A Model for the Propagation and Control of Pulmonary Tuberculosis Disease in Kenya

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Introduction
Mathematical modeling is one of the valuable tools that explore the transmission dynamics of infectious diseases and assess the impact of various control interventions.Terefore, mathematical modeling of infectious diseases is used to guide public health policies and inform decision-making during epidemics [1,2].Tis paper employs a classical SEIR transmission mechanism to formulate a novel model for pulmonary tuberculosis, aiming to accurately depict its natural progression.Te SEIR model, a widely used compartmental model in epidemiology, is utilized to forecast the spread of infectious diseases with latent or exposed phases [3].During the latent phase of a disease, individuals are neither infectious nor symptomatic [4].Te evolution of infectious diseases, taking into consideration the SEIR model, has been discussed by several authors [5][6][7][8][9].
Research suggests that around one-quarter of the global population harbors latent pulmonary tuberculosis infections [10].Consequently, an epidemiological model for pulmonary tuberculosis must account for this latent population and hence adopts the classical SEIR framework.Within the SEIR model, the population is categorized into four compartments: Susceptible (S), Exposed (E), Infectious (I), and Recovered (R) [2].
Pulmonary tuberculosis is a contagious disease that primarily afects the lungs.It is caused by microorganisms known as Mycobacterium tuberculosis, which spread when an infectious individual coughs, sneezes, speaks, or sings [4,10].Te disease exerts a heavy toll on the human population, causing a signifcant number of deaths worldwide.Every year, approximately 10,000,000 people fall ill with pulmonary tuberculosis across the globe, with about 15% of them succumbing to the disease.In 2022, approximately 10,600,000 people fell ill with tuberculosis worldwide, resulting in 1.6 million deaths [11].In Kenya, data from Civil Registration Services show that pulmonary tuberculosis accounted for about 5.4% of all reported deaths in 2019 [12].Terefore, it is imperative to scale up preventive measures aimed at reducing the transmission of infections to the susceptible population.Te United Nations' "Agenda 2030 Sustainable Development Goals (SDGs)" has identifed tuberculosis as one of the communicable diseases that need to be eradicated worldwide by 2030 for sustainable development [13].
Various mathematical models for pulmonary tuberculosis disease have been formulated, focusing on diverse strategies to control the transmission of infections, such as vaccination, early diagnosis, social protection, and treatment of drug-resistant strains [14][15][16].Athithan et al. [17] analyzed a nonlinear model of pulmonary tuberculosis considering case detention and treatment, and their results showed that sustaining treatment and case detention hold great promise in controlling the spread of infections.Salpeter et al. [18] conducted a study on the mathematical modeling of pulmonary tuberculosis with estimates of reproduction number and infection delay function.Teir outcome revealed that the risk of infection reactivation decreases rapidly and then gradually, for the frst ten years after infection.Houben et al. [19] formulated a mathematical model to estimate the global burden of latent tuberculosis infection, and their results estimated that approximately a quarter of the world's population was infected with latent tuberculosis in 2014.Vaccination and efective contact rates on the spread of pulmonary tuberculosis were assessed using a mathematical model in [20,21].Teir results showed that vaccination coverage is not sufcient to control pulmonary tuberculosis, and the efective contact rate has a higher impact on the spread of infections.Aparicio et al. [22] explored the strengths and limitations of homogeneous and heterogeneous mixing in tuberculosis epidemics through mathematical modeling, and their results indicated that a decrease in pulmonary tuberculosis incidence was due to a reduction in progression rates.Kasereka et al. [23] simulated a mathematical model of pulmonary tuberculosis transmission in the Democratic Republic of Congo, revealing that monitoring contacts, detection of latent infection, and treatment are the optimal strategies to reduce the transmission of infections in the population.
Te main purpose of this study is to investigate the impact of various intervention strategies on controlling the transmission of infections in the population.We formulated and analyzed a deterministic pulmonary tuberculosis model incorporating an asymptomatic infectious population.Te justifcation for incorporating an asymptomatic infectious population stems from a report by the National TB Prevalence Survey of 2016 in Kenya, which showed that 26% of prevalent cases diagnosed during their survey were asymptomatic infectious [24].We investigated the efects of screening asymptomatic and latently infected populations on controlling the transmission of infections to the susceptible population.Tis paper is organized as follows: Section 2 presents the model formulation, while in Section 3 mathematical analyses have been performed.Section 4 provides the numerical simulations of the model to illustrate the impact of various intervention strategies on controlling the transmission of infections.Finally, Section 5 presents the conclusion.

