Modelling the potential impact of TB-funded prevention programs on the transmission dynamics of TB

Abstract

Tuberculosis (TB) continues to be a major global health challenge, with millions of new cases and deaths each year despite the massive efforts and funding put in the fight against the disease. In this paper, we develop a mathematical model to evaluate the impact of TB-funded prevention programs on the transmission dynamics of TB. The model incorporates stages of TB infection (latent and active), and accounts for the effects of treatment, funding and TB-funded prevention programs. Our analysis shows that increased funding and enhanced prevention programs reduce the number of active TB cases, thereby decreasing the reproduction number and TB endemicity. Specifically, higher funding rates lead to improved prevention and treatment outcomes, resulting in the lowering of the effective reproduction number $({\mathcal{R}}_0)$ and reduced transmission. The model's steady states are determined and it is shown that the model has a disease-free equilibrium that is locally asymptotically stable whenever ${\mathcal{R}}_0<1$ and multiple endemic equilibria for ${\mathcal{R}}_0^c<{\mathcal{R}}_0<1$ and a unique endemic equilibrium for ${\mathcal{R}}_0>1$. The model is shown to exhibit a backward bifurcation that vanishes as the funding for TB is increased. The paper also highlights that treatment alone, while beneficial, is less effective than a combined strategy involving funding and prevention. Numerical simulations are carried out and the influences of various parameters on the effective reproduction number are investigated. The implications of TB-funded prevention programs on TB dynamics and control of TB are discussed and valuable insights for policymakers in designing effective TB control programs are highlighted.

1. Introduction

The Global TB Report 2023 reveals that TB claimed 1.3 million lives in 2022, including 167,000 deaths among people living with HIV. As the second leading cause of infectious disease-related deaths between COVID-19 and HIV/AIDS-TB continue to pose a significant global health challenge. That year, an estimated 10.6 million new TB cases were reported worldwide, affecting 5.8 million men, 3.5 million women, and 1.3 million children, underscoring its impact across all demographics and regions. Multidrug-resistant TB (MDR-TB) remains a critical public health concern, with only 40 % of those affected receiving treatment in 2022. Since 2000, global TB control efforts have saved approximately 75 million lives. However, achieving the targets set at the 2018 UN High-Level Meeting on TB requires an annual investment of 13 billion for prevention, diagnosis, treatment, and care (WHO, 2023).

Tuberculosis (TB) is a life-threatening infectious disease caused by the bacterium Mycobacterium tuberculosis (CDC, 2023a) and has affected humanity for centuries. The bacterium is an obligatory aerobic, rod-shaped, nonmotile aerobic bacterium that needs oxygen to survive. It can enter the body through the inhalation of droplet nuclei and be detected in the lungs’ well-aerated upper lobes. Once inside the body, Mycobacterium tuberculosis mainly affects the upper lobes of the lungs, which are well-ventilated (Cruz-Knight & Blake-Gumbs, 2013). TB spreads through the air from individuals with active pulmonary TB. These infected individuals release TB bacteria into the air when coughing, sneezing, or spitting. It only takes a small number of bacteria inhaled for a person to become infected. Approximately 10 % of individuals infected with Mycobacterium tuberculosis will develop active TB, while the remaining 90 % will have a latent infection where the bacteria remain in the body without causing symptoms or being contagious. However, people with weakened immune systems, such as those with HIV or diabetes, face a higher risk of progressing to active TB (Ullah et al., 2020). Symptoms of active TB include a persistent cough (sometimes with blood), chest pain, fatigue, weakness, weight loss, fever, and night sweats. While both latent TB infection and active TB disease are treatable, untreated latent TB can progress to active disease. Without appropriate treatment, active TB can be fatal (CDC, 2023b).

To prevent tuberculosis (TB), it is crucial to implement effective TB prevention that requires funding programs. TB prevention programs consist of a series of organized strategies and actions designed to decrease both the incidence and spread of TB. These programs typically involve detecting and treating individuals with latent TB infection to stop it from developing into active TB, ensuring adherence to treatment through directly observed therapy (DOT), and applying infection control measures in healthcare settings and communities. Additionally, these programs often focus on educating the public, providing vaccinations where relevant, and enhancing access to medical care for at-risk populations (WHO, 2021).TB funding refers to the financial resources dedicated to supporting TB prevention, diagnosis, treatment, and research. Funding for TB control efforts comes from governments, international organizations, non-governmental organizations (NGOs), and private sector contributors. Sufficient financial support is crucial for implementing effective TB programs, advancing research on diagnostics and treatments, and addressing the needs of high-burden countries and vulnerable populations. In 2023, the Global Fund accounted for 76 % of all international TB financing, investing 9.2 billion in prevention and treatment programs. In 2022 alone, 1.5 million individuals exposed to TB received preventive therapy, 6.7 million underwent treatment, 118,000 were treated for drug-resistant TB, and 331,000 TB patients living with HIV received antiretroviral therapy alongside their TB treatment. In countries where the Global Fund operates, TB-related deaths dropped by 36 % between 2002 and 2022. Without these interventions, TB deaths would have increased by 129 % over the same period (Grobal, 2023).

Studying disease spread through mathematical models is a widely recognized approach for understanding how diseases progress over time. In this context, various mathematical models and statistical techniques have been developed to investigate TB and its transmission dynamics. In (Elaydi & Lozi, 2024), a tuberculosis (TB) model that includes treatment was presented, while in (Sang et al., 2023), a model with DeAngelis incidence and distributed delays to explore interactions like latent periods, reactivation, and TB treatment was developed. A TB model that incorporates age-related latency and delays in treatment was developed in (Guo et al., 2023). Oshinubiet al (Abiodun Oguntolu et al., 2022; Oshinubi et al., 2023). modelled the importance of vaccination and treatment access in reducing TB incidence. The management of TB through diagnosis, treatment was modelled in (Egonmwan & Okuonghae, 2019; Yeketi Ayinla et al., 2021). Ullah et al. (Ullah et al., 2020), found that effective treatment reduces TB cases, while incomplete treatment increases infections. The link between TB and diabetes, and the impact that diabetes has on TB spread, was investigated in (Malik et al., 2018). Optimal control models developed to examine control methods like social distancing, case detection, and treatment for TB eradication were presented in (Kereyu & Demie, 2021; Kim Jung et al., 2018).

