Research Article | Open Access
A Mathematical Model of Treatment and Vaccination Interventions of Pneumococcal Pneumonia Infection Dynamics
Streptococcus pneumoniae is one of the leading causes of serious morbidity and mortality worldwide, especially in young children and the elderly. In this study, a model of the spread and control of bacterial pneumonia under public health interventions that involve treatment and vaccination is formulated. It is found out that the model exhibits the disease-free and endemic equilibria. The disease-free equilibrium is stable if and only if the basic reproduction number and the disease will be wiped out of the population. For the endemic equilibrium is globally stable and the disease persists. We infer the effect of these interventions on the dynamics of the pneumonia through sensitivity analysis on the effective reproduction number from which it is revealed that treatment and vaccination interventions combined can eradicate pneumonia infection. Numerical simulation to illustrate the analytical results and establish the long term behavior of the disease is done. The impact of pneumonia infection control strategies is investigated. It is revealed that, with treatment and vaccination interventions combined, pneumonia can be wiped out. However, with treatment intervention alone, pneumonia persists in the population.
Pneumonia is the major cause of respiratory morbidity of more than 2 million children under 5 years of age mostly in low-income countries [1–6]. It is an infection of the lungs that is caused by bacteria, viruses, fungi, or other pathogens. Most common cause of bacterial pneumonia is the Streptococcus pneumoniae, also known as pneumococcus [6–8]. It is characterized primarily by inflammation in the air sacs (alveoli) in the lungs that are filled with fluid or pus making it difficult to breathe.
It is reported that about 30%–70% of young children carry S. pneumoniae in their nasopharynx, and up to 40% of the carriers are colonized with penicillin-nonsusceptible S. pneumoniae. Pneumococcus spreads through microaspiration of oropharyngeal organisms and inhalation of aerosols containing bacteria or viruses especially in children that carry the bacteria in their throats without being sick. It may also spread via airborne droplets from cough or sneeze of an infected person. Children become severely ill with high fever and rapid breathing. Infants usually suffer convulsions, unconsciousness, hypothermia, lethargy, and feeding problems . The data on carriage among adults  is limited and most studies suggest that children are the source of transmission to adults in the family .
The risk factors associated with the spread of pneumonia include smoking history and passive smoking, malnutrition, crowded living conditions, lack of exclusive breastfeeding, indoor air pollution, heart disease, alcoholism and drug abuse, acidosis, diabetes, and antecedent viral infection [12, 13]. The overdiagnosis of pneumonia and underdiagnosis of asthma have led significantly to untreated respiratory morbidity and mortality among children less than five years in low-income countries. This has been due to some similarities of symptoms of pneumonia and asthma that often make it difficult to separate the two diseases without proper diagnostic tools .
Pneumonia classification depends on its origin and mode of transmission. Some of the pneumonia classifications include community-acquired, health care-associated, hospital-acquired, ventilator-associated, and walking pneumonia .
Pneumonia is preventable through vaccination, proper diagnosis, screening, environmental control measures, and appropriate treatment of other diseases [4, 5]. Vaccination is the most effective way to prevent certain bacterial and viral pneumonia in both children and adults. The two types of vaccines available against S. pneumoniae are the pneumococcal polysaccharide vaccine (PPV), based on purified capsular (PS) and pneumococcal conjugate vaccine (PCV), obtained by chemical conjugation of the capsular (PS) to a protein carrier . PCVs were developed for use in children only and PPV for vaccination of the at-risk adults and the elderly [17, 18].
Newborn babies can be protected from pneumonia infection through early recognition and treatment at the level of the community or the primary-care health facility, testing pregnant mothers for Group B streptococcus and chlamydia trachomatis, and giving antibiotic treatment and vaccination with PPV that has a proven record of safety in pregnant and breastfeeding mothers for pneumococcal pneumonia prevention in infants. Suctioning the mouth and throat of babies with meconium-stained amniotic fluid decreases the rate of aspiration pneumonia . Environmental measures for pneumonia prevention include reduction of indoor air pollution by encouraging good hygiene in crowded homes and smoking cessation that reduces risks of pneumonia infections among children and adults. Since the bacteria and viruses can also be spread to your hands and then to your mouth, it is important to wash hands with soap when around a person with pneumonia infection.
