Abstract
Vaccination and treatment are the most effective ways of controlling the transmission of most infectious diseases. While vaccination helps susceptible individuals to build either a long-term immunity or short-term immunity, treatment reduces the number of disease-induced deaths and the number of infectious individuals in a community/nation. In this paper, a nonlinear deterministic model with time-dependent controls has been proposed to describe the dynamics of bacterial meningitis in a population. The model is shown to exhibit a unique globally asymptotically stable disease-free equilibrium , when the effective reproduction number , and a globally asymptotically stable endemic equilibrium , when ; and it exhibits a transcritical bifurcation at . Carriers have been shown (by Tornado plot) to have a higher chance of spreading the infection than those with clinical symptoms who will sometimes be bound to bed during the acute phase of the infection. In order to find the best strategy for minimizing the number of carriers and ill individuals and the cost of control implementation, an optimal control problem is set up by defining a Lagrangian function to be minimized subject to the proposed model. Numerical simulation of the optimal problem demonstrates that the best strategy to control bacterial meningitis is to combine vaccination with other interventions (such as treatment and public health education). Additionally, this research suggests that stakeholders should press hard for the production of existing/new vaccines and antibiotics and their disbursement to areas that are most affected by bacterial meningitis, especially Sub-Saharan Africa; furthermore, individuals who live in communities where the environment is relatively warm (hot/moisture) are advised to go for vaccination against bacterial meningitis.
1. Introduction
Meningitis is an inflammation of the meninges which are membranes that surround the spinal cord and the brain [1]. It is often caused by viruses, bacteria, and protozoa. Bacterial meningitis is common in children and young adults. This disease mostly spreads in communities/societies that live in close quarters (e.g., police staff, police cells, college students, military staff, and prisons) [2]. Bacterial meningitis is generally caused by germs such as Listeria monocytogenes, Streptococcus pneumoniae, Group B Streptococcus, Neisseria meningitidis, and Haemophilus influenzae, which spreads from one person to another [3]. This infection varies by age groups: Group B Streptococcus, Streptococcus pneumoniae, Listeria monocytogenes, and Escherichia coli are mostly found in newborn babies; Streptococcus pneumoniae, Neisseria meningitidis, Haemophilus influenzae type b (Hib), and Group B Streptococcus are common in babies and children; Neisseria meningitidis and Streptococcus pneumoniae are predominant in teens and young adults; and Streptococcus pneumoniae, Neisseria meningitidis, Haemophilus influenzae type b (Hib), Group B Streptococcus, and Listeria monocytogenes are commonly found in older adults [3]. Bacterial meningitis is characterized by intense headache and fever, vomiting, sensitivity to light, and stiff neck, which result in convulsion, delirium, and death.
It is estimated that meningococcal meningitis causes over 10,000 deaths annually in Sub-Saharan Africa [4]. About 4,100 cases of bacterial meningitis occurred between 2003 and 2007 in the United States [3, 5]. Between 5% to 40% of children and 20% to 50% of adults with this condition die [6]. Infections from bacterial meningitis can cause permanent disabilities such as brain damage, hearing loss, and learning disabilities [3]. The illness of bacterial meningitis becomes worse when symptoms are not detected early enough; even with proper treatment, the individual could die [2].
Prevention of bacterial meningitis can be achieved through vaccination and/or preventing contact with infectious individuals. Vaccination is the most effective way of protecting children against certain types of bacterial meningitis [3]. Vaccines that can prevent meningitis include Haemophilus influenza type B (Hib), pneumococcal conjugate, and meningococcal vaccine [6, 7]. The conjugate meningitis A vaccine, MenAfrivac, is recommended to protect people in Sub-Saharan Africa against the most common type, serotype A [8]. In the United States, the primary means of preventing meningococcal meningitis is antimicrobial chemoprophylaxis [9]. Empirical therapy includes ceftriaxone or cefotaxime and vancomycin for Streptococcus pneumoniae [2]. There is a vaccine against meningococcal disease which is 85%–100% effective in preventing four kinds of bacteria (serogroups A, C, Y, and W-135) that cause about 70% of the disease in the United States [2].
