MATHEMATICAL MODELING AND STABILITY ANALYSES ON THE TRANSMISSION DYNAMICS OF BACTERIAL MENINGITIS

Bacterial meningitis has posed a serious threat to lives and livelihood of people, especially those in the meningitis belt. This study presents a deterministic compartmental model of the disease based on SusceptibleVaccinated-Carrier-Infected-Treated-Recovered (SVCITR). This transmission process is made up of seven mutually exclusive epidemiological compartments for the transmission dynamics of the disease. The invariant region, positivity of the solutions and stability of the equilibrium points were examined using quantitative analysis. The basic reproduction number, R0 was computed using the next generation matrix approach and this was used as a threshold to establish the local and global stabilities of the model. The simulation results from the numerical simulation of the model demonstrate the effects of the model parameters on each compartment. The results show that getting people vaccinated is crucial to the control of the disease. Furthermore, the sensitivity analysis of R0 was performed in order to determine the effect of each of the model parameters in controlling the disease. Hence, reducing the values of the parameters with negative sensitivity index will help to curtail the spread of the disease.


INTRODUCTION
Meningitis, a disease of the Central Nervous System is an acute inflammation of the three protective membranes covering the brain and spinal cord called the meninges. It is an epidemic-prone disease affecting a substantial proportion of the world's population. The causative agents of meningitis are viruses, bacteria, parasites and fungi but the global burden of disease is seen with bacterial meningitis, WHO [19]. Both viral and bacterial meningitis may have similar symptoms at the onset of infection, but bacteria meningitis usually presents more severe symptoms as time goes on. This is largely due to the infection of the fluid surrounding the meninges. Injuries, cancer, drugs and other infections can also lead to meningitis. It is therefore pertinent to know the particular cause of meningitis to aid in effective treatment.
About 4 out of every 5 cases of bacteria meningitis are caused by Streptococcus pneumoniae, Haemophilus influenzae type B and Neisseria meningitidis (Nm), Amidu et al. [1].
Bacterial meningitis is one of the most dangerous infections due to repeated occurence of the infection and the sequelae of delibitating effects among survivors even after treatment.
Regional outbreaks can occur at any time, though the Meningitis belt stands at a higher risk. The Meningits belt spans from the Atlantic Ocean to the Red Sea -a semi-arid area of sub-Saharan Africa, Woods et al. [21]. The proximity and inflammation of the protective membranes to the brain and spinal cord can make bacterial meningitis very fatal. It can lead to permanent disability, coma, swelling of the brain and even death if not treated immediately. Therefore, the condition is considered as a medical emergency, Martinez et al. [14]. Case fatality rates which is often between 1 to 2 days after the onset of symptoms may be as high as 50-80% when not treated and about 8-15% when treated. Also, about 10-20% of survivors have serious permanent health problems like epilepsy, hearing impairment or mental retardation. In totality, about 10% of all bacterial meningitis results in death, Letsa et al. [12]. The average incubation period for bacterial meningitis is four (4) days, but symptoms may develop over several hours after exposure to the bacteria usually between 2 to 10 days, Asamoah et al. [3].
Most people have a good recovery from bacterial meningitis; however many recover from the acute phase of the disease only to experience some difficulties while trying to get back to their everyday routine, Kaburi et al. [10]. Bacterial meningitis can result in severe health complications such as headaches, decreased appetite, paralysis, irritability, memory problems, stroke, hearing loss, brain damage, kidney failure, seizures and septicemia (body-wide infection and shock). These complications are often permanent. The longer one has the infection without treatment, the greater the risk of these complications. With prompt treatment, even patients with severe meningitis can have good recovery, Nuoh et al. [16].
Bacterial meningitis is preventable due to the availability of effective vaccines against most of the disease causing agents -S. pneumonia, H. influenza type b and N. meningitidis serogroups: A, B, C, W135 and Y. These vaccines are used for prevention, that is routine immunization and in prompt reactive vaccination during outbreaks, McCarthy, Sharyan and Sheikhi Moghaddam [15]. There is also treatment with antibiotics such as benzyl penicillin, ampicillin, ceftriaxone and chloramphenicol, Trestioreanu et al. [17].
Wiah and Adetunde [20]  Martcheva and Crispino-O'Connell [13] used an age-structured mathematical model to study the transmission dynamics of meningococcal infection. The conditions that give rise to the stability of the disease-free steady state and the existence of an endemic state were examined.
The contribution of the carrier class to the transmission of the disease was established from the numerical simulation.
The pattern of the transmission dynamics of meningococcal meningitis was investigated using deterministic compartmental models. The results from the numerical simulation of the model showed that seasonal vibration and temporary immunity were due to the irregular epidemics which often occur in the meningitis belt, Irving, Blyuss, Colijn and Trotter [9]. Martinez et al. [14] presented a novel mathematical model for the transmission of meningococcal meningitis using cellular automata. Their results established that both the individual and global behaviours of the disease could be determined. This result agreed favourably with the empirical predictions.
The dynamics of bacterial meningitis in a given population was presented using time-dependent controls, nonlinear deterministic model. The results indicate that effective contact rate and infectious carriers have a great effect in transmitting the disease. The model was extended as an optimal control problem in order to determine the best strategies for the control of the disease.
The solution of the optimal control problem showed that the best strategies for controlling bacterial meningitis is the combination of vaccination of susceptible population with other interventions, Asamoah et al. [3].
Other researchers such as Karachaliou, Conlan, Preziosi and Trotter [11], Blyuss [4], Yusuf [22], Dukić et al. [6], Agusto and Leite [2] have also presented works on meningitis. Most of the mathematical models developed represent the different types of Bacterial Meningitis. Classes, R 1 (t) and R 2 (t). The Susceptible Class is made up of the individuals who are not yet infected and have also not been vaccinated against the disease. This is generated by the recruitment of individuals at a rate α and by loss of immunity from previous vaccination. The Susceptible class is reduced by natural death, vaccination or by infection through effective contact with infected individuals at the rate The parameter β is the effective transmission probability per contact and the parameter η 1 ≤ 1 is a modification parameter, Agusto and Leite [2].
The Vaccinated Class are the individuals who have taken the vaccine as a form of protection from the disease. This population is increased by vaccination of susceptible individuals. Often, individuals develop immunity within two (2) weeks after taking the Meningitis vaccines and should protect one for three (3) to five (5) years. Since the vaccines confer varying degrees of immunity to its recipients, the vaccinated individuals may become infected, but at a lower rate than the unvaccinated. The vaccinated class is therefore decreased by been exposed to the disease or by vaccine waning and by natural death. We note that all the model parameters are assumed to be non-negative.  (1) Every individual in the studied population who has not been infected is susceptible to the disease.
(2) A vaccinated individual who loses immunity becomes susceptible with no vaccine protection.
(3) The vaccine is only administered to the susceptible population.
(4) There is a natural death rate from each compartment.
(5) Recovered individuals cannot be re-infected. (2) The total population is given as Substituting (2) into (4) yields Integrating both sides, we have where c is the constant of integration.
From Equations (6) and (7), we get This implies N(t) ∈ 0, α µ . Therefore, the feasible set of solution of the model equations enter and remain in the region: We note that α µ is the upper bound of N(t). However, if N > α µ then N(t) will decrease to α µ and the solutions (S,V,C, I, T, R 1 , R 2 ) will enter Ω or approach it asymptotically, as such, the region will attract all solutions in ℜ 7 + . Hence, the model is well posed mathematically and epidemiologically since the region Ω is positively invariant and attracting.

