations dS dt = µN + γI + φR− β(1− u2)S(C + I)− µS − u1S, dC dt = β(1− u2)S(C + I)− (a+ α+ µ)C, dI dt = aC − (ρ+ γ + µ)I, dR dt = ρI + αC − (φ+ µ)R+ u1S, (1) where N = S+C+I+R with initial conditions S(0) ≥ 0, C(0) ≥ 0, I(0) ≥ 0 and R(0) ≥ 0. The host population is divided into susceptible S, an asymptomatic carrier C, ill I, and immune R. The parameters used in the model are defined in [3] as follows. µ : the natural death rate; ρ : the recovery rate; γ : the rate of death from disease; β : the transmission rate; α : the rate of loss of carriage; a : the rate at which carriers become ill; φ : the rate of loss of immunity. To minimize the carrier individuals and, therefore, to minimize the total number of ill individuals, controls on vaccination and treatment are used. These controls are represented as functions of time and assigned reasonable upper and lower bounds as indicated in the subsequent discussion in this paper. The first control function u1(t) is inserted into the model to represent the fraction of individuals who are vaccinated and considered as immune individuals, and the second control function u2(t) is inserted to represent the effectiveness of treatment. Therefore, (1 − u2(t)) represents the fraction of individuals developing the disease and hence increasing the value of u2(t) causes reduction in the number of ill individuals. The functions u1(t) and u2(t) are bounded and Lebesgue integrable functions. We define the objective functional as J(u1, u2) = min u1,u2 ∫ tf 0 ( C + A 2 u21 + B 2 u22 ) dt, (2) where tf is the specified and finite final time and the constants A and B are the balancing cost factors. Our goal is to find two optimal controls, u∗1 and u∗2, such that J(u∗1, u ∗ 2) = min Ψ J(u1, u2), (3) where Ψ = {(u1, u2) ∈ L1(0, tf )|ai ≤ ui ≤ bi, i = 1, 2}, (4) and ai, bi; i = 1, 2, are positive constants representing the bounds on the two controls. S. A. Mohamed et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6129 5 of 17 3. Analysis of the optimal control problem In this section, we use the Pontryagin’s Maximum Principle [23] to derive the necessary conditions that an optimal problem must satisfy. This principle is used to convert (1), (2), and (3) into a problem of minimizing a pointwise Hamiltonian, H, with respect to u1 and u2: H = C + A 2 u21 + B 2 u22 + 4∑ i=1 λigi, (5) where gi and λi, i = 1, ..., 4, are the right hand side of the differential equation and the associated adjoint for the ith state variable of system (1). By applying Pontryagin’s Maximum Principle [23] and existence result for the optimal control pair from [24], we obtain the following theorem. Theorem 1. There exists optimal control pair (u∗1, u ∗ 2) and solution (S∗, C∗, I∗, R∗) for the state system (1) that minimizes J(u1, u2), given by (2), over Ψ. Furthermore, there exists a vector of adjoint functions, (λ1(t), ..., λ4(t)), such that dλ1 dt = (β(1− u2)(C ∗ + I∗) + u1)λ1 − β(1− u2)(C ∗ + I∗)λ2 − u1λ4, dλ2 dt = −1− (µ− β(1− u2)S ∗)λ1 − (β(1− u2)S ∗ − (a+ α+ µ))λ2 − aλ3 − αλ4, dλ3 dt = −(µ+ γ − β(1− u2)S ∗)λ1 − β(1− u2)S ∗λ2 + (ρ+ γ + µ)λ3 − ρλ4, dλ4 dt = (ϕ+ µ)(λ4 − λ1), (6) with transversality conditions λi(tf ) = 0, i = 1, ..., 4, (7) and the following characterization u∗1 = min ( max ( a1, S∗ (λ1 − λ4) A ) , b1 ) , u∗2 = min ( max ( a2, βS∗ (C∗ + I∗) (λ2 − λ1) B ) , b2 ) . (8) Proof. By using a results from [24], we can obtain the existence of an optimal control pair due to the convexity of integrand of J with respect to (u1, u2), a priori boundedness of the state solutions, and the Lipschitz property of the state system with respect to the S. A. Mohamed et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6129 6 of 17 state variables. By applying the Pontryagin’s Maximum Principle [23], we obtain dλ1 dt = −∂H ∂S , λ1(tf ) = 0, dλ2 dt = −∂H ∂C , λ2(tf ) = 0, dλ3 dt = −∂H ∂I , λ3(tf ) = 0, dλ4 dt = −∂H ∂R , λ4(tf ) = 0, (9) evaluated at the optimal