Trotter and Ramsay [10] outlined some recommendations on the use of conjugate vaccines in Europe based on the experience with meningococcal C conjugate (MCC) vaccines. In areas with limited health infrastructure and resources, there are a number of antibiotics including penicillin, ampicillin, and chloramphenicol that can be used to treat the infection meningitis.
Mathematical models have been shown to help increase the understanding of the spread and control of infectious diseases. Martínez et al. [2] studied the spread of meningococcal meningitis with the use of a discrete mathematical model, based on cellular automata where the population was divided into five classes: susceptible, asymptomatic infected, infected with symptoms, carriers, recovered, and died classes. Broutin et al. [11] studied the dynamics of meningococcal meningitis in nine African countries by adopting some mathematical tools to time series analysis and wavelet method, the results of their studies suggest that “international cooperation in Public Health and cross disciplines studies are highly recommended to help in controlling this infectious disease.” Miller and Shahab [12] studied the cost effectiveness of immunisation strategies for the control of epidemic meningococcal meningitis. The research work in [13] gives a detailed description of the use of antibiotics for the prevention and treatment of meningitis infection. Irving et al. [14] used deterministic compartmental models to investigate how well simple model structures with seasonal forcing were able to qualitatively capture the patterns of meningitis infection. They demonstrated that the complex and irregular timing of epidemics could be caused by the interaction of temporary immunity conferred by carriage of the bacteria together with seasonal changes in the transmissibility of infection. Actually, there have been a significant number of studies of various types of Meningitis in Africa and Europe without the use of optimal control analysis (see [15–28]).
It is obvious that mathematical modelling has become crucial in investigating the epidemiological behaviour of meningitis. Furthermore, mathematical modelling helps to identify the risk factors for diseases, so as to find out why everyone does not have the same infection uniformly [29].
The application of optimal control in disease modelling gives valuable information on how to apply control measures. Through vaccination, treatment, public education, and so forth, many infectious diseases have been controlled [29]. Since the introduction of optimal control theory in disease modelling, there have been a considerable number of studies of infectious diseases using optimal control analysis (see [30–41]). With the significant influence of optimal control theory in disease modelling, this paper presents an optimal control model for bacterial meningitis in the presence of vaccination and treatment due to public health education. The model is qualitatively analyzed and numerically simulated in order to help give policy direction on how to control the spread of the disease.
The rest of the paper is organized as follows. Section 2 presents the model formulation and analysis. Section 3 presents the analysis of the optimal control problem, leading to the existence and characterization of the control measures. Section 4 contains the numerical simulations and discussion. Section 5 presents the conclusion of the study.