Very few mathematical models address the impact of prevention programs on tuberculosis (TB), and none have accounted for the influence of funding for TB prevention programs. In this paper, we developed a simple model to evaluate how funding of prevention programmes affect TB transmission dynamics in the human population with the aim of quantifying the role of funding in TB prevention. This model includes various stages of TB infection, such as latent and active TB, and examines how treatment, funding levels, and prevention strategies impact disease transmission. The insights gained from this model are crucial for managing funding and refining control strategies over time. Although simple, this model offers valuable insights by incorporating the role of funding on prevention programs for TB.

The paper is structured as follows: Section 1 introduces the research, followed by model formulation in Section 2. Section 3 discusses the fundamental properties and analysis of the model, while Section 4 covers numerical simulations. Finally, Section 5 presents the conclusion.

2. Model formulation

Concerning the spread of TB infections, In this section, we formulate a mathematical model with eight compartments, namely: Susceptible S(t), latent slow L_s(t), latent fast L_f(t), infectious I(t), those on treatment H(t), level prevention programs P(t), level of funding F(t) and recovered R(t), at time t. In particular, the total population at time t, is denoted by N and defined as

$$N=S+L_s+L_f+I+H+R.$$

Susceptible Individuals refer to individuals who do not carry tuberculosis infections but, through contact, can be infected by individuals in compartment I. Individuals in L_s and L_f are infected but not contagious, Individuals in I are infected and contagious, and individuals at H are infectious and decided to go for treatment. The recovered are those who would have had the disease and are treated, but do not have lifelong immunity. We also assume that populations interact uniformly. The prevention programs P are structured strategies designed to lower the rates and transmission of tuberculosis. Other programs that focus on identifying and treating active cases, screening and managing latent TB infections, promoting vaccination, raising public awareness, enhancing infection control, providing support and follow-up, and addressing social factors are not considered in this paper. Furthermore, the program is supported by funding F, from various sources, including government agencies, international organizations, nongovernmental organizations (NGOs), kind contributions, research grants, and public-private partnerships.

Let Λ denote the number of individuals recruited into a susceptible compartment are assumed to be susceptible. Susceptible individuals are infected through contact or interaction with infectious individuals with the force of infection λ(t), given by

$$\lambda (t)=\frac{\beta I}{\left(1+\frac{P}{P_{\mathit{max}}}\right)N},$$

where β represents the transmission rate among infectious interacting with susceptible individuals.

After being infected, a proportion q of the susceptible individuals slowly progress to L_s, and the remaining proportion of (1 − q) progresses to L_f. After some time, some of the individuals in L_s progress to through reinfection to L_f with η as the relative reinfection parameter, and the remaining individuals in L_f progress to active TB at a rate of γ₂ and those in L_s progress to active TB at a rate γ₁. Infectious individuals join the treatment class at a rate ω. After treatment, individuals join the recovered class at a rate ϕ. We also assume that infected individuals cannot heal naturally. Recovered individuals relapse to the infectious class at a rate k. We assume that individuals can either die naturally at a rate μ or due to a disease-induced mortality rate δ.

The funding growth rate is influenced by an increase in infectious individuals at a rate ρ and a decrease at a rate μ_f, with F₀ representing the baseline rate of funding inflow. The prevention programs grow at a rate α and decrease at a rate μ_p, with P₀ denoting the constant rate at which the prevention programs are implemented within the population.

The model parameters in Fig. 1 are summarized in Table 1.

Fig. 1.

A compartmental model diagram for tuberculosis (TB) transmission dynamics incorporating the impact of prevention programs and funding. The model consists of eight compartments: susceptible S(t), latent slow L_s(t), latent fast L_f(t), infectious I(t), on treatment H(t), recovered R(t), prevention programs P(t), and funding F(t). Funding and prevention programs evolve dynamically based on infection levels and support structures. The continuous arrows represent movement of individuals between compartments, and the dot-dashed arrows represent how the infected population influences the growth of funding on the implementation and intensity of prevention programs.

Table 1.

Summary of the model parameters and their respective units.

Parameter	Description	Units
Λ	Recruitment rate	Individuals per time
μ	Natural mortality rate	Per time
δ	Death rate due to TB	Per time
β	Transmission rate	Per time
q	Proportion of susceptible individuals who progress Ls	Dimensionless
η	Relative reinfection parameter	Dimensionless
γ1	Progression rate from Ls to active TB,I	Per time
γ2	Progression rate from Lf to I	Per time
k	The rate of progression from R to I	Per time
ω	Treatment rate for individuals with active TB	Per time
ϕ	Recovered rate of treated individuals	Per time
ρ	Funding growth rate	Funding per individual per time
μf	Funding outflow rate	Per time
F0	Constant rate of funding inflow	Funding per time
α	Prevention program growth rate	Programs per funding per time
μp	Rate of decline of prevention programs	Per time
P0	Baseline rate of prevention programs	Programs per time

From the model assumptions and the parameter descriptions, we have the following equations:

$$\left\{\begin{array}{ll}\frac{dS}{dt}& =\mathrm{\Lambda }-\mu S-\lambda S,\\ \frac{d{L}_s}{dt}& =q\lambda S-\lambda \eta L_s-(\mu +{\gamma }_1)L_s,\\ \frac{d{L}_f}{dt}& =(1-q)\lambda S+\eta \lambda L_s-(\mu +{\gamma }_2)L_f,\\ \frac{dI}{dt}& ={\gamma }_1{L}_s+{\gamma }_2{L}_f+kR-(\mu +\delta +\omega )I,\\ \frac{dH}{dt}& =\omega I-(\mu +\varphi )H,\\ \frac{dR}{dt}& =\varphi H-(\mu +k)R,\\ \frac{dP}{dt}& =P_0+\alpha F-{\mu }_{p}P,\\ \frac{dF}{dt}& =F_0+\rho I-{\mu }_{f}F,\end{array}\right)$$ (1)

with the initial conditions S ≥ 0, L_s ≥ 0, L_f ≥ 0, I ≥ 0, H ≥ 0, R ≥ 0, P ≥ 0, and F ≥ 0.

The derivatives of the summing up of the human population, we have

$$\frac{dN}{dt}=\mathrm{\Lambda }-\mu N-\delta I.$$

In this article, we assume that Λ = μN + δI, following the assumptions in (Mulone & Straughan, 2009).

Considering the varying dimensions of the human population, prevention programs, and funding, we rescale these factors by defining:

$s=\frac{S}{N},l_s=\frac{L_s}{N},l_f=\frac{L_f}{N},i=\frac{I}{N},h=\frac{H}{N},r=\frac{R}{N},f=\frac{F}{F_{\mathit{max}}},p=\frac{P}{P_{\mathit{max}}},f_0=\frac{F_0}{F_{\mathit{max}}},p_0=\frac{P_0}{P_{\mathit{max}}},\text{and}\lambda (t)=\hat{\lambda }=\beta i\left(\frac{1}{1+p}\right)$.