control pair (u∗1, u ∗ 2) and the corresponding states S,C, I,R, which results in the stated adjoint system (6). The characterization (8) can be derived by considering the optimality conditions ∂H ∂u1 = 0 and ∂H ∂u2 = 0, (10) and solving for u∗1 and u∗2, subject to the constraints (4). The characterization for u∗1 can be obtained from the first equation of (10) ∂H ∂u1 = Au1 − S∗λ1 + S∗λ4, (11) by considering three cases: ∂H ∂u1 < 0 =⇒ u1(t) = a1 =⇒ a1 < S∗(λ1 − λ4) A , ∂H ∂u1 = 0 =⇒ u1(t) = S∗(λ1 − λ4) A =⇒ a1 ≤ S∗(λ1 − λ4) A ≤ b1, ∂H ∂u1 > 0 =⇒ u1(t) = b1 =⇒ b1 > S∗(λ1 − λ4) A , and hence, u∗1 =  a1 ; S∗(λ1−λ4) A < a1, S∗(λ1−λ4) A ; a1 ≤ S∗(λ1−λ4) A ≤ b1, b1 ; S∗(λ1−λ4) A > b1. (12) Therefore, the characterization of u∗1 in compact form can be written as u∗1 = min ( b1,max ( a1, S∗(λ1 − λ4) A )) . (13) In a similar manner, we can derive the characterization of u∗2 which in compact form as u∗2 = min ( b2,max ( a2, βS∗(C∗ + I∗)(λ2 − λ1) B )) . (14) S. A. Mohamed et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6129 7 of 17 With the existence of the optimal control pair established, we point out that the uniqueness of the optimal control pair follows from the uniqueness of the optimality system the pair must satisfy, which consists of the state system (1) with the its initial conditions, the adjoint system (6) with the transversality conditions (7), and the optimality condition (8). As mentioned in [17], one can obtain the uniqueness of the optimal control for small tf due to a-priori boundedness of the state and adjoint functions and the resulting Lipschitz structure of the ODEs. The finiteness of tf guarantees the uniqueness of the optimality system. This finiteness restriction on the length on the time interval is due to the opposite time orientations of (1), (6), and (7) where the state problem has initial values and the adjoint problem has final values. Next, we discuss the numerical solutions of the optimality system and the correspond- ing results of varying the optimal controls u1 and u2 along with the interpretations for different scenarios. 4. Numerical results In this section, the optimality system consisting of the equations of (1) and (6), is solved iteratively using a fourth order Runge-Kutta method to obtain the optimal strategies regarding the vaccination and treatment. The state system is solved forward in time while the adjoint system is solved backward in time. The two controls are updated at the end of each iteration using (8). The iterations continue until a required convergence is achieved. The readers may note that the main goal of this paper is to design the optimal control strategy and not to find a robust numerical solver. The results obtained by using RK-4 are already convincing and therefore we did not attempt to use variants of RK, e.g, ETDRK or any other methods that we are already familiar with. By using different combinations of the balancing factors A and B, several vaccination and treatment schedules can be derived. Keeping the bounds for the controls fixed, time period is 90 days, β = 0.24658, α = 0.071233, a = 0.00054795, ϕ = 0.00068493 and rest of the parameters unchanged, we illustrate different scenarios by varying the values of the balancing factors. In the simulations below, we use the following initial conditions, as given in [3], S(t0) = 9000, C(t0) = 996, I(t0) = 2, and R(t0) = 2. When A = 10 and B = 10, the two controls are plotted as functions of time in Figure 1. In Figure 2, we can see the effect of this scenario, the number of the susceptible individuals S(t) is dropped very quickly in the first 10 days when using the two controls as the treated individuals return back to the susceptible class and these individuals will be vaccinated and then removed to the recovered class R(t). This explains the increment in the number of the recovered individuals as indicated in the last simulation in the same figure. Similarly, one