Appropriate antibiotics are used for treatment of bacterial pneumonia. Pneumonia treatment depends on the underlying cause of the pneumonia infection. Appropriate antibiotics are used for treatment of bacterial pneumonia. Effective and timely treatment together with better diagnostic tools and education prevents antibiotic resistance . According to Wardlaw et al. , treatment alone could save at least 600,000 children’s lives annually at a cost of US $600 million if antibiotic treatment is universally delivered to children with pneumonia. Amoxicillin is recommended as a suitable alternative because of its proven efficacy against S. pneumoniae and severe pneumonia cases should be hospitalized.
Vaccines are effective in reduction of the number of new cases and severity of the disease [21–23]. Childhood pneumonia is preventable through immunization with the effective two vaccines: Hib conjugate vaccine (HibCV) against the Haemophilus influenzae type b (Hib) and pneumococcal conjugate vaccine (PCV) against pneumococcus . PCVs have additional protective qualities that enhance their use as they may reduce nasopharyngeal acquisition of vaccine-specific serotypes of S. pneumoniae, and this in turn reduces the incidence of pneumococcal pneumonia among nonvaccinated individuals [25, 26]. This is referred to as indirect or herd immunity. In this study, we focus on treatment and vaccination of S. pneumoniae among children with PCVs.
Mathematical models of infectious diseases have been recognized as powerful tools that can provide important insights into our understanding of epidemiological processes, the course of infection within a host, the transmission dynamics in a host population, and formulation or implementation of disease control programs [27, 28]. Compartmental mathematical models involving vaccination strategy for infectious disease control have been considered in [29, 30]. Greenhalgh et al.  and Lamb et al.  modeled the transmission of pneumonia among young children to explore the relationship between sequence types and serotypes. Other epidemic models to study pneumonia have been considered (see, e.g., [2, 33–35] and the references therein).
In this study, a deterministic compartmental model to investigate the effect of treatment and vaccination against S. pneumoniae transmission dynamics among children less than five years is formulated. The population studied is divided into a set of distinct compartments according to the disease status. The vaccination strategy consists of vaccination of a proportion of the newborn babies.
This paper is structured as follows. In Section 2, we formulate the model based on the assumptions and definitions of variables and parameters. In Section 3, the pneumonia model with treatment intervention is studied for its boundedness and positivity of solutions and equilibrium points and their stability. In Section 4, the model is extended to investigate the effect of treatment and vaccination interventions combined on the spread of pneumonia. In Section 5, sensitivity analysis of the effective reproduction number is done. Numerical simulation of the model is carried out in Section 6. Finally, in Section 7, we discuss the results and make a conclusion.
2. Formulation of the Model
The model consists of four compartments categorizing individuals based on their status with respect to the disease. The assumptions and definitions of variables and parameters are given in Sections 2.1 and 2.2, respectively.
(1)The model assumes a homogeneous mixing of individuals in the population where all individuals have equal likelihood of catching the infection if they are exposed to the disease.(2)All recovered individuals clear the bacteria from the body and thus do not participate in transmitting the disease.(3)Newborns are given additional dose of vaccine to elicit booster optimal levels of response.(4)All treated individuals get vaccinated after completing the dose.(5)Vaccinated children do not evolve to the susceptible population because of booster vaccine doses.