This leads to the following system of equations:

$$\left\{\begin{array}{ll}\frac{ds}{dt}& =\mu +\delta i-\hat{\lambda }s-\mu s,\\ \frac{d{l}_s}{dt}& =q\hat{\lambda }s-\hat{\lambda }\eta l_s-a_0{l}_s,\\ \frac{d{l}_f}{dt}& =(1-q)\hat{\lambda }s+\hat{\lambda }\eta l_s-a_1{l}_f,\\ \frac{di}{dt}& ={\gamma }_1{l}_s+{\gamma }_2{l}_f+kr-a_{2}i,\\ \frac{dh}{dt}& =\omega i-a_{3}h,\\ \frac{dr}{dt}& =\varphi h-a_{4}r,\\ \frac{dp}{dt}& =p_0+k_{1}f-{\mu }_{p}p,\\ \frac{df}{dt}& =f_0+k_{2}i-{\mu }_{f}f,\end{array}\right)$$ (2)

where, $a_0=\mu +{\gamma }_1,a_1=\mu +{\gamma }_2,a_2=\mu +\delta +\omega ,a_3=\mu +\varphi ,a_4=\mu +k,k_1=\frac{\alpha F_{\mathit{max}}}{P_{\mathit{max}}}$ and $k_2=\frac{\rho N}{F_{\mathit{max}}}$, with initial conditions

$$s>0,l_s\ge 0,l_f\ge 0,i\ge 0,h\ge 0,r\ge 0,p>0,\text{and}f>0.$$

Note that the re-scaling gives

$$n=s+l_s+l_f+i+h+r=1.$$

3. Model analysis

We demonstrate that all solutions of system (2) are non-negative and bounded for t > 0, thus maintaining biological relevance within the feasible region Ω.

3.1. Positivity of solutions

The positivity of the solution to our new model is captured in the following lemma.

Lemma 1

Supposes(0) > 0, l_s(0) ≥ 0, l_f(0) ≥ 0, i(0) > 0, h(0) ≥ 0, r(0) ≥ 0, p(0) > 0, and f(0) > 0, then the solutionss(t), l_s(t), l_f(t), i(t), h(t), r(t), p(t)and f(t)of system 2 are non-negative for all timet ≥ 0.

Proof. Let (s(t), l_s(t), l_f(t), i(t), h(t), r(t), p(t), f(t)) be the solution of system (2). From the first equation of system (2), we have:

$$\begin{array}{ll}\frac{ds}{dt}& =\mu +\delta i-\mu s-\hat{\lambda }s,\\ & \ge -(\mu +\hat{\lambda })s.\end{array}$$ (3)

By integrating both sides and applying the initial conditions, we obtain:

$$s(t)\ge s(0)e^{-\left({\int }_0^t\hat{\lambda }(\tau )d\tau +\mu t\right)},$$

for any t ≥ 0. Since s(0) > 0, we conclude that s(t) > 0.

Additionally, from the second equation of system (2), we have:

$$\begin{array}{ll}\frac{d{l}_s}{dt}& =q\hat{\lambda }s-\hat{\lambda }\eta l_s-a_0{l}_s,\\ & \ge -(\hat{\lambda }\eta +a_0)l_s.\end{array}$$ (4)

Integrating both sides and applying the initial conditions gives:

$$l_s(t)\ge l_s(0)e^{-\left({\int }_0^t\eta \hat{\lambda }(\tau )d\tau +a_{0}t\right)},$$

for any t ≥ 0. Since l_s(0) ≥ 0, it follows that l_s(t) ≥ 0.

Similarly, it can be shown that:

$$l_f(t)\ge 0,i(t)\ge 0,h(t)\ge 0,r(t)\ge 0,p(t)>0,\text{and}f(t)>0.$$

3.2. Feasible region

Consider the biologically feasible region given by

$$\mathrm{\Omega }=\left\{\chi \in {\mathbb{R}}_{+}^8:n=1,p\le \frac{p_0}{{\mu }_p}+\frac{k_1}{{\mu }_p{\mu }_f}(f_0+k_2),f\le \frac{1}{{\mu }_f}(f_0+k_2)\right\},$$

where χ = (s(t), l_s(t), l_f(t), i(t), h(t), r(t), p(t), f(t)).

Lemma 2

The solution of system (2) with non-negative initial conditions remains bounded for all t ≥ 0 within the biologically feasible region Ω.

Proof. From system (2), we observe that n = 1.

$$\frac{dn}{dt}=0.$$

Additionally, from system 2 we have that

$$\begin{array}{ll}\frac{df}{dt}& =f_0+k_{2}i-{\mu }_{f}f,\\ & \le \left(f_0+k_2\right)-{\mu }_{f}f,\text{since}i\le 1.\end{array}$$

By solving the differential inequality, we have

$$f(t)\le \frac{1}{{\mu }_f}(f_0+k_2)+\left(f(0)-\frac{1}{{\mu }_f}(f_0+k_2)\right)e^{-{\mu }_{f}t}.$$

Therefore, as t → ∞, we have that $f(t)\le \frac{f_0+k_2}{{\mu }_f}$.

Lastly, from system 2 we have that

$$\begin{array}{ll}\frac{dp}{dt}& =p_0+k_{1}f-{\mu }_{p}p,\\ & \le p_0+k_1\left(\frac{f_0+k_2}{{\mu }_f}\right)-{\mu }_{p}p,\text{since}f\le \frac{f_0+k_2}{{\mu }_f}.\end{array}$$

Solving the differential inequality gives,

$$p(t)\le \left(\frac{p_0}{{\mu }_p}+\frac{k_1}{{\mu }_p{\mu }_f}(f_0+k_2)\right)+\left(p(0)-\frac{p_0}{{\mu }_p}+\frac{k_1}{{\mu }_p{\mu }_f}(f_0+k_2)\right)e^{-{\mu }_{p}t}.$$

Thus, as t → ∞ we have that

$$p(t)\le \frac{p_0{\mu }_f+k_1{f}_0+k_1{k}_2}{{\mu }_p{\mu }_f}.$$

Hence s(t), l_s(t), l_f(t), i(t), h(t), r(t), p(t) and f(t) are bounded in the biologically feasible region Ω.