Pulmonary Tuberculosis Model Formulation
Incorporating Asymptomatic Infectious Population Individuals are recruited into the population at a constant rate of π.A fraction of the recruits is vaccinated at a constant rate of P and enters the vaccinated class, whereas the rest become susceptible.Te susceptible individuals become latently infected after efective contact with any of the following populations: symptomatic infectious, asymptomatic infectious, symptomatic infectious undergoing treatment, and asymptomatic infectious undergoing treatment.Te force of infection is represented by , where β represents the transmission rate of pulmonary tuberculosis infections, while η 1 , η 2 , and η 3 are the dimensionless transmission coefcients accounting for the relative infectiousness of asymptomatic infectious individuals, symptomatic infectious individuals undergoing treatment, and asymptomatic infectious individuals undergoing treatment, respectively, with η 3 < η 2 < η 1 .Tis hierarchy assumes that asymptomatic infectious individuals (I a ) are more infectious than symptomatic individuals undergoing treatment (T s ), who are, in turn, more infectious than asymptomatic infectious individuals undergoing treatment (T a ).Te dimensionless transmission coefcients, η 1 , η 2 , and η 3 are considered to be less than 1.Te model assumes that vaccination is not 100% efective, and thus the vaccinated class has a chance of being latently infected at a force of infection given by λ , where ρ is the vaccine efcacy, such that 0 ≤ ρ ≤ 1. Te latent infected individuals, denoted by E(t), are either screened at a constant rate of θ 1 and moved to the latent infected undergoing treatment class T E (t) or progress to the asymptomatic infectious class I a (t), at a rate of χ 1 .Alternatively, they may develop severe disease and transition to the symptomatic infectious class I s (t), at a constant rate of χ 2 .Asymptomatic infectious individuals are either screened at a rate of θ 2 and moved to the asymptomatic infectious undergoing treatment class T a (t), or they progress to severe disease and join the symptomatic infectious class I s (t) at a rate of ω.Symptomatic infectious individuals are identifed for treatment at a rate of θ 3 and moved to the symptomatic infectious undergoing treatment class T s (t).Te rate of disease-induced deaths due to pulmonary tuberculosis for individuals in the symptomatic infectious class is given by δ 1 , whereas the rate of death due to the disease for symptomatic infectious individuals undergoing treatment is given by δ 2 .Treatment for diferent forms of pulmonary tuberculosis is assumed to be successful, and thus latent infected individuals undergoing treatment, asymptomatic infectious individuals undergoing treatment, and symptomatic infectious individuals undergoing treatment recover at rates of ξ 1 , ξ 2 , and ξ 3 , respectively, and move to the recovered class R. Te model assumes that recovered individuals become susceptible again after their immunity wanes at a rate of σ.Te rate at which individuals die from causes other than pulmonary tuberculosis is denoted by μ.It is worth noting that all parameters are positive constants.
Te compartmental model illustrating the interaction of the human population in various classes is depicted in Figure 1.

Model Analysis
3.1.Invariant Region.Te model system (1) deals with the human population, and thus we need to demonstrate that its solutions are bounded for all time t > 0.

