Abstract

The membranes that encompass the brain and spinal cord become inflamed by the potentially fatal infectious disease called pneumococcal meningitis. Pneumonia and meningitis “coinfection” refers to the presence of both conditions in a single host. In this work, we accounted for the dynamics of pneumonia and meningitis coinfection in communities by erroneously using a compartment model to analyze and suggest management techniques to stakeholders. We have used the next generation matrix approach and derived the effective reproduction numbers. When the reproduction number is less than one, the constructed model yields a locally asymptotically stable disease-free equilibrium point. Additionally, we conducted a sensitivity analysis to determine how different factors affected the incidence and transmission rate, which revealed that both the pneumonia and meningitis transmission rates are extremely sensitive. The performance of our numerical simulation demonstrates that the endemic equilibrium point of the pneumonia and meningitis coinfection model is locally asymptotically stable when . Finally, as preventative and control measures for the coinfection of pneumonia and meningitis illness, the stakeholders must concentrate on reducing the transmission rates, reducing vaccination wane rates, and boosting the portion of vaccination rates for both pneumonia and meningitis.

1. Introduction

The word “epidemiology” is derived from the Greek term “demos,” which means “people,” and “logos,” which means “the study of”. In other words, the term “epidemiology” has its roots in the study of the experiences of a population. Despite the fact that many definitions have been provided, the one that best encapsulates the fundamental ideas and public health perspective of epidemiology is: “Epidemiology is the study of the prevalence and causes of health-related conditions or incidents in particular populations, as well as the application of this information to the prevention or treatment of health issues” [1, 2]. In epidemiology, the frequency and distribution of health events in a population are studied.

By the term “frequency,” we not only mean the number of health events, such as the number of cases of meningitis or diabetes in a population, but also the correlation between that number and the size of the population [2, 3].

The underlying premise of epidemiology is that disease does not develop in a community at random but rather develops only when an individual has the proper confluence of risk factors or determinants. Individuals are the “patients” of clinicians, whereas communities are the “patients” of epidemiologists. In light of this, while dealing with a patient who is unwell, the clinician and the epidemiologist have separate duties. When a patient with diarrheal illness first shows up, for instance, both parties are concerned with making the right diagnosis [35].

A potentially fatal infection called pneumococcal meningitis inflames the membranes that cover the brain and spinal cord. The meninges, which are these layers, serve to shield the brain from damage and infection [5]. Millions of people have died as a result of pneumonia, an airborne disease that is caused by breathing harmful organisms, primarily Streptococcus pneumonia. Other illnesses, including meningitis, ear infections, and sinus infections, are also brought on by these bacteria [57]. Moreover, these illnesses can afflict people of all ages, from infants to the elderly. Pneumonia is particularly hazardous when the immune system is weakened, as in infants or the elderly, or when it is concomitant with another illness like meningitis [7, 8]. Pneumonia is a common coinfection that occurs at the time of admission in cases of bacterial meningitis and is independently linked to a poor prognosis and death [9].

The most frequent pneumococcal infection in children are caused by 13 different varieties of pneumococcal bacteria. There are vaccines named PCV13, which can protect against these types, and PPSV23, which can protect against 23 other types. For the sake of this investigation, we have studied prior work by other researchers who used mathematical modeling to explain the transmission and spread of coinfections with pneumonia and meningitis, such as [7, 1013]. The majority of these investigations were carried out to identify community-level infectious disease control methods. To our knowledge, no one has created and examined the three kinds of vaccine independently in a mathematical model of meningitis and pneumonia coinfection in a specific community. As a result, this recently proposed study considers the dynamics of meningitis and pneumonia in communities, utilizing a deterministic compartmental model to analyze and recommend appropriate management techniques to actors. Therefore, we are driven and intrigued to investigate the three vaccine kinds for meningitis and pneumonia coinfection in this work by developing a mathematical model of meningitis and pneumonia coinfection combined with regulating techniques. We have laid up the basic framework for this investigation as follows: in Section 2, we outline and develop the compartmental mathematical model of coinfection with meningitis and pneumonia. The model analysis includes the equilibrium points, fundamental reproduction numbers, and stability analysis of the submodels and the main model also presented in Section 2. Numerical simulations and sensitivity analysis are presented in Section 3. The study’s discussion and conclusion were then finalized.

2. Mathematical Model Formulations and Its Qualitative Analysis

In this section we have proposed a mathematical model which depend upon the assumption and present the qualitative properties of the constructed model.