3.3. Model steady states analysis

3.3.1. Disease-free equilibrium (DFE)

We obtain the disease-free equilibrium by assuming that the disease-induced states l_s = l_f = i = h = r = 0. This point is obtained by setting the right-hand side of the model 2 equal to zero. Therefore, the DFE is given by

$$E_0^{\ast }=\left(s^{\ast },l_s^{\ast },l_f^{\ast },i^{\ast },h^{\ast },r^{\ast },p^{\ast },f^{\ast }\right)=\left(1,0,0,0,0,0,\frac{p_0{\mu }_f+k_1{f}_0}{{\mu }_p{\mu }_f},\frac{f_0}{{\mu }_f}\right).$$ (5)

3.3.2. The effective reproduction number

In epidemiology, the effective reproduction number ${\mathcal{R}}_0$ represents the average number of secondary infections generated by a single infectious individual in a fully susceptible population. It indicates the expected number of new infections from an initial case in a population where everyone is susceptible ((Dietz, 1993) (Liana & Chuma, 2023)). By applying the next-generation matrix FV⁻¹ as outlined in (Van den Driessche & Watmough, 2002) and using the disease-free equilibrium given in equation (5), we calculate the epidemiological effective reproduction number ${\mathcal{R}}_0$ for model 2 as follows:

$$F{V}^{-1}=\left[\frac{\partial {\mathcal{F}}_i(E_0)}{\partial x_j}\right]\times {\left[\frac{\partial {\mathcal{V}}_i(E_0)}{\partial x_j}\right]}^{-1}$$

where, ${\mathcal{F}}_i$ is the rate of appearance of new infections in compartment i, and ${\mathcal{V}}_i$ is the rate of other transitions between compartment i and other infected compartments ((Van den Driessche, 2017)). Taking into account the equations for infected compartments, l_s, l_f, i, h, r, p, and f from the model system (2) we have:

$$\left\{\begin{array}{ll}\frac{d{l}_s}{dt}& =q\hat{\lambda }s-\hat{\lambda }\eta l_s-a_0{l}_s,\\ \frac{d{l}_f}{dt}& =(1-q)\hat{\lambda }s+\hat{\lambda }\eta l_s-a_1{l}_f,\\ \frac{di}{dt}& ={\gamma }_1{l}_s+{\gamma }_2{l}_f+kr-a_{2}i.\\ \frac{dh}{dt}& =\omega i-a_{3}h,\\ \frac{dr}{dt}& =\varphi h-a_{4}r,\\ \frac{dp}{dt}& =p_0+k_{1}f-{\mu }_{p}p,\\ \frac{df}{dt}& =f_0+k_{2}i-{\mu }_{f}f,\end{array}\right)$$ (6)

Now, from the equation system (6), we get the following matrices from which we evaluate the spectral radius of $\mathcal{F}{\mathcal{V}}^{-1}$ given by $\rho (\mathcal{F}{\mathcal{V}}^{-1})$, $\mathcal{F}$ and $\mathcal{V}$ can be partially differentiated with respect to l_s, l_f, i, h, r, p, and f at the DFE to produce

$$\mathcal{F}=\left(\begin{array}{lllllll}0& 0& \frac{q\beta {\mu }_p{\mu }_f}{p_0{\mu }_f+{\mu }_p{\mu }_f+k_1{f}_0}& 0& 0& 0& 0\\ 0& 0& \frac{(1-q)\beta {\mu }_p{\mu }_f}{p_0{\mu }_f+{\mu }_p{\mu }_f+k_1{f}_0}& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\end{array}\right),\mathcal{V}=\left(\begin{array}{lllllll}{a}_0& 0& 0& 0& 0& 0& 0\\ 0& a_1& 0& 0& 0& 0& 0\\ -{\gamma }_1& -{\gamma }_2& a_2& 0& -k& 0& 0\\ 0& 0& -\omega & a_3& 0& 0& 0\\ 0& 0& 0& -\varphi & a_4& 0& 0\\ 0& 0& 0& 0& 0& -k_1& {\mu }_p\\ 0& 0& -k_2& 0& 0& 0& {\mu }_f\end{array}\right).$$

Upon computation and simplification, the effective reproduction number $({\mathcal{R}}_0=\rho (\mathcal{F}{\mathcal{V}}^{-1}))$ is found to be;

$$\begin{array}{ll}{\mathcal{R}}_0& =R_{l_s}+R_{l_f}\text{with}\\ R_{l_s}& =\frac{\beta {\mu }_p{\mu }_{f}q{\gamma }_1}{a_0{a}_2\left(1-{\psi }_4\right)\left((p_0+{\mu }_p){\mu }_f+k_1{f}_0\right)}\text{and}{R}_{l_f}& =\frac{\beta {\mu }_p{\mu }_f(1-q){\gamma }_2}{a_1{a}_2\left(1-{\psi }_4\right)\left((p_0+{\mu }_p){\mu }_f+k_1{f}_0\right)},\end{array}$$ (7)

where, $R_{l_s}$ and $R_{l_f}$ represents the contribution of individuals in class l_s and l_f with the presence of prevention programs and funding.

Based on Theorem 2 from ((Van den Driessche & Watmough, 2002)), we can describe the stability of the disease-free equilibrium E∗ as follows:

Theorem 3

The disease-free equilibrium,$E_0^{\ast }$, is locally asymptotically stable if${\mathcal{R}}_0<1$and unstable otherwise.

3.3.3. The endemic equilibrium

Solving the last equation of the model system 2 for f∗, we obtain

$$f^{\ast }=\frac{k_2{i}^{\ast }+f_0}{{\mu }_f}.$$

Solving second last equation of the model system 7 for p∗, we obtain

$$p^{\ast }=\frac{k_1{k}_2{i}^{\ast }+p_0{\mu }_f+k_1{f}_0}{{\mu }_p{\mu }_f}.$$

Likewise, solving the fifth, sixth, first, second, and third equations of system (2) respectively for $h^{\ast },r\ast ,s^{\ast },l_s^{\ast }$, and $l_f^{\ast }$ in terms of i∗ gives