Theorem 1. Given the positive initial conditions, the feasible region is defned as follows:
Proof.Te sum of all equations in system (1) represents the total human population in the model and satisfes the following equation: ( By integrating inequality (3) and applying the initial conditions, we obtain As t ⟶ ∞ in inequality (3), the population N(t) ⟶ π/μ, implying that 0 ≤ N(t) ≤ π/μ.Tus, the feasible solution of the system enters and remains in the region: Terefore, the basic model is well-posed epidemiologically and mathematically, and hence it is sufcient to study its dynamics in Ω. □

Te Basic Reproduction Number and the Control Reproduction Number.
Te basic reproduction number is a threshold parameter that governs the spread of disease.It is defned as the average number of secondary infections caused by a single infectious individual during his entire infectious period in a population that is entirely susceptible [25].
Te control reproduction number is defned as the expected number of secondary infections produced by an index-infected individual in a population that is not entirely susceptible due to the presence of control measures [26].
In the absence of pulmonary tuberculosis, E � I a � I s � T E � T a � T s � R � 0. Terefore, system (1) has a diseasefree equilibrium given by We use the next-generation matrix to obtain the control reproduction number as given by [27].
where f and v are matrices representing the new infections and transition terms, respectively, given as follows: Te Jacobian matrices of the new infections and transition terms at the disease-free equilibrium are given, respectively, as follows: Ρπ F � Te dominant eigenvalue corresponding to the spectral radius ρ(FV − 1 ) of the matrix FV − 1 is the control reproduction number (R cVST ) with vaccination, screening, and treatment of all forms of pulmonary tuberculosis as the intervention strategies.R cVST is given as follows: where Without vaccination intervention, the fraction of recruits vaccinated, P, equals zero, and consequently, the parameter, ρ, for vaccine efcacy becomes zero since there will be no vaccinated population.Substituting P � ρ � 0 in (10) gives the control reproduction number (R cST ) with screening and treatment as the only intervention strategies.R cST is thus given as follows: Considering the presence of latent infected and asymptomatic infectious individuals in the population without their screening, the parameters θ 1 , θ 2 , and ξ 2 become zero.Substituting θ 1 � θ 2 � ξ 2 � 0 in (10) gives the control reproduction number (R cVT s ) with vaccination and treatment of the symptomatic infectious population as the only intervention strategies.R cVT s is thus given as follows: Discrete Dynamics in Nature and Society When there is no vaccination of recruits and screening of both latent infected and asymptomatic infectious populations, the parameters θ 1 , θ 2 , ξ 2 , P, and ρ become zero.Substituting θ 1 � θ 2 � ξ 2 � P � ρ � 0 in (10) gives the control reproduction number (R cT s ) with the treatment of the symptomatic infectious population as the only intervention strategy.R cT s is thus given as follows: Considering no intervention measures in place, that is, when there is no vaccination of the recruits, screening of both latent and asymptomatic infectious populations, and treatment of all forms of pulmonary tuberculosis, the parameters P, ρ, θ 1 , θ 2 , θ 3 , ξ 2 , ξ 3 , and δ 2 become zero.(10) gives the basic reproduction number (R o ) given as follows:

Local Stability of Disease-Free Equilibrium
Theorem .Te disease-free equilibrium point is locally Proof.To prove the local stability of the disease-free equilibrium, we obtain the Jacobian matrix of the system at the disease-free equilibrium as follows: where 6 Discrete Dynamics in Nature and Society Evaluating the eigenvalues of ( 16), we obtain Either or By the Routh-Hurwitz criteria, (21) has strictly negative roots given as follows: Te characteristic polynomial of ( 20) is obtained as follows: where M 1 , M 2 , M 3 , M 4 , M 5 , M 6 , and M 7 are determined using Mathematica software as follows: By the Routh-Hurwitz criteria, From M 7 > 0, we have S 0 � (1 − P)π/μ, and V 0 � Pπ/μ in inequality (26) and rearranging yield However, Terefore, ( 27) can be expressed as follows: From ( 29), the disease-free equilibrium is locally asymptotically stable if R cVST < 1. Tis implies that each infectious individual infects, on average, less than one susceptible person during the infectious period, resulting in the disease dying out [27]. ) is evaluated by equating the model system of (1) to zero.Te steady-state solution for the model equations is as follows: Discrete Dynamics in Nature and Society where (31)