2.1. Baseline Model Formulation and Assumptions

We have developed the new model by expanding the model which was developed previously by another researcher based on the following hypotheses. Under this study, we consider homogenous population and that the factors such as sex, social status, and race do not affect the probability of being infected. The model subdivides the human total population into nine mutually-exclusive compartments, namely, susceptible population pneumonia-only infectious meningitis-only infectious , meningitis and pneumonia coinfectious treated class , meningitis vaccinated (t), pneumonia vaccinated (t), class of people who take both vaccines of pneumonia and meningitis (PCV13 Pneumococcal conjugate vaccine) group , and recovered class . The recovery from natural immunity and the effects of vertical transmission to pneumonia and meningitis were assumed to be insignificant in this study. Epidemiologically, individuals in the removed/recovered compartment do not attain permanent immunity so that we are assigned such case by the parameter.

In this study the mass action-incidence rate of new infections are used, and the modification parameters and are the factors that describe the fact of how infectiousness level of pneumonia increases the susceptibility level to meningitis disease and vice versa, respectively. The meningitis disease is assumed to be transmitted after effective contact between the susceptible and meningitis infectious classes with effective contact rate where is a composite parameter that measures the meningitis contact rate and the probability of transmission upon contact . Additionally, pneumonia disease is assumed to be transmitted after effective contact between the susceptible and pneumonia infectious classes with effective contact rate , where is a composite parameter which measure the pneumonia infectious contact rate and the probability of transmission upon contact . Individuals can get meningitis by contact rate from a meningitis-only infected or coinfected person with force of infection of meningitis and join compartment where is the modification parameter.

An individual can get pneumonia with contact rate of from a pneumonia-only infected or coinfected person with force of infection of pneumonia and join the compartment with modification parameter . Pneumonia-only infected individuals also can get an additional meningitis infection with force of infection and modification parameter and join coinfected compartment . The coinfected compartment increases because of individuals that come from meningitis-only infected compartment are infected by pneumonia with force of infection and modification parameter . Since the coinfected individuals are aware of the disease, they remain there in the treated compartment even if they are free from either pneumonia or meningitis until they are free from all the diseases. The parameters used in the model are described in Table 1.

Using the above assumptions and parameters, we have constructed the following schematic diagram that is given in Figure 1.

From the diagram given in Figure 1, the corresponding dynamical systems will be as follows from the assumptions of the model and using the above basic model assumption we have the following flow chart.

This system of differential equation is the mathematical representation of full meningitis and pneumonia model which is the combination of the two diseases. In the next section, we have studied the qualitative behavior of the constructed model. For simplification of our work, we split the full meningitis-pneumonia coinfection model into submodels, which are meningitis-only and pneumonia-only models. First, we will study the qualitative behavior of the submodel and then qualitative behavior of the full model is followed.

2.2. Positivity of Solutions and Invariant Region of the Only Pneumonia-Infected Model

In this subsection, we have considered the model of pneumonia only in the absence of meningitis disease. This procedure will help us to summarize and conclude some properties of the full coinfected model depending on the properties of sub models. To gate this submodel from the full model, we set, , and we have the following dynamical system.

The corresponding dynamical systems are as follow.

For the dynamical systems to be epidemiologically meaningful as well as well-posed, we need to prove that all the state variables of dynamical systems are nonnegative.

Theorem 1. All the populations of the system with positive initial conditions are positive.
Proof: assume are positive for time and for all nonnegative parameters.
First, let us take such that S and }.
From the first equation of system (2), we do have .
There, is positive. Following the same procedure, all the remaining state variables are nonnegative. Therefore, from proof, we can conclude that whenever the initial values of the systems are all nonnegative, then all the solutions of our dynamical system are positive.

Theorem 2. The total human population of the dynamical system (2) is positively closed in the closed invariant set.
Furthermore, the system’s nonnegative solutions are all constrained, and it may exhibit the persistence property under any nonnegative initial concentration conditions [14].
Proof: assume the total population of the model is . To get an invariant region, which shows boundedness of solution, it can be obtained as follows. .
Therefore, the dynamical system that we do have is bounded.

2.3. Existence and Stability of Disease-Free Equilibrium Point

The disease-free equilibrium point is obtained by making all the equations equal to zero, provided that and the obtained disease-free equilibrium point is given by

2.3.1. Effective Reproduction Number

The reproduction number is the number of secondary cases produced by one infectious individual joining in a completely susceptible population during its infectious period [1517].

Using the next generation matrix method, we have obtained the effective reproduction number of pneumonia-infected-only submodel, which is .

Theorem 3. The disease-free equilibrium point of the model in system (2) is locally asymptotically stable if the effective reproduction number and is unstable if.
Proof: from the Jacobean matrix of the model (2), with respect to at the disease-free equilibrium point, we have the following characteristics equation. Where and Hence, all the parameters are nonnegative, and all the eigenvalues of the corresponding Jacobean matrix are negative. But .Therefore, the disease-free equilibrium point is locally asymptotically stable if and only if , otherwise it is unstable, that is, if.