$$\left\{\begin{array}{ll}{h}^{\ast }& =\frac{\omega i^{\ast }}{a_3},\\ r^{\ast }& =\frac{\varphi \omega i^{\ast }}{a_3{a}_4},\\ s^{\ast }& =\frac{(\delta i^{\ast }+\mu )\left(\left(p_0+{\mu }_p\right){\mu }_f+k_1\left(k_2{i}^{\ast }+f_0\right)\right)}{\left(\mu p_0+(\beta i^{\ast }+\mu ){\mu }_p\right){\mu }_f+\mu k_1\left(k_2{i}^{\ast }+f_0\right)},\\ l_s^{\ast }& =\frac{q\beta i^{\ast }(\delta i^{\ast }+\mu ){\mu }_p{\mu }_f\left((p_0+{\mu }_p)\right){\mu }_f+k_1(k_2{i}^{\ast }+f_0)}{((\mu p_0+(\beta i^{\ast }+\mu ){\mu }_p){\mu }_f+\mu k_1(k_2{i}^{\ast }+f_0))(\beta {\mu }_p{\mu }_f\eta i^{\ast }+a_0((p_0+{\mu }_p){\mu }_f+k_1(k_2{i}^{\ast }+f_0)))},\\ l_f^{\ast }& =\frac{\beta i^{\ast }(\delta i^{\ast }+\mu ){\mu }_p{\mu }_f(\beta \eta i^{\ast }{\mu }_p{\mu }_f+(1-q)a_0((p_0+{\mu }_P){\mu }_f+k_1(k_2{i}^{\ast }+f_0)))}{a_1((\mu p_0+(\beta i^{\ast }+\mu ){\mu }_p){\mu }_f+\mu k_1(k_2{i}^{\ast }+f_0))(\beta \eta i^{\ast }{\mu }_p{\mu }_f+a_0((p_0+{\mu }_p){\mu }_f+k_1(k_2{i}^{\ast }+f_0)))}.\end{array}\right)$$ (8)

Substituting the values of $l_s^{\ast },l_f^{\ast }$, and r∗ into the fourth equation of system (7) in terms of i∗ leads to the following third-order polynomial equation in terms of i∗:

$$i^{\ast }\left({\xi }_2{i^{\ast }}^2+{\xi }_1{i}^{\ast }+{\xi }_0\right)=0,‘$$ (9)

where

$$\left\{\begin{array}{ll}{\xi }_2& =\beta {\mu }_p{\mu }_f((a_1{k}_1{k}_{2}k\eta \mu \varphi \omega (1-{\psi }_1)+a_3{a}_{4}q\delta {\gamma }_1)+\beta \eta (a_{1}k\varphi \omega (1-{\psi }_1)+\delta a_3{a}_4{\gamma }_2){\mu }_p{\mu }_f)+a_0{k}_1{k}_2{\psi }_2,\\ {\xi }_1& =a_0\left(a_{1}k\varphi \omega (1-{\psi }_1)(2\mu k_1{k}_2+\beta {\mu }_p{\mu }_f)((p_0+{\mu }_p){\mu }_f+k_1{f}_0)+G_1\right)+\beta {\mu }_p{\mu }_f{G}_2,\\ {\xi }_0& =\mu a_0{a}_{1}k\varphi \omega (1-{\psi }_1){\left((p_0+{\mu }_p){\mu }_f+k_1{f}_0\right)}^2\left(1-{\mathcal{R}}_0\right),\end{array}\right)$$ (10)

$$\left\{\begin{array}{ll}{G}_1& =(1-q)(\beta a_3{a}_4{\gamma }_2{\mu }_p{\mu }_f(\delta {\mu }_f(p_0+{\mu }_p)+k_1(\mu k_2+\delta f_0))),\\ G_2& =\beta \eta \mu a_3{a}_4{\gamma }_2{\mu }_p{\mu }_f+a_1(k\eta \mu \varphi \omega (p_0+{\mu }_p){\mu }_f+k_1{f}_0)+a_3{a}_{4}q\delta {\gamma }_1(1-{\psi }_3)(\mu p_0+{\mu }_p{\mu }_f)+k_1{G}_3,\\ G_3& =q\mu k_2{\gamma }_1+q\delta {\gamma }_1(1-{\psi }_3)f_0,\\ {\psi }_1& =\frac{a_2{a}_3{a}_4}{k\varphi \omega },\\ {\psi }_2& =((1-q)\beta \delta a_3{a}_4{\gamma }_2{\mu }_p{\mu }_f)+a_{1}k\varphi \omega (1-{\psi }_1)(\mu k_1{k}_2+\beta {\mu }_p{\mu }_f),\\ {\psi }_3& =\frac{\eta \mu a_2}{q\delta {\gamma }_1}.\\ {\psi }_4& =\frac{k\varphi \omega }{a_2{a}_3{a}_4}.\end{array}\right)$$ (11)

Fig. 2 illustrates the backward bifurcation for the parameter values given in the caption. It shows a locally stable disease-free equilibrium (DFE) and both unstable and stable endemic equilibria (EE) for ${\mathcal{R}}_0<1$. Effective disease control requires a globally stable DFE when ${\mathcal{R}}_0<1$. Here, the DFE is globally stable only below the threshold $R_0^c\approx 0.7$, complicating disease elimination as this value is significantly below unity for the selected parameters.

Fig. 2.

The backward bifurcation for the dynamics of TB with the following parameter values: Λ = 0.8, μ = 0.014, δ = 0.04, q = 0.2, η = 0.001, γ₁ = 0.002, γ₂ = 0.04, k = 0.21, ω = 0.21, ϕ = 0.3, μ_p = 0.45, μ_f = 0.7, p₀ = 0.26, k₁ = 0.4, k₂ = 0.5, f₀ = 0.12, β = 0.7. The units of the infected population are given as a proportion of the total population.

Fig. 3 illustrates the impact of varying funding levels for tuberculosis (TB) prevention programs on TB dynamics. The results show that increasing funding for prevention programs helps eliminate the backward bifurcation, which in turn reduces the infection rate.

Fig. 3.

The effects of varying k₂ as the funding to the elimination of the disease. The black dot represents the critical point.

Solving (9) gives i∗ = 0, which corresponds to the disease-free equilibrium. The endemic equilibrium is obtained by solving the following quadratic equation in i,

$${\xi }_2{i^{\ast }}^2+{\xi }_1{i}^{\ast }+{\xi }_0=0.$$ (12)

Table 2 below shows various possibilities for the existence of the endemic equilibrium. Here, i∗ denotes the number of positive roots.

Table 2.

Shows the number of possible positive solutions for i∗.

	ξ2 > 0
	ξ1 > 0		ξ1 < 0
	ξ0 > 0(R0 < 1)	ξ0 < 0(R0 > 1)	ξ0 > 0(R0 < 1)	ξ0 < 0(R0 > 1)
i∗	0	1	2	1

Here we summarise the results on the existence of the endemic equilibrium of system (2) in Theorem (4).