2.2. Variables and Parameters
The model variables and parameter definitions represented are given as follows: : susceptible individuals who are at risk of acquiring pneumonia infection at time . : carrier individuals who carry the pneumonia bacteria and can transmit the infection at time . : infective individuals capable of transmitting the infection to individuals at risk at time . : recovered individuals who have been treated of pneumonia at time . : vaccinated individuals at time . : per capita natural mortality rate of individuals. : per capita recruitment rate into susceptible population. : proportion of susceptible individuals that joins the carriers. : per capita disease induced mortality rate. : per capita recovery rate of carriers. : force of infection of susceptible individuals. : per capita recovery rate of infected individuals. : rate at which carriers develop symptoms. : rate at which treated individuals become susceptible. : rate at which susceptible individuals get vaccinated. : rate at which treated individuals are vaccinated. : transmission coefficient for the carrier subgroup. : rate of transmission. : probability that a contact is efficient enough to cause infection. : rate of contact.
Based on assumptions and definitions of variables and parameters mentioned above, the following system of ordinary equations is obtained.
3. Pneumonia Model under Treatment
The population dynamics of the pneumonia model with treatment intervention is given by the following system of four ordinary nonlinear differential equations: together with The initial conditions are , and the force of infection iswhere is the transmission rate, is the contact rate, is the probability that a contact is efficient enough to cause infection, and is the transmission coefficient for the carrier subgroup.
3.1. Positivity and Boundedness of the Solutions
The positivity of solutions describes the nonnegativity of solutions of system (1).
Lemma 1. Let the initial population be Then, the solution set of system (1) is positive for all
Proof. From the first equation of system (1), This impliesBy separation of variables, (6) is integrated to obtainwhere is a constant of integration. Applying the initial conditions to (7) gives Hence,
Similarly, it can be shown that the other equations of system (1) are also positive for all . Thus, the solutions of the model are positive for all values of .
It is important to establish whether system (1) is well-posed and biologically meaningful. Now, we study the invariant region which describes the region in which the solution to system (1) makes biological sense. It is assumed that all the state variables and parameters of the model are nonnegative for all .
In the absence of pneumonia,Inequality (10) is referred to as the threshold population level. Therefore, the feasible solution set of system (1) enters and remains in the region; where denotes the nonnegative cone of including its lower dimensional faces. In this case, whenever , then , implying that the host population reduces asymptotically to the carrying capacity. However, whenever , every solution with initial conditions in remains in that region for . Thus, the region is positively invariant, that is, for all values of , the solution remains positive and thus the model is well-posed and biologically meaningful.
3.2. Equilibria of the Model
We analyze the model for pneumonia transmission to determine the basic reproduction number and other threshold parameters for pneumonia dynamics. The equilibria of system (1) are obtained by setting the right-hand side of system (1) equal to zero. The disease-free equilibrium is given by
Theorem 2. There is a unique disease-free equilibrium for the model represented by system (1).
Proof. This theorem is proved by substituting into system (1). The results show that all the derivatives are equal to zero, hence the disease-free equilibrium.
3.3. Basic Reproduction Number
To establish the linear stability of , we use the next generation operator approach on system (1) to compute the basic reproduction number . This is determined using the approach by van den Driessche and Watmough . For the notation of the matrices and , we have where This gives and, thus,Thus, the eigenvalues for the matrix are The spectral radius is given by , which gives the basic reproduction number The endemic equilibrium is defined as a steady state solution for system (1). This occurs when there is a persistence of the disease. Hence, the endemic equilibrium is determined by setting the right-hand side of system (1) equal to zero as follows: together withFrom the second equation of system (18)Plugging in (21) into the first and third equations of system (18) gives Substituting for in (20) into the second equation of system (18) gives From the fourth equation of system (18), we have Substituting for in (22) gives From (23), we obtain Substituting for in (24) givesSolving (28) yields ButTherefore, Substituting for and in (26) gives Finally, we have Therefore, we have the endemic equilibrium , where
Lemma 3. For , a unique endemic equilibrium exists and there is no endemic equilibrium otherwise.