can see the sharp decrease in the carrier population C(t). Without using any control strategies, we find that the number of carriers C(t) increases for the first 17 days until it reaches its maximum at 3811 individuals and then it declines until ≈ 84 at the end of the interval. However, when we use the control strategies described above, we can see that starting C(t) with 996 individuals, it drops immediately and continue decreasing without fluctuations until it reaches ≈ 2 individuals indicating the effectiveness of the strategy S. A. Mohamed et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6129 8 of 17 used. In the third plot of Figure 2 we notice that starting with I(t) = 2 individuals, the number of population in this class increases until it reaches its maximum ≈ 12 by the 23rd day without incorporating the controls. When end of interval reached, the value of I(t) is ≈ zero. However, when controls are used, it is shown that the number of infectious individuals increases in the first 7 days, but remains below the curve for the infectious without controls, and then it declines until it reaches I(t) ≈ 0 by the end of the time interval. 0 10 20 30 40 50 60 70 80 90 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Time (Days) T h e c o n tr o l u 1 (t )− V a c c in a ti o n 0 10 20 30 40 50 60 70 80 90 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Time (Days) T h e c o n tr o l u 2 − C h e m o p ro h y la x is T re a tm e n t Figure 1: Values of the optimal controls when A = 10 and B = 10. 0 10 20 30 40 50 60 70 80 90 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 Time (Days) S u s c e p ti b le I n d iv id u a ls ( S (t )) With Controls Without Controls 0 10 20 30 40 50 60 70 80 90 0 500 1000 1500 2000 2500 3000 3500 4000 Time (Days) C a rr ie r In d iv id u a ls ( C (t )) With Controls Without Controls 0 10 20 30 40 50 60 70 80 90 0 2 4 6 8 10 12 Time (Days) In fe c ti o u s I n d iv id u a ls ( I( t) ) With Controls Without Controls 0 10 20 30 40 50 60 70 80 90 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Time (Days) R e c o v e re d I n d iv id u a ls ( R (t )) With Controls Without Controls Figure 2: Effect of optimal vaccination and treatment strategies when A = 10 and B = 10. S. A. Mohamed et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6129 9 of 17 Figure 3 shows the different profiles of the two controls u1 and u2 for different values of the balancing factors A and B. As expected, increasing the values of these constants reduces the values of the controls and hence their effect is also reduced, whereas decreasing their values causes the controls last longer at their upper bounds before they decrease again but still at larger scales than before. 0 10 20 30 40 50 60 70 80 90 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Time (Days) T h e c o n tr o l u 1 (t )− V a c c in a ti o n A=1, B=1 A=50, B=50 A=1000, B=5000 A=10000, B=20000 A=50000, B=30000 0 10 20 30 40 50 60 70 80 90 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Time (Days) T h e c o n tr o l u 2 (t )− T re a tm e n t A=1, B=1 A=50, B=50 A=1000, B=5000 A=10000,B=20000 A=50000, B=30000 Figure 3: Plots for controls u1 and u2 for various values of the balancing factors A and B. The effects of these controls on other state variables are shown in Figure 4, where we use the same values of A and B used in Figure 3. As can be seen, the simulations verify the fact that increasing the values of the two controls reduces C(t) and I(t) and causes S(t) to decrease and R(t) to increase because in this case more susceptibles are vaccinated and more infectious are successfully treated. S. A. Mohamed et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6129 10 of 17 0 10 20 30 40 50 60 70 80 90 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 Time (Days) S u s c e p ti b le I n d iv id u a ls ( S (t )) A=1, B=1 A=50, B=50 A=1000, B=5000 