Proof.
Considering the first equation of the model where f (t) = (λ + θ + µ)dt and c is the constant of integration.
Similarly, considering the second equation of the model Applying the same technique to the remaining equations of the system, the third equation yields The fourth equation yields The fifth equation yields The sixth equation yields Lastly, the seventh equation yields This completes the proof of the theorem.

EXISTENCE OF EQUILIBRIA AND BASIC REPRODUCTION NUMBER
Equating the system of equations in (2) to 0 and substituting C = I = T = R 1 = R 2 = 0, we obtain the system Solving the system simultaneously, the DFE is obtained as: , 0, 0, 0, 0, 0 where, b 1 = ω + µ.

Endemic Equilibrium Point.
The EEP of the model is defined as This yields the system of equations which results in, , , where, From the force of infection in Equation (1), which can be written as (20) λ * N * − β (η 1 C * + I * ) = 0 Substituting all the state solutions into Equation (20) and simplifying leads to the equation where,

The Basic Reproduction Number. The basic reproduction number is a fundamental
threshold in mathematical study of epidemiology. It helps to forecast the transmission potential of a disease. The basic reproduction number associated with (2) is given as: where, The matrices F and V are obtained as follows: Thus, the next generation matrix: The eigenvalues of the matrix, G are Consequently, the Basic Reproduction Number, which is the spectral radius of G is given as R 0 provides the expected number of newly infected individuals that would arise from introduction of a single case of bacterial meningitis into a completely susceptible population.

Proof.
The Jacobian matrix, J evaluated at E 0 is given as The eigenvalues of the Jacobian matrix, J are −µ (of multiplicity 2), (of multiplicity 2). Clearly, all the eigenvalues of the Jacobian matrix are strictly negative provided Thus for stability, the negativity condition imposed yields Therefore, for Equation (28) to be valid, R 0 must be less than 1. Hence the DFE is LAS.