For the disease to be endemic, and ; that is, From inequality (35), we have Using the fact , we obtainFrom inequality (36), we have and, substituting for in inequality (38), inequality (39) yields Thus, a unique endemic equilibrium exists when .
3.4. Local and Global Stability of the Disease-Free Equilibrium
Lemma 4. The disease-free equilibrium of system (1) is locally asymptotically stable whenever and unstable whenever .
The threshold quantity is a measure of the number of secondary infections caused by a single individual in his /her entire lifetime as an infective . It is an important parameter that plays a big role in the control of the disease. The reduction of the disease from the population targets the parameters that will bring its value to less than unity.
When the reproduction number is less than unity, then the disease-free equilibrium is locally asymptotically stable, and thus there is a possibility that the disease will be wiped out of the population.
The Jacobian matrix for system (1) is given byEvaluating the Jacobian matrix (41) at the disease-free equilibrium givesThe disease-free equilibrium is asymptotically stable if and only if the and the .
Thus, from the Jacobian matrix (42), Since the parameters , , , and are all positive, then Therefore
On the other hand, can never be negative and the numerator is positive; that is, This implies that since and Thus, The conditions and above imply that is locally asymptotically stable whenever
Theorem 5. The disease-free equilibrium is globally asymptotically stable in if and unstable if .
Proof. Consider the Lyapunov function defined by Its derivative along the solutions to system (1) is but
Thus, is necessary and sufficient for disease elimination. All the model parameters are positive, so that if with if and only if Hence, is a Lyapunov function on and the largest compact invariant set in is the singleton Therefore, by LaSalle’s invariance principle , every solution to system (1), with initial conditions in , approaches as if Hence the disease-free equilibrium of the pneumonia model with treatment intervention is globally asymptotically stable.
3.5. Local and Global Stability Analysis of the Endemic Equilibrium
We study the local stability of the endemic equilibrium by applying the Routh-Hurwitz criterion.
Theorem 6. If , the endemic equilibrium of system (1) is locally asymptotically stable in .
Proof. We evaluate the Jacobian matrix (41) at the endemic equilibrium to obtain where is defined as the force infection at the endemic equilibrium. We obtain a characteristic equation , where is a unit matrix Thus, the characteristic equation becomes where Thus, from Routh-Hurwitz criterion  we have the matrix According to the Routh-Hurwitz criterion, for , the endemic equilibrium is locally asymptotically stable if
The global stability of the endemic equilibrium is analyzed using the following constructed Lyapunov function.
Theorem 7. If , the endemic equilibrium of system (1) is globally asymptotically stable.
Proof. Let the Lyapunov function bethus collecting positive terms together and negative terms together from the above whereThus if , then we obtain that , noting that if and only if Therefore, the largest compact invariant set in is the singleton , where is the endemic equilibrium of system (1).
Thus, by LaSalle’s invariance principle , it implies that is globally asymptotically stable in if .
4. Pneumonia Model under Treatment and Vaccination Interventions
In the this section, the model formulated in Section 3 is extended to investigate the impact of treatment and vaccination interventions on the transmission dynamics of pneumonia. The dynamics of the modified pneumonia model is described by the following system of five ordinary nonlinear differential equations: with initial conditions . The total population size is given by and is changing at the rate The susceptible individuals become infected at rate which is the force of infection; that is, the number of infected individuals produced by adequate contact and is given by where , is the transmission rate, is the rate of contact, and is the probability that a contact is efficient enough to cause infection.
4.1. Analysis of the Model
The equilibria of system (57) is obtained by setting the right-hand side of the equations to be equal to zero. The disease-free equilibrium is given by
4.2. Effective Reproduction Number
To establish the stability of , we use the next-generation operator approach on system (57) to compute the effective reproduction number . Using the notation of the matrices and , we have Evaluating at the disease-free equilibrium we to obtain Let and ; then and, thus,Now we have The eigenvalues for the matrix are given by The spectral radius is given by , which gives the effective reproduction number as but Therefore, is referred to as the effective reproduction number rather than the basic reproduction number because vaccination and treatment have been included in the model . It is defined as the expected number of secondary cases caused by a typical infected individual entering an entirely susceptible population at equilibrium.