A=10000,B=20000 A=50000, B=30000 0 10 20 30 40 50 60 70 80 90 0 200 400 600 800 1000 1200 Time (Days) C a rr ie r In d iv id u a ls ( C (t )) A=1, B=1 A=50, B=50 A=1000, B=5000 A=10000,B=20000 A=50000, B=30000 0 10 20 30 40 50 60 70 80 90 0 0.5 1 1.5 2 2.5 3 3.5 Time (Days) In fe c ti o u s I n d iv id u a ls ( I( t) ) A=1, B=1 A=50, B=50 A=1000, B=5000 A=10000,B=20000 A=50000, B=30000 0 10 20 30 40 50 60 70 80 90 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Time (Days) R e c o v e re d I n d iv id u a ls ( R (t )) A=1, B=1 A=50, B=50 A=1000, B=5000 A=10000,B=20000 A=50000, B=30000 Figure 4: Effect of optimal vaccination and treatment strategies for various values of the balancing factors A and B. Another scenario is considered to compare the effect of applying the vaccination and treatment together, or applying the vaccination only, or applying the treatment only, on the solutions of the state system (1) when the values for the balancing factors A and B are chosen to be 5000. As can be seen in Figure 5, when β = 0.13699 using both controls u1 and u2 together having the same effect as using the first control u1 only. We further notice that inserting both controls u1 and u2 together is better than using the control u2 only. This suggests that the effect of vaccination and treatment applied together on the model is the same as applying the vaccination control only. Finally, the use these controls together is better as compared to the use of treatment only control. Therefore, under the conditions given in these simulations, it is better to consider a strategy that depends on the vaccination only. S. A. Mohamed et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6129 11 of 17 0 10 20 30 40 50 60 70 80 90 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 Time (Days) S u s c e p ti b le I n d iv id u a ls ( S (t )) u 1 >0, u 2 >0 u 1 >0, u 2 =0 u 1 =0, u 2 >0 0 10 20 30 40 50 60 70 80 90 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Time (Days) C a rr ie r In d iv id u a ls ( C (t )) u 1 >0, u 2 >0 u 1 >0, u 2 =0 u 1 =0, u 2 >0 0 10 20 30 40 50 60 70 80 90 0 1 2 3 4 5 6 7 Time (Days) In fe c ti o u s I n d iv id u a ls ( I( t) ) u 1 >0, u 2 >0 u 1 >0, u 2 =0 u 1 =0, u 2 >0 0 10 20 30 40 50 60 70 80 90 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Time (Days) R e c o v e re d I n d iv id u a ls ( R (t )) u 1 >0, u 2 >0 u 1 >0, u 2 =0 u 1 =0, u 2 >0 Figure 5: Effect of optimal vaccination and treatment strategies used together as compared to using only vaccination or treatment strategies. Other parameters are taken as A = 5000, B = 5000 and β = 0.13699. To further corroborate this assertion, we consider three other values of β and repeat the simulations, results of which are indicated in Figure 6 (for β = 0.24658), Figure 7 (for β = 0.35616) and Figure 8 (for β = 0.54795). It is clear that using only the second control u2 does not give better results as compared to using u1 only or using both of u1 and u2. S. A. Mohamed et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6129 12 of 17 0 10 20 30 40 50 60 70 80 90 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 Time (Days) S u s c e p ti b le I n d iv id u a ls ( S (t )) u 1 >0 and u 2 >0 u 1 >0 and u 2 =0 u 1 =0 and u 2 >0 0 10 20 30 40 50 60 70 80 90 0 500 1000 1500 2000 2500 3000 3500 4000 Time (Days) C a rr ie r In d iv id u a ls ( C (t )) u 1 >0 and u 2 >0 u 1 >0 and u 2 =0 u 1 =0 and u 2 >0 0 10 20 30 40 50 60 70 80 90 0 2 4 6 8 10 12 Time (Days) In fe c ti o u s I n d iv id u a ls ( I( t) ) u 1 >0 and u 2 >0 u 1 >0 and u 2 =0 u 1 =0 and u 2 >0 0 10 20 30 40 50 60 70 80 90 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Time (Days) R e c o v e re d I n d iv id u a ls ( R (t )) u 1 >0 and u 2 >0 u 1 >0 and u 2 =0 u 1 =0 and u 2 >0 Figure 6: Effect of optimal vaccination and treatment strategies used together as compared to using only vaccination or treatment strategies. Other