2.3.2. Global Stability of Disease-Free Equilibrium Point of the Model

To verify the global stability of the disease-free equilibrium point of the pneumonia monoinfection model, we have used an adopted method of Castillo-Chavez et al. used by others scholar such as [18, 19].

Lemma 4. If the pneumonia monoinfection model can be written as where be the components of noninfected individuals and be the components of infected individuals including treated class and denotes the disease-free equilibrium point of dynamical system (2).
Assume (i)For , is globally asymptotically stable (GAS)(ii), for where is an M-matrix, i.e., the off diagonal elements of are nonnegative and is the region in which the system makes biological sense. Then the fixed point is globally asymptotically stable equilibrium point of the system (2) whenever.

Lemma 5. The disease-free equilibrium point of the pneumonia monoinfection model (2) is globally asymptotically stable if and the two sufficient conditions given in Lemma 4 are satisfied.
Proof: here we are applying Lemma 5 on the pneumonia monoinfection model (2) and we have gotten the following matrices. and Since , we have , thus, the disease-free equilibrium point is globally asymptotically stable if . Biologically, whenever , the only pneumonia infection disease dies out while the total population increases [18].

2.4. The Existence and Stability of Endemic Equilibrium Point

The endemic equilibrium point of the dynamical system of (2) is obtained by making the right side of the system equal to zero, providing that . We have supposed that the endemic equilibrium point of the model is denoted by and the corresponding force of infection is.

For simplification of algebraic manipulation, we have assumed the parameters in the model by another variable as follows,

, , ,

and . Now the equation of force of infection can be rearranged as

but if .

Therefore, there is a unique endemic equilibrium point for pneumonia monoinfected model as given by where

Theorem 6. The endemic equilibrium point of system (2) is locally asymptotically stable for the reproduction number .
Proof: to show that the local stability of the endemic equilibrium point, we have used the method of the Jacobian matrix and the Routh Hurwitz stability criteria.
Then the corresponding characteristic equation is obtained from the determinant of where , , , , , , where , , , , .

To apply the Routh-Hurwitz stability criteria, it is obligatory to check if the necessary condition of all the coefficients have the same sign or not. Since is positive in sign, all , , , and should be positives in sign. All the coefficients of the characteristic’s polynomial are positives whenever . We have observed that the first column of the Routh Hurwitz array has no sign change, thus the root of the characteristics equation of the dynamical system are negative. Hence, the endemic equilibrium point of the dynamical system is locally asymptotically stable.

2.5. Positivity of Solutions and Invariant Region of the Only Meningitis-Infected Model

We have made from the full pneumonia and meningitis coinfection model to obtain this submeningitis-only model, and got the following dynamical system.

The above dynamical systems are needed to be epidemiologically meaningful as well as well-posed. To prove that, we have intimated that all the state variables of dynamical systems are nonnegative.

Theorem 7. All the populations of the system with positive initial conditions are positive.
Proof: assume and are positive for time and for all nonnegative. First let us take .
From the first equation of system (17), we do have Therefore, is positive. Subsequent to the same procedure, the remaining state variables are nonnegative. Therefore, from the stated proof, we can conclude that whenever the initial values of the systems are all nonnegative, then all the solutions of our dynamical system are positive.

Theorem 8. All the populations of the system with positive initial conditions are nonnegative
The total human population of the dynamical system (17) is positively closed in the closed invariant set . Furthermore, the system’s nonnegative solutions are all constrained, and it may exhibit the persistence property under any nonnegative initial concentration conditions [14].
Proof: to get an invariant region, boundedness of solution is obtained as follow. Therefore, the dynamical system that we do have is bounded.

2.6. Existence and Stability of Disease-Free Equilibrium Point

The disease-free equilibrium point is obtained by making all the equations in the system equal to zero, provided that providing that . Therefore, the disease-free equilibrium point is

2.6.1. Effective Reproduction Number

The reproduction number can be defined as a number of secondary cases produced by one infectious individual joining in a completely susceptible population during its infectious period [16, 17, 20].

To compute the reproduction number, first distinguishing the new infected from all other changes in the host population is mandatory as follows.

Let the rate of appearance of new infected in compartment,

the rate of transfer of individuals in to compartment

the rate of transfer of individuals out of compartment.

And then but and , where and V are matrix with is number of infected compartment. is the next generation matrix, and the spectral radius of next generation matrix is needed for the reproduction number we are seeking for.

Thus and and

Therefore, the effective reproduction number of meningitis monoinfected submodel is

Theorem 9. The disease-free equilibrium point of the model in system (17) is locally asymptotically stable if the effective reproduction number , and it is unstable if .
Proof:
Using the Jacobean matrix of the model (17) with respect to at the disease-free equilibrium point, we have the characteristic equation Hence, all the parameters are nonnegative, all the eigenvalues of the corresponding Jacobean matrix are negative other than .
For Therefore, the disease-free equilibrium point of the meningitis monoinfected model is locally asymptotically stable if the effective reproduction number and is unstable if .