4.3. Local Stability of the Disease-Free Equilibrium
The Jacobian matrix for the system is given byThe disease-free equilibrium point is discussed by examining the Jacobian matrix (70) at the steady point Now, at the disease-free equilibrium the Jacobian matrix is given byFor stability of the disease-free equilibrium, it is required that the and the Thus, from the Jacobian matrix (72), it is clearly seen that The determinant of the Jacobian matrix is also given by Therefore, the disease-free equilibrium of the pneumonia model under treatment and vaccination interventions is locally asymptotically stable. This is established by the fact that the and the
Proposition 8. for any given parameters.
Proof. implies that Thus
The above result leads us to the following theorem.
Theorem 9. The disease-free equilibrium of the pneumonia model under treatment and vaccination interventions is locally asymptotically stable if and unstable if
The proof of the theorem follows from the Jacobian matrix (72).
The endemic equilibrium is defined as a steady state solution for system (57). This occurs when there is a persistence of the disease. Hence, can be determined as below. Consider system (57) with right-hand side equal to zero to obtain together withSolving system (77) together with (78) gives the endemic equilibrium , where where
4.4. Local Stability of the Endemic Equilibrium
The Jacobian matrix for system (57) is given byEvaluating the Jacobian matrix (81) at the endemic equilibrium gives where We now obtain the characteristic equation , where is a unit matrix. where Thus the characteristic equation becomes where The necessary and sufficient conditions for the local asymptotic stability of endemic equilibrium are that the Hurwitz determinants are all positive for the Routh-Hurwitz criteria. For a fifth-degree polynomial , these criteria are given by from which we can conclude that the endemic equilibrium is locally asymptotically stable.
5. Sensitivity Analysis
Intervention strategies to reduce the mortality and morbidity due to pneumonia should target the parameters that have a high impact on the effective reproduction number, . Sensitivity analysis is used to obtain the sensitivity index that is a measure of the relative change in a state variable when a parameter changes. We compute the sensitivity indices of to the model parameters with the approach used by Chitnis et al. . These indices show the importance of each individual parameter in the disease transmission dynamics and prevalence.
Definition 10. The normalized forward sensitivity index of a variable, , that depends differentiability on index on a parameter, , is defined as We use the formula for to derive an expression for the sensitivity of given by to each of the ten parameters given in Table 1. In the following example, we obtain the sensitivity index of with respect to : The same method is used to obtain the indices of , and
The parameters given in Table 1 are ordered from most sensitive to the least sensitive. The parameter values , , , , , , , , , and are used to determine the sensitivity indices.
5.1. Interpretation of Sensitivity Indices
It is noted from the sensitivity indices given in Table 1 that the value of increases when the parameter values , , , and increase while other parameter values are kept fixed. This implies an increase in the endemicity of the disease since the indices have positive signs. On the other hand, when the parameter values , , , , , and are decreased while the rest of the parameter values are kept fixed, the value of decreases. This shows a decrease in the disease endemicity because the indices have negative signs. The transmission rate and recruitment rate are the most sensitive parameters. The transmission coefficient of the carrier subgroup and proportion of susceptible population that become carriers are the other key parameters that are sensitive.
6. Numerical Simulation
We illustrate the analytical results of the model by carrying out numerical simulation of the models using a set of estimated parameter values obtained from literature. The system is simulated using ODE solvers coded in MATLAB programming language. Simulation of the pneumonia model under treatment intervention alone and the model with treatment and vaccination interventions combined is carried out to investigate the impact of the key parameters on the spread of pneumonia and how their influence can be controlled. The parameter values are presented in Table 2.