Theorem 4

System (2) has

T₁: a unique endemic equilibrium$E^{\ast }=(s^{\ast },l_s^{\ast },l_f^{\ast },i^{\ast },h^{\ast },r^{\ast },p^{\ast },f^{\ast })$if and only ifξ₂ > 0, and ξ₀ < 0, ${\mathcal{R}}_0>1$, or when ${\xi }_1^2-4{\xi }_2{\xi }_0=0$,

T₂: two endemic equilibria

$$E_1^{\ast }=(s_1^{\ast },{l_s^{\ast }}_1,{l_f^{\ast }}_1,i_1^{\ast },h_1^{\ast },r_1^{\ast },p_1^{\ast },f_1^{\ast }),$$

and

$$E_2^{\ast }=(s_2^{\ast },{l_s^{\ast }}_2,{l_f^{\ast }}_2,i_2^{\ast },h_2^{\ast },r_2^{\ast },p_2^{\ast },f_2^{\ast }),$$

whenξ₂ > 0, ξ₁ < 0 and ξ₀ > 0, ${\mathcal{R}}_0<1$ where

$$i^{\ast }=\left(\frac{-{\xi }_1\pm \sqrt{{{\xi }_1}^2-4{\xi }_2{\xi }_0}}{2{\xi }_2}\right),$$

T₃: no endemic equilibrium otherwise.

3.3.4. Bifurcation analysis

According to (T₁) from Theorem (4), a unique endemic equilibrium exists whenever ${\mathcal{R}}_0>1$. Additionally, (T₂) highlights the potential for backward bifurcation. In epidemic models, backward bifurcation means a stable disease-free equilibrium (DFE) can coexist with a stable endemic equilibrium even when the effective reproduction number is less than unity, as discussed in (Gumel, 2012). This phenomenon suggests that having ${\mathcal{R}}_0<1$ alone is not sufficient for disease elimination; instead, the success of elimination efforts depends on the critical parameter value at the bifurcation point, as detailed in (Hadeler & Van den Driessche, 1997).

A critical value ${\mathcal{R}}_0^c$ in the context of backward bifurcation refers to the threshold value of a key parameter, typically the basic reproduction number, at which the system's behaviour changes from having only a stable disease-free state to allowing the coexistence of both disease-free and endemic states. It marks the lowest value at which a stable endemic equilibrium can exist, meaning that even if the reproduction number is brought below one, the disease may still persist in the population. This highlights the need for stronger interventions to completely eliminate the disease (Martcheva, 2015). To find this critical value, we set the discriminant of equation (12) to zero and solve for ${\mathcal{R}}_0$, denoted as ${\mathcal{R}}_0^c$, and given by

$${\mathcal{R}}_0^c=1-\frac{{\xi }_1^2}{4\left(\mu a_0{a}_{1}k\varphi \omega (1-{\psi }_1){\left((p_0+{\mu }_p){\mu }_f+k_1{f}_0\right)}^2\right){\xi }_2}.$$

Therefore a backward bifurcation exists for ${\mathcal{R}}_0^c<{\mathcal{R}}_0<1$. To confirm its existence in a model (2), we apply the Centre Manifold Theory as outlined in (Castillo-Chavez & Song, 2004). This theory helps determine whether a forward bifurcation occurs (where the disease-free equilibrium is locally asymptotically stable when ${\mathcal{R}}_0<1$, and the endemic equilibrium is stable when ${\mathcal{R}}_0>1$) or a backward bifurcation arises (where both the disease-free and endemic equilibria coexist when ${\mathcal{R}}_0<1$). To apply the Centre Manifold Theory, we first introduce the following change of variables:

$$y_1=s,y_2=l_s,y_3=l_f,y_4=i,y_5=h,y_6=r,y_7=p,y_8=f.$$

Model (2) can be rewritten as

$$\left\{\begin{array}{ll}{f}_1=\frac{d{y}_1}{dt}& =\mu +\delta y_4-(\hat{\lambda }+\mu )y_1,\\ f_2=\frac{d{y}_2}{dt}& =q\hat{\lambda }{y}_1-\hat{\lambda }\eta y_2-a_0{y}_2,\\ f_3=\frac{d{y}_3}{dt}& =(1-q)\hat{\lambda }{y}_1+\hat{\lambda }\eta y_2-a_1{y}_3,\\ f_4=\frac{d{y}_4}{dt}& ={\gamma }_1{y}_2+{\gamma }_2{y}_3+k{y}_6-a_2{y}_4,\\ f_5=\frac{d{y}_5}{dt}& =\omega y_4-a_3{y}_5,\\ f_6=\frac{d{y}_6}{dt}& =\varphi y_5-a_4{y}_6,\\ f_7=\frac{d{y}_7}{dt}& =p_0+k_1{y}_8-{\mu }_p{y}_7,\\ f_8=\frac{d{y}_8}{dt}& =f_0+k_2{y}_4-{\mu }_f{y}_8,\end{array}\right)$$ (13)

where $\hat{\lambda }=\beta y_4\left(\frac{1}{1+y_7}\right)$.

Theorem 5

Model system (13) undergoes a backward bifurcation close to ${\mathcal{R}}_0=1$ whenever the computed bifurcation coefficients a and b are positive.

Proof. Let the bifurcation parameter β = β∗ and ${\mathcal{R}}_0=1$. The Jacobian matrix, $J(E_0^{\ast }){|}_{\beta ={\beta }^{\ast }}$, of the transformed system (13) is given by

$$J(E_0^{\ast }){|}_{\beta ={\beta }^{\ast }}=\left(\begin{array}{llllllll}-\mu & 0& 0& \delta -\frac{{\beta }^{\ast }{\mu }_p{\mu }_f}{{\mu }_p{\mu }_f+p_0{\mu }_f+k_1{f}_0}& 0& 0& 0& 0\\ 0& -a_0& 0& q\left(\frac{{\beta }^{\ast }{\mu }_p{\mu }_f}{p_0{\mu }_f+{\mu }_p{\mu }_f+k_1{f}_0}\right)& 0& 0& 0& 0\\ 0& 0& -a_1& (1-q)\left(\frac{{\beta }^{\ast }{\mu }_p{\mu }_f}{p_0{\mu }_f+{\mu }_p{\mu }_f+k_1{f}_0}\right)& 0& 0& 0& 0\\ 0& {\gamma }_1& {\gamma }_2& -a_2& 0& k& 0& 0\\ 0& 0& 0& \omega & -a_3& 0& 0& 0\\ 0& 0& 0& 0& \varphi & -a_4& 0& 0\\ 0& 0& 0& 0& 0& 0& -{\mu }_p& k_1\\ 0& 0& 0& k_2& 0& 0& 0& -{\mu }_f\end{array}\right).$$