2.6.2. Global Stability of Disease-Free Equilibrium Point

We utilized the approach developed by Castillo-Chavez et al. and used it to confirm the overall stability of the disease-free equilibrium point of the meningitis monoinfection model [18, 19].

Lemma 10. If the pneumonia monoinfection model can be written as where be the components of noninfected individuals and be the components of infected individuals including the treated class, and denotes the disease-free equilibrium point of the dynamical system (3).
Assume (i)For , is globally asymptotically stable (GAS)(ii), for where is an M-matrix, i.e., the off diagonal elements of are nonnegative and is the region in which the system makes biological senseThen the fixed point is globally asymptotically stable equilibrium point of the system (17) whenever .

Lemma 11. The disease-free equilibrium point of the pneumonia monoinfection model (17) is globally asymptotically stable if and the two sufficient conditions given in Lemma 10 are satisfied.
Proof: here we are applying Lemma 11 on the meningitis monoinfection model (17) and we have gotten the following matrices Here after some steps of calculations, we have determined that Since , we have , thus, the disease-free equilibrium point of model (17) is globally asymptotically stable if . Biologically, whenever , the meningitis monoinfection disease dies out while the total population increases [18].

2.7. Existence and Stability of Endemic Equilibrium Point

The endemic equilibrium point of the dynamical system of (3) is obtained by making the right side of the system equal to zero, providing that . We have supposed that the endemic equilibrium point of the model is denoted by and the corresponding force of infection is . For simplification of algebraic manipulation, we have assumed the parameters in the model by another variable as follows.

and . Now the equation of force of infection can be rearranged as

Therefore, the unique endemic equilibrium point for the meningitis monoinfected model is given by where

Theorem 12. The endemic equilibrium point of system (17) is locally asymptotically stable for the reproduction number .
Proof: To show the local stability of the endemic equilibrium point we have used the method of Jacobian matrix and Routh Hurwitz stability criteria.
From Jacobian matrix, we have obtained the following characteristic equation Where , , , , , , Where , , ,
.

To apply Routh-Hurwitz stability criteria, it is the must to check the necessary condition of all the coefficients have the same sign or not. Since is positive in sign, all , , and should be positives in sign. All the coefficients of the characteristic’s polynomial are positives whenever.

We have observed that the first column of the Routh Hurwitz array has no sign change, thus the root of the characteristics equation of the dynamical system are negative. Hence, the endemic equilibrium point of the dynamical system is locally asymptotically stable.

2.8. Positivity and Boundedness of Full Pneumonia and Meningitis Coinfected Model

The corresponding dynamical system of the full pneumonia and meningitis coinfection model is given in Equation (1).

The constructed model is expected to be meaningful epidemiologically as well as well-posed. We need to prove that all the state variables of the dynamical system are positive.

Theorem 13. All the population of the system with positive initial conditions are nonnegative
Proof: Assume , , and are positive for time and for all nonnegative parameters. Let us define .
Since all are continuous, we can say for . If then positivity holds.
Nevertheless, if then all the state variables are zeros.
From the first equation of system (1) we do have Following same procedure, all the remaining state variables are nonnegative.
Therefore, from proof, we can conclude that whenever the initial values of the systems are all nonnegative, then all the solutions of our dynamical system are positive.

Theorem 14. The total human population is assumed to be and the dynamical system (1) is positively invariant in the closed invariant set . Furthermore, the system’s nonnegative solutions are all constrained, and it may exhibit the persistence property under any nonnegative initial concentration conditions [24]. Proof: to get an invariant region, which shows boundedness of solution, is obtained as follow. Therefore, the dynamical system that we have constructed is bounded.

2.9. Disease-Free Equilibrium Point and Its Stability

The disease-free equilibrium point of full pneumonia and meningitis coinfection model is obtained by making all the right-hand-side of equation in system (1), providing that all the infectious classes are equal to zero.

2.9.1. Effective Reproduction Number

The reproduction number is the average number of people that become infected because of the entry of one infectious person into a completely susceptible population in the absence of intervention. Moreover, reproduction number is utilized to determine the effect of the control measures and to understand the capability of the spread of the infection to disseminate in the entire community when the control strategies are applied [15, 17, 21].

The reproduction number of pneumonia and meningitis confection model denoted by , which is manipulated by the Van den Driesch, Pauline, and James Warmouth next generation matrix approach [20], is the largest eigenvalue of the next generation matrix , where is the rate of appearance of new infection in compartment , is the transfer of infections from one compartment to another, and is the disease-free equilibrium point.

The

Then and