parameters are taken as A = 5000, B = 5000 and β = 0.24658. S. A. Mohamed et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6129 13 of 17 0 10 20 30 40 50 60 70 80 90 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 Time (Days) S u s c e p ti b le I n d iv id u a ls ( S (t )) u 1 >0, u 2 >0 u 1 >0, u 2 =0 u 1 =0, u 2 >0 0 10 20 30 40 50 60 70 80 90 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Time (Days) C a rr ie r In d iv id u a ls ( C (t )) u 1 >0, u 2 >0 u 1 >0, u 2 =0 u 1 =0, u 2 >0 0 10 20 30 40 50 60 70 80 90 0 5 10 15 Time (Days) In fe c ti o u s I n d iv id u a ls ( I( t) ) u 1 >0, u 2 >0 u 1 >0, u 2 =0 u 1 =0, u 2 >0 0 10 20 30 40 50 60 70 80 90 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Time (Days) R e c o v e re d I n d iv id u a ls ( R (t )) u 1 >0, u 2 >0 u 1 >0, u 2 =0 u 1 =0, u 2 >0 Figure 7: Effect of optimal vaccination and treatment strategies used together as compared to using only vaccination or treatment strategies. Other parameters are taken as A = 5000, B = 5000 and β = 0.35616. S. A. Mohamed et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6129 14 of 17 0 10 20 30 40 50 60 70 80 90 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 Time (Days) S u s c e p ti b le I n d iv id u a ls ( S (t )) u 1 >0, u 2 >0 u 1 >0, u 2 =0 u 1 =0, u 2 >0 0 10 20 30 40 50 60 70 80 90 0 1000 2000 3000 4000 5000 6000 7000 Time (Days) C a rr ie r In d iv id u a ls ( C (t )) u 1 >0, u 2 >0 u 1 >0, u 2 =0 u 1 =0, u 2 >0 0 10 20 30 40 50 60 70 80 90 0 2 4 6 8 10 12 14 16 18 Time (Days) In fe c ti o u s I n d iv id u a ls ( I( t) ) u 1 >0, u 2 >0 u 1 >0, u 2 =0 u 1 =0, u 2 >0 0 10 20 30 40 50 60 70 80 90 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Time (Days) R e c o v e re d I n d iv id u a ls ( R (t )) u 1 >0, u 2 >0 u 1 >0, u 2 =0 u 1 =0, u 2 >0 Figure 8: Effect of optimal vaccination and treatment strategies used together as compared to using only vaccination or treatment strategies. Other parameters are taken as A = 5000, B = 5000 and β = 0.54795. 5. Conclusion In this paper, we considered two optimal strategies associated with the vaccination and treatment on a mathematical model of meningitis in order to reduce the numbers of carrier and infectious populations. Several scenarios are dealt with. In the first one, we incorporated the two strategies: vaccination and treatment. We have shown that both controls have great impact on reducing the carriers and infectious individuals and hence on increasing the recovered individuals. In the second scenario it is shown that we can use different combinations of the balancing factors to achieve the results required, and all results obtained numerically were satisfactory. Scenario three involved a comparison of application of a single control vs application of both controls. We found that it is better to apply both vaccination and treatment strategies together than applying only the treatment. On the other hand, for the set of parameters and initial values given for the basic model, we have found that using vaccination and treatment have the same effect as using the vaccination only. From the cost effectiveness aspect, this suggests that one can adopt a single strategy, the vaccination, to effectively reduce the populations of carriers and infectious individuals. Currently, we are busy investigating the applicability of numerical methods such as S. A. Mohamed et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6129 15 of 17 Nonstandard Finite Difference Methods to solve the state and adjoint systems as these methods are convenient when dealing with models describing populations because they preserves the positivity of the solution. This is an essential requirement when modeling epidemic diseases. Acknowledgements The authors acknowledge the anonymous referees for their valuable comments and suggestions. Funding H. A. 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