The right eigenvectors $w={[w_1,w_2,w_3,w_4,w_5,w_6,w_7,w_8]}^T$ is obtained and found to be

$$\begin{array}{ll}{w}_1& =-\frac{((k+\mu )(q{\gamma }_2\delta +\mu ({\gamma }_2+\delta +\mu )+{\gamma }_1({\gamma }_2+(1-q)\delta +\mu ))(\mu +\varphi )+({\gamma }_1+\mu )({\gamma }_2+\mu )(k+\mu +\varphi )\omega ){\mu }_f}{k_2(k+\mu )(((1-q){\gamma }_2\mu )+{\gamma }_1({\gamma }_2+q\mu ))(\mu +\varphi )},\\ w_2& =\frac{q({\gamma }_2+\mu )((k+\mu )(\delta +\mu )(\mu +\varphi )+\mu (k+\mu +\varphi )\omega ){\mu }_f}{k_2(k+\mu )(((1-q){\gamma }_2\mu )+{\gamma }_1({\gamma }_2+q\mu ))(\mu +\varphi )},\\ w_3& =\frac{(1-q)({\gamma }_1+\mu )((k+\mu )(\delta +\mu )(\mu +\varphi )+\mu (k+\mu +\varphi )\omega ){\mu }_f}{k_2(k+\mu )(((1-q){\gamma }_2\mu )+{\gamma }_1({\gamma }_2+q\mu ))(\mu +\varphi )},\\ w_4& =\frac{{\mu }_f}{k_2},w_5=\frac{\omega {\mu }_f}{k_2{a}_3},\\ w_6& =\frac{\varphi \omega {\mu }_f}{k_2{a}_4{a}_3},w_7=\frac{k_1}{{\mu }_p},w_8=1.\end{array}$$ (14)

Also its left eigenvectors U = [u₁, u₂, u₃, u₄, u₅, u₆, u₇, u₈] corresponding to $J(E_0^{\ast }){|}_{\beta ={\beta }^{\ast }}$ are calculated and determined to be

$$\begin{array}{ll}{u}_1& =0,u_2=\frac{{\gamma }_1{a}_4}{k{a}_0},u_3=\frac{{\gamma }_2{a}_4}{k{a}_1},u_4=\frac{a_4}{k},\\ u_5& =\frac{\varphi }{a_3},u_6=1,u_7=u_8=0.\end{array}$$ (15)

To determine the values of a and b, as outlined in Theorem 4.1 by (Castillo-Chavez & Song, 2004), we need to compute the non-vanishing partial derivatives of f₂, f₃, f₄, f₅, and f₇ at the disease-free equilibrium (DFE), given that u₁ = u₇ = u₈ = 0. For the system in (3.3.4), the associated non-zero partial derivatives of f₂, f₃, f₄, f₅, and f₇ at the DFE are calculated as follows:

$$\begin{array}{ll}\frac{{\partial }^2{f}_2}{\partial y_1\partial y_4}& =\frac{{\partial }^2{f}_2}{\partial y_4\partial y_1}=\frac{q{\beta }^{\ast }{\mu }_p{\mu }_f}{{\mu }_p{\mu }_f+p_0{\mu }_f+k_1{f}_0},\frac{{\partial }^2{f}_2}{\partial y_2\partial y_4}=\frac{{\partial }^2{f}_2}{\partial y_4\partial y_2}=\frac{-{\beta }^{\ast }\eta {\mu }_p{\mu }_f}{{\mu }_p{\mu }_f+p_0{\mu }_f+k_1{f}_0},\\ \frac{{\partial }^2{f}_3}{\partial y_1\partial y_4}& =\frac{{\partial }^2{f}_3}{\partial y_4\partial y_1}=\frac{(1-q){\beta }^{\ast }{\mu }_p{\mu }_f}{{\mu }_p{\mu }_f+p_0{\mu }_f+k_1{f}_0},\frac{{\partial }^2{f}_3}{\partial y_2\partial y_4}=\frac{{\partial }^2{f}_3}{\partial y_4\partial y_2}=\frac{{\beta }^{\ast }\eta {\mu }_p{\mu }_f}{{\mu }_p{\mu }_f+p_0{\mu }_f+k_1{f}_0}.\end{array}$$ (16)

Therefore,

$$\begin{array}{ll}a& =u_2\sum _{i,j=1}^8{w}_i{w}_j\frac{{\partial }^2{f}_2}{\partial y_i\partial y_j}+u_3\sum _{i,j=1}^8{w}_i{w}_j\frac{{\partial }^2{f}_3}{\partial y_i\partial y_j}+u_4\sum _{i,j=1}^8{w}_i{w}_j\frac{{\partial }^2{f}_4}{\partial y_i\partial y_j}\\ & +u_5\sum _{i,j=1}^8{w}_i{w}_j\frac{{\partial }^2{f}_5}{\partial y_i\partial y_j}+u_6\sum _{i,j=1}^8{w}_i{w}_j\frac{{\partial }^2{f}_6}{\partial y_i\partial y_j},\\ & =\left(\frac{2w_4{\beta }^{\ast }{\mu }_p{\mu }_f}{{\mu }_p{\mu }_f+p_0{\mu }_f+k_1{f}_0}\right)\left(\frac{{\mu }_f{N}^{\ast }}{k{k}_2({\gamma }_1+\mu )({\gamma }_2+\mu )(\mu +\varphi )({\gamma }_1({\gamma }_2+\mu q)+{\gamma }_2\mu (1-q))}\right)\end{array}$$ (17)

where $N^{\ast }=\left(\eta \mu q({\gamma }_2-{\gamma }_1)({\gamma }_2+\mu )((\delta +\mu )(k+\mu )(\mu +\varphi )+\mu \omega (k+\mu +\varphi ))+({\gamma }_1({\gamma }_2+\mu q)+{\gamma }_2\mu (1-q))(\omega ({\gamma }_1+\mu )({\gamma }_2+\mu )(k+\mu +\varphi )+(k+\mu )(\mu +\varphi )(\mu ({\gamma }_2+\delta +\mu )+{\gamma }_1({\gamma }_2+\mu +\delta (1-q)+{\gamma }_2\delta q)))\right)$ Also to determine the value of b,the associated non-zero partial derivatives of f₂, f₃, f₄, f₅, and f₇ at the DFE are

$$\frac{{\partial }^2{f}_2}{\partial y_4\partial {\beta }^{\ast }}=\frac{q{\mu }_p{\mu }_f}{{\mu }_p{\mu }_f+p_0{\mu }_f+k_1{f}_0},\frac{{\partial }^2{f}_2}{\partial y_4\partial {\beta }^{\ast }}=\frac{(1-q){\mu }_p{\mu }_f}{{\mu }_p{\mu }_f+p_0{\mu }_f+k_1{f}_0}.$$ (18)