2. Model Formulation and Analysis
Adopting the epidemiological studies of a meningitis model as presented in [4], we consider four mutually exclusive compartments to indicate individuals with unique natures (i.e., susceptibles, , carriers, , ill individuals, , and recovered individuals, ) in relation to the disease. It is assumed that the susceptible compartment, , is populated through recruitment at the rate, (thus migration and/or birth rate), and is the rate of effective contact of carriers and/or infected (ill) individuals in the susceptible population. The carrier compartment consists of individuals that have the infection and do not show any clinical symptoms but contribute to the spread of the disease. When a susceptible individual is exposed to this infection, that individual can harbour the bacterium for weeks or even months [42]; but, in a normal circumstance, an individual develops symptoms of the infection within 3 to 7 days after exposure [3]. Carriers are assumed to develop clinical symptoms (i.e., move to ill individuals compartment, ) at rate . Ill individuals who are seriously infected are assumed to have no natural recovery except when given treatment on time. From an epidemiological perspective, individuals in the removed/recovered compartment, , do not attain permanent immunity. After vaccination, immunity develops within 7–10 days and remains effective for approximately 3–5 years [2]. Therefore, it is assumed that the immunity acquired from developing the diseases or carrying the bacteria or through vaccination is of the same intensity (they all lead to the recovered compartment from which people return to the susceptible compartment at a given unique rate ). Research indicates that carriers may recover naturally from the infection without treatment, and we denote such natural recovery rate as . Infected individuals are assumed to die from disease at rate . Since natural death is inevitable, is assumed to be the natural death rate of individuals in all the compartments. Vaccination and treatment due to public health education have been shown to be strategies of control of diseases. Therefore we introduced this two control measures in the model as and , respectively, here ; thus vaccination is comprised of both reactive vaccination and preventative vaccination. The effectiveness of both control measures in minimizing the disease is denoted by and , respectively. If , it signifies that vaccination and treatment have no effect on the model; if , it also signifies that vaccination and treatment are perfectly effective (i.e., 100% effectiveness) (see [43]). If , it signifies that both vaccine and treatment are imperfect [44]. In view of this, it is assumed that administering treatment to the ill individuals leads to recovery at rate . It is also assumed that the vaccinated individuals develop partial immunity at rate . From an epidemiological perspective treatment is not given to carriers in real life (since we do not know who carries the bacteria or not), but it is assumed that carriers of meningitis are just like people with the HIV virus who do not know their status unless they go for medical test; hence this paper seeks to encourage individuals to go for regular test of this bacterial disease; therefore it is assumed that a certain portion of carriers could be treated before any symptoms of the infection show up, which results in as shown in Figure 1. The vaccine is assumed to be imperfect and thus has a failure rate of . Therefore the force of new infections is given byEquation (1) is often referred to as standard incidence rate of new infections, which is normalized by the total population . Table 1 gives a full description of parameters used in the model.
The set of differential equations and flow diagram corresponding to the bacterial meningitis dynamics and disease pathway with control terms is given in system (2) and Figure 1.Adding the equations in model (2) gives the rate of change of total population asLet System (2) is well-posed with all solutions in remaining in if initial conditions are positive. It can easily be shown that if the initial conditions start outside the solutions tend to .
2.1. Equilibrium Points
To obtain the equilibrium points of system (2), we assume that the control measures are time-independent (see [45] for similar analysis).
2.1.1. Disease-Free Equilibrium
To obtain the disease-free equilibrium, , , and the right-hand-side of system (2) are set to zero. If susceptible individuals are assumed to receive vaccination against the disease at a constant rate, then the disease-free equilibrium will be given by
2.1.2. Effective Reproduction Number
Using the next generation matrix method [46], the effective reproduction number of the bacterial meningitis model with vaccination and treatment is obtained aswhere
To determine how the two control measures impact on the reproduction number, we make a plot of , on the plane for arbitrary constant values of model parameters in Figure 2. From the figure, it is shown that decreases with increasing . So an increase in vaccination levels decreases the need for treatment. Figure 2 also shows the region in which the vaccination and treatment values should lie for the disease control.
2.1.3. Endemic Equilibrium
System (2) can be shown to have a unique endemic equilibrium of the form , where with and
Remark 1. (i) If , then system (2) will have only one equilibrium: the disease-free equilibrium.
(ii) If , then system (2) will have two equilibria: the disease-free equilibrium, , and the endemic equilibrium, .
(iii) The case is a critical threshold point where the disease-free equilibrium loses its local asymptotic stability. Thus gives the idea of transcritical bifurcation where the stability of system (2) moves between and [37].
2.2. Stability Analysis
In analyzing the local stability of the disease-free equilibrium, the Routh-Hurwitz criteria are used, and, for the global stability of the two equilibria, the direct Lyapunov technique is employed.
2.2.1. Local Stability Analysis of
Theorem 2. The disease-free equilibrium of system (2) is locally asymptotically stable if and unstable if .