Global Stability of Endemic Equilibrium Point.
We consider the method of the Lyapunov function to prove the global stability of the endemic equilibrium point.We propose a logarithmic Lyapunov function L defned by Te derivatives of L along the solution of the model system (1) give Substituting the expressions dS/dt, dV/dt, dE/dt, dI a /dt, dI S /dt, dT E /dt, dT a /dt, dT S /dt, and dR/dt from the model system (1) into (33) and simplifying give where Discrete Dynamics in Nature and Society Terefore, the largest compact invariant set in is the singleton endemic equilibrium point B * .Tus, from LaSalle's invariance principle [28], we conclude that as t ⟶ ∞, the solution of the model system (1) approaches the endemic equilibrium B * when the control reproduction number R * cVST > 1. Terefore, the endemic equilibrium point B * is globally asymptotically stable in the invariant set Ω if Q < W.

Sensitivity Analysis on Control Reproduction Numbers.
In this section, we present a sensitivity analysis of the reproduction numbers to determine the relative importance of the various parameters responsible for pulmonary tuberculosis transmission and prevalence in the population.We employed the normalized forward sensitivity index for this model as used by [29].Te normalized sensitivity index which measures the relative change in a parameter K with respect to the reproduction number R c is given by Te parameter values in Table 1 are used to calculate the sensitivity indices of the reproduction numbers for the parameters β, P, χ 1 , χ 2 , θ 1 , θ 2 , and θ 3 .
Te calculated sensitivity indices of reproduction numbers are given in Table 2.A positive sensitivity index shows that the reproduction number is an increasing function of the corresponding parameter whereas a negative sensitivity index shows that the reproduction number is a decreasing function of the corresponding parameter.Tus, increasing a parameter with a positive sensitivity index holding other parameters constant increases the reproduction number whereas increasing a parameter with a negative sensitivity index while other factors are held constant decreases the reproduction number [37].
From Table 2, it is observed that β, χ 1 , and χ 2 have positive sensitivity index values, and thus an increase in these parameters results in a corresponding increase in the number of the infected population.On the other hand, θ 1 , θ 2 , θ 3 , and P have negative sensitivity index values, and thus an increase in these parameters results in a decrease in the number of infected populations.For instance, if the transmission rate, β, is increased by 10%, the reproduction numbers increase by 10%.On the other hand, increasing screening rate of latent infected, θ 1 , by 10% decreases R cVST by 6.673668% while increasing screening rate of asymptomatic infectious, θ 2 , by 10% decreases R cVST by 0.751639%.

Efects of Various Intervention Strategies on the Control
Reproduction Number.Equations ( 10), ( 12)-( 14) represent the control reproduction numbers with various intervention strategies: R cVST for vaccination, screening, and treatment of all infected cases; R cST for screening and treatment of all infected cases; R cVT S for vaccination and treatment of symptomatic infectious population; and R cT S for treatment of symptomatic infectious population alone.Figure 2 illustrates the efects of varying reproduction numbers with respect to transmission rate, β.It is observed that R cVST < R cST < R cVT S < R cT S .Tis indicates that a combination of vaccination, screening, and treatment of all infected cases is the most efective control measure in reducing infection transmission in the population.Following this, a combination of screening and treatment of all infected cases proves efective, while the combination of vaccination and treatment of symptomatic infectious ranks as the third appropriate strategy.Furthermore, it is noted that treating symptomatic cases alone is the least efective strategy in reducing infection transmission in the population.