Global
Stability of the Disease-free Equilibrium. The global asymptotic stability of the model in (2) is investigated by following Castillo-Chavez, Feng and Huang [5]. The model is denoted by: where X = (S,V, R 1 , R 2 ) denotes the uninfected population and Y = (C, I, T ) denotes the infected population.
Theorem 5.3. The Disease-Free Equilibrium is said to be globally asymptotically stable in Ω if R 0 < 1 and the following two conditions hold: C1: For dX dt = F(X, 0), E 0 is globally asymptotically stable. 0)] is the Jacobian of G(X,Y ) obtained with respect to (C, I, T ) and evaluated at (X * , 0).

Proof.
C1: From the model, it follows that: , 0, 0, 0, 0, 0 This can be verified using the method of integrating factors. From Equation (30), we have: which can be written in standard from as The integrating factor is given as I.F. = e b 1 dt = e b 1 t .
Multiplying Equation (32) through by the integrating factor yields Let I = Se b 1 t dt. Integrating by parts, we have So, Therefore, Furthermore, from Equation (30), we have, Therefore, Equation (41) can be put in standard form as The integrating factor is given as Multiplying Equation (42) through by the integrating factor gives where c is the constant of integration. Therefore, From Equation (46), S → αb 1 µ(b 1 +θ ) as t → ∞; and this implies the global convergence of Equation (30) in Ω.
C2: G(X,Y ) is given as where λ is the force of infection defined in Equation (1).

MODEL PARAMETER ESTIMATION
6.1. Initial Conditions. Ghana's demographic data for the year 2017 is adopted for our simulation. Since the disease is endemic in the northern part of Ghana, the total population of the northern part as at 2017 was 4953293, GSS [8], as such the initial total population, N(0) = 4953293. It is known that 10 − 20% of every population is carrier of Meningitis, WHO [18], so the average which is 15% is adopted as the case in Ghana. This gives a carrier population of about 742993.95. 10% of the population is assumed to be vaccinated against the disease.   We note that the set of parameter values in Table (3) yields a basic reproduction number less than unity (R 0 = 0.091) which implies that with effective vaccination and treatment, this disease which is considered to be endemic could be eradicated.

SENSITIVITY ANALYSIS
In modeling infectious diseases, it is pertinent to ascertain the major model parameters influencing the disease's transmission. Sensitivity analysis is therefore performed to determine the model's robustness predictions to parameter values.
Definition 7.1. The normalized forward sensitivity index of R 0 , that depends differentiably on a parameter ψ, is defined by In particular, the sensitivity index is a local estimate to establishing an efficient way of reducing Therefore, all the partially differentiable model parameters with respect to R 0 , their values and sensitivity indices are given in Table 4. A positive sensitivity index suggests that the parameter is directly proportional to the value of R 0 . Thus, an increase in any of the values of ω, β , η 1 and σ by some percentage will increase the value of R 0 , thereby increasing the spread of the disease, and vice versa. However, the parameters with a negative sensitivity index means that these parameters are inversely proportional to the value of R 0 . Therefore, when the value of any of these parameters, δ , µ, κ, τ, r, θ is increased while holding all other parameters constant, it will reduce the value of R 0 and, hence, contribute to the eradication of the disease and vice versa. For instance, increasing the modification parameter of the infectiousness of the carrier, η 1 by 10% will lead to a 7.768% increase on R 0 while increasing the treatment rate, κ by 10% will result in a reduction of 5.410% on R 0 .

NUMERICAL RESULTS AND DISCUSSIONS
The numerical solutions of the model (2) is obtained by using MATLAB ODE45 Algorithm for solving non-stiff system of ordinary differential equations with the initial conditions and parameter values estimated.   figure (5(a)) and infected population in figure (5(b)). Also, the higher the treatment rate, the more people get fully recovered in figure (5(c)) and the less people recover with complications as seen in figure (5(d)).

CONCLUSION
The transmission dynamics of bacterial meningitis with a focus on vaccination and effective treatment in curtailing the spread of the disease is presented. The basic reproduction number of the model is computed using the Next Generation matrix. The equilibrium solutions of the model are obtained and used to establish criteria for the model's stability. Using the basic reproduction number, R 0 , as a threshold given R 0 < 1, the disease-free equilibrium point is established to be both locally and globally asymptotically stable. This study relates to the fact that Bacterial meningitis is a vaccine preventable disease. This is because as the vaccine uptake rate increases, the vaccinated population increases and remains stable. The numerical simulations established that the disease can be eradicated with effective and efficient vaccination and treatment since that led the basic reproduction number below unity. The contributions of the model parameters on R 0 using the normalized sensitivity index was examined. The results indicate that the transmission probability, β is an effective contributor to R 0 , as such very essential in the spread and control of the disease. Therefore, control mechanisms that can reduce the transmission probability significantly will most definitely curtail the endemicity of the disease.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.