Proof. Evaluating the Jacobian matrix of system (2) at the disease-free equilibrium gives The characteristic polynomial of the Jacobian matrix is given bywhere The Routh-Hurwitz conditions [46] that guarantee that the eigenvalues of the characteristic polynomial in (10) have negative real parts are given by These conditions are easily seen to be satisfied when . Thus, the disease-free equilibrium of system (2) is locally asymptotically stable when and unstable when . This completes the proof.
2.2.2. Global Stability of
Theorem 3. The disease-free equilibrium of system (2) is globally asymptotically stable if and unstable if .
Proof. Let , with positive constants, and , be a Lyapunov function defined as Taking the time derivative of the Lyapunov function we obtain Substituting , , , and in (2) into (14) gives Since this implies that Equating the coefficient of in (17) to zero gives Choosing and and plugging and into (17), we have Additionally if and only if . Hence, the largest compact invariant set in is the singleton set . Therefore, from LaSalle’s invariance principle, we conclude that is globally asymptotically stable in if [37, 49].
2.2.3. Global Stability of
Theorem 4. The endemic equilibrium of system (2) is globally asymptotically stable whenever .
Proof. Suppose , and then the existence of the endemic equilibrium point is assured. Using the common quadratic Lyapunov function as illustrated in [50], we consider the following candidate Lyapunov function: The time derivative of in (21) is given by Plugging the equations in system (2) into (22) yields Now setting we have Further simplification gives It has therefore been shown that is negative, and additionally at (i.e., if , , , and ), . It follows from LaSalle’s invariant principle [51] that all solutions of system (2) approach as if . Therefore, the endemic equilibrium is globally asymptotically stable in whenever [37, 49]. This completes the proof.
2.3. Sensitivity Analysis
Sensitivity analysis is used to determine the response of a model to variations in its parameter values. In the present case, the focus is given to determining how changes in the model parameters impact the effective reproduction number. This is done through the normalized forward-sensitivity index. We also use the Latin hypercube sampling and the partial rank correlation coefficients (PRCC) to plot scatter diagrams and Tornado plots to determine the relative importance of the parameters in for the disease transmission and prevalence (see also [52]).
Definition 5. The normalized forward-sensitivity index of to any parameter, say , as given in [46] can be defined as
The sensitivity indexes of with respect to its parameters are computed as follows:
Similarly, we can compute the sensitivity indexes of with respect to the remaining parameters in , in the same manner. Using the parameter values , , , , , , , , and , with and , the sensitivity indexes of are shown in Table 2.
The corresponding Tornado plots based on a random sample of points for the twelve parameters in are shown in Figure 3(a). The positive values in Table 2 show a promotion of the propagation of the disease. Therefore an increase in the values of , , , , and will have an increase in the spread of the disease. For example, indicates that increasing the effective contact rate by 10% increases the number of secondary infections by 10%. The negative values in Table 2 indicate a reduction in the effective reproduction number if the values of the corresponding parameters are increased. Thus, a reduction in the values of vaccination , treatment , and natural recovery will lead to an increase in the number of secondary infections in the population.
(a) The Tornado plots for the elven parameters in
(b) The effect of effective contact rate on
(c) The effect of vaccination on
(d) The effect of treatment on
Figure 3(a) shows the Tornado plots for the twelve parameters in . It can be seen that, in controlling the spread of bacterial meningitis in a population, more susceptible individuals should be given vaccination. Figure 3(a) also suggests that carriers are likely to have more contacts with the susceptible population than the ill individuals who will typically be bound to their beds during the acute phase of the disease. Therefore, the probability of ill individuals transmitting the infections to susceptibles may be lower than that of carriers who are able to mix well with others within the population. Figures 3(b), 3(c), and 3(d) show the regression plots of effective contact rate (), vaccination rate (), and treatment rate (, respectively. Figure 3(b) shows that transmission rate has a positive correlation in the spread of bacterial meningitis. Figure 3(c) shows that vaccination has a negative correlation in the spread of bacterial meningitis and hence vaccination increases the immunity of individuals against the meningitis infection, thereby reducing the spread of the infection. Figure 3(d) shows that individuals who receive treatment after being infected with bacterial meningitis have a higher chance of recovery and that reduces the spread of the infection and deaths due to bacterial meningitis.