Efects of Varying Screening and Treatment Rates on the Control Reproduction Number.
In the model fowchart depicted in Figure 1, θ 1 represents the rate of screening latent infected individuals, with Figure 3 illustrating the efects of screening the latent infected population on the control reproduction number.Similarly, according to the model fowchart shown in Figure 1, θ 2 represents the rate of screening the asymptomatic infectious population, with Figure 4 displaying the efects of screening this population on the control reproduction number.It is observed that increasing screening rates for both latent infected and asymptomatic infectious populations reduce the control reproduction number, consequently decreasing the rate of infection transmission.Screening and treating latent infections reduce reactivation, subsequently decreasing infection transmission in the population.Te asymptomatic infectious population, experiencing no symptoms, continues their daily routines without seeking medical intervention.Tis behavior leads to more interactions with susceptible individuals, contributing to a high rate of infection transmission in the population.Terefore, it is prudent to screen and treat asymptomatic infectious individuals since they spread infections incessantly without being noticed.
In the model fowchart depicted in Figure 1, θ 3 represents the rate of treating the symptomatic infectious population, while Figure 5 shows the efects of varying treatment on the control reproduction number for this population.It is observed that increasing the treatment rate of the symptomatic infectious population reduces the control reproduction number.Tis reduction is attributed to the decrease in the population that is infectious, consequently reducing the rate of infection transmission in the population.

Conclusion
In this paper, a mathematical model of pulmonary tuberculosis incorporating an asymptomatic infectious population has been formulated.Te asymptomatic infectious population was targeted since they spread infections incessantly to susceptible populations without being noticed, thus contributing to a high rate of infection transmission.Tis was based on research fndings from a survey conducted in Kenya in 2016, which showed that 26% of active  pulmonary tuberculosis cases were asymptomatic infectious and did not seek medical care.Te model considered intervention strategies that include vaccination, screening of both latent infected and asymptomatic infectious populations, and treatment of all forms of pulmonary tuberculosis disease.Qualitative as well as numerical analyses have been performed to determine efective intervention strategies that reduce the transmission of pulmonary tuberculosis infections in the population.However, the limitations of this model, which will be considered in our next research paper, include the impact of natural immunity on the progression of individuals from latent infection to pulmonary tuberculosis disease and the waning of vaccine efcacy.
Qualitative and numerical results demonstrate that increasing the screening of asymptomatic and latently infected individuals reduces the transmission of infections to the susceptible population.Te numerical analysis indicates that the combination of vaccination, screening, and treatment of all forms of pulmonary tuberculosis disease is the most efective intervention in decreasing disease transmission.Furthermore, the results suggest that a combination of screening and treatment of all forms of pulmonary tuberculosis disease is more efective than a combination of vaccination and treatment of symptomatic infectious individuals alone.Treating the symptomatic population alone is identifed as the least efective intervention for curtailing infection transmission in the susceptible population.
Terefore, this study recommends that more attention should be directed toward screening and treatment of latent infected and the asymptomatic infectious populations.Screening and treating latent infections reduce the development of pulmonary tuberculosis disease and consequently decreases the rate of infection transmission in the population.Additionally, screening and treating the asymptomatic infectious population reduce the incessant spread of infections to susceptible individuals and, consequently, decreases the rate of infection transmission.
Te limitations of this study, which could be considered in future studies, include the optimal control theory in the presence of vaccination, screening, and treatment, as well as the stochastic-and fractional-order approaches of the model.

Figure 1 :
Figure 1: Schematic model diagram for the propagation and control of pulmonary tuberculosis disease in Kenya.

Figure 5 :
Figure 5: Efects of varying the treatment rate for symptomatic infectious individuals on the control reproduction number.

7 Figure 6 :
Figure 6: Efects of varying the screening rate for asymptomatic individuals on the symptomatic infectious population.

Figure 7 :
Figure 7: Efects of varying the screening rate for latent infected individuals on the asymptomatic infectious population. 

Table 1 :
Baseline parameters used in simulation.

Table 2 :
Sensitivity indices of reproduction numbers with respect to some model parameters.Efects of various intervention strategies on the control reproduction number.Figure 4: Efects of varying the screening rate for asymptomatic infectious individuals on the control reproduction number.