3. Optimal Control Problem
Since the goal of this paper is to find the best ways to control the spread of meningitis, we define the following optimal control problem:
subject to model (2).
The admissible control set is Lebesgue measurable, which is defined by
Our objective is to find which minimizes the associated cost of the vaccination and the associated cost of the treatment over the specified time interval, as well as minimizing the number of infections at a terminal time (see also [37]). The coefficients and are constants that are introduced to maintain a balance in the size of and , respectively. and are the corresponding weights associated with the cost of vaccination and treatment , respectively. The higher bounds (maximum) attainable for the control measures and are and , respectively. We fix the control measures and to lie between and so that and . Therefore the attainment of and depends on the number of resources available [37]. These resources may include the human effort, material resources, cost of producing vaccine and disbursement, infrastructural resources, the number of health facilities in the community, and the number of hospital beds at the health facilities. The cost of hospitalization, medical test, diagnosis, drug cost, and so forth (see [38–40]) can be associated with treatment. The cost of vaccination may include the cost of the vaccine, the cost of production, the cost of disbursement, the vaccine storage cost, and other related overheads [37]. The severity of the side effects and overdoses of the vaccination and treatment is taken care of by squaring the control measures, and is the final time during the optimal simulation.
3.1. Existence of the Optimal Control
Model (2) can be written aswhere
and is the times derivative of . System (31) is nonlinear with a bounded coefficient.
Setting and gives
Thereforewhere
so that
The function is therefore uniformly Lipschitz continuous. From the definition of the control measures and and the constraint on the state variables, such that , , , and , we observe that a solution of system (31) exists [40, 53, 54]. From the objective functional and its associated constraints in model (2), we can find the optimal solution for our model. Firstly, we find the Lagrangian () and Hamiltonian () for the control problem [55]. The Lagrangian of the optimal problem is given by
Our focus is to find the minimal value of the Lagrangian function, which is done by a pointwise minimization of the Hamiltonian () defined as follows (using Pontryagin’s maximum principle):where , are the adjoint variables associated with , , , and , defined by
Theorem 6. There exists an optimal control pair such that
Proof. We start our proof by considering the properties of the existence of the optimal control (see [56]). Following [57], the set of control measures with corresponding state variables are positive. The set is convex and closed by definition. Therefore, our optimal system is closed and bounded which ascertains the compactness required for the existence of the optimal control. Additionally, the integrand in the objective functional (29), , is convex on the control set . Furthermore, we can state that there exists a positive constant [58], and nonnegative numbers and such that the objective functional has a lower bound of so thatsince the control measures and the state variables are bounded, this leads us to a compact proof of existence of the optimal control.
3.2. Characterization of the Optimal Control
We will apply Pontryagin’s maximum principle to the Hamiltonian function above to derive the necessary condition of optimality for our control problem.
Theorem 7. Let , , , and be optimal state solutions with corresponding optimal control variables and for the objective functional and its constraints in model (2) with . Then, there exist four adjoint variables , , , and that satisfywith transversality conditionsTherefore, the optimal control pair is given by
Proof. We use the Hamiltonian function in (38) in order to obtain the adjoint relations and the transversality conditions. We set the state variables in the Hamiltonian function to , and , and differentiating the Hamiltonian with respect to , and , respectively, yields (42). Also, differentiating the Hamiltonian with respect to the control measures and in the interior of , we obtain the optimality conditions below:Plugging and into (45) and solving the optimal control pair , we haveThe two control measures which are bounded with lower bounds zero and upper bounds , where , giveUsing (47), the optimal control measures are characterized as (44), completing the proof.
Therefore, our optimality system is given by