So,

$$\begin{array}{ll}b& =u_2\sum _{i=1}^8{w}_i\frac{{\partial }^2{f}_2}{\partial y_i\partial {\beta }^{\ast }}+u_3\sum _{i=1}^8{w}_i\frac{{\partial }^2{f}_3}{\partial y_i\partial {\beta }^{\ast }},\\ & =u_2{w}_4\left(\frac{q{\mu }_p{\mu }_f}{{\mu }_p{\mu }_f+p_0{\mu }_f+k_1{f}_0}\right)+u_3{w}_4\left(\frac{(1-q){\mu }_p{\mu }_f}{{\mu }_p{\mu }_f+p_0{\mu }_f+k_1{f}_0}\right)>0.\end{array}$$ (19)

The bifurcation coefficient b remains positive. Thus, according to Theorem (4.1) of (Castillo-Chavez & Song, 2004), model 2 experiences a backward bifurcation at ${\mathcal{R}}_0=1$ whenever the bifurcation coefficient a > 0. This implies the condition γ₂ > γ₁. □

4. Numerical simulations

4.1. Parameter estimation

To estimate the unknown parameters of the model, we employed a calibration process grounded in data obtained from South Africa. Specifically, we used annual TB incidence data spanning from 1990 to 2019, sourced from the Institute for Health Metrics and Evaluation (IHME) (Institute for Health Metrics and Evaluation (IHME), 2024), and total population data from the World Bank (World Bank, 2024). The incidence data covers all age groups. To maintain consistency and eliminate potential biases due to population growth over time, we assumed a constant population by averaging the total population over the 30 years. Each year's TB incidence value was then divided by this average to express the incidence as a proportion of the population, facilitating direct comparison with the model output. All model parameters were estimated using this normalized data, except for the natural human death rate, denoted by μ, which was derived from demographic statistics. Based on World Bank data (World Bank, 2022), the average life expectancy in South Africa is 64.2 years, and we thus set:

$$\mu =\frac{1}{64.2}\text{per year}.$$

The mathematical model, as described in Equation (2), can be expressed in the general form:

$$\frac{dx}{dt}=g(t,x,\theta ),x(t_0)=x_0$$ (20)

where x is the vector of state variables, and θ is the vector of parameters to be estimated. The estimated parameter values obtained through this process are summarized in Table 3.

Table 3.

Estimated parameter values used in the model.

Parameter	Estimated Value (year−1)	Source
β	0.6651	Fitted
γ1	0.01355	Fitted
γ2	0.07437	Fitted
q	0.5405	Fitted
η	0.09408	Fitted
δ	0.1200	Fitted
ω	0.04114	Fitted
ϕ	0.1371	Fitted
μ	0.0156	World Bank (2022)
k	0.02074	Fitted
p0	0.5316	Fitted
f0	0.5286	Fitted
k1	9.400 × 10−4	Fitted
k2	0.9907	Fitted
μp	0.3277	Fitted
μf	0.8958	Fitted

4.2. Model fitting

The model was fitted to the normalized TB incidence data using a least-squares approach. To solve the underlying system of ordinary differential equations, we used MATLAB's ode45 solver, which is suitable for non-stiff systems. The discrepancy between the model output and the observed data was quantified using the sum of squared errors (SSE), given by:

$$\mathrm{\Phi }(\theta )=\sum _{i=1}^m{\left(x_i-{\hat{x}}_i\right)}^2$$ (21)

Here, ${\hat{x}}_i$ denotes the observed data at time t_i, and x_i = x(t_i, θ) represents the corresponding model prediction. The goal of the optimization process is to find the parameter set θ that minimizes this error:

$$\underset{\theta }{\min }\mathrm{\Phi }(\theta )$$ (22)

subject to the model dynamics defined in Equation (20). Parameter optimization was performed using MATLAB's fminsearch function, which applies the Nelder–Mead simplex method. This iterative method searches for the parameter values that best align the model output with the observed data. The quality of the model fit is demonstrated in Fig. 4, which shows the comparison between the model predictions and the observed TB incidence data.

Fig. 4.

Model fitting results showing the comparison between the predicted TB incidence (solid curve) and the normalized observed data (dots). The model was calibrated by minimizing the sum of squared errors (SSE) between the model output and the data using a least-squares approach. The system of ordinary differential equations was solved using MATLAB's ode45 solver, and parameter optimization was carried out using MATLAB's fminsearch function based on the Nelder–Mead simplex method.

4.3. Sensitivity analysis

Sensitivity analysis is a crucial method for evaluating model performance and identifying the parameters that exert the greatest or least influence on its output (Marino et al., 2008). This study employs the Partial Rank Correlation Coefficient (PRCC), a well-established global sensitivity analysis technique recognized for its reliability and efficiency in sampling-based approaches. PRCC effectively measures the impact of variations in individual parameters on model output while accounting for the linear dependencies of other parameters (McKay et al., 2000).

To compute the PRCC values, the Latin Hypercube Sampling (LHS) method is applied to the input parameters. This technique utilizes stratified sampling without replacement, enhancing the robustness of the analysis. Each model parameter follows a uniform distribution, with independent sampling performed for each. A total of 1000 simulations are conducted, and PRCC values are calculated for the model outputs at designated time points. The resulting parameter indices of ${\mathcal{R}}_0$ are presented in Table 3 and illustrated in Fig. 5. PRCC values range between −1 and 1, where a positive value indicates a direct proportional relationship between the parameter and the effective reproduction number${\mathcal{R}}_0$, whereas a negative value signifies an inverse relationship. Parameters γ₁, γ₂, β, q, μ_f, μ_p, ϕ, and k exhibit positive PRCC values, implying that an increase in these parameters will lead to a rise in ${\mathcal{R}}_0$. Conversely, parameters ω, μ, δ, f₀, p₀, and k₁ have negative PRCC values, indicating that increasing these parameters will contribute to a reduction in ${\mathcal{R}}_0$.

Fig. 5.

Tornado diagram illustrating the most influential parameters affecting TB dynamics, namely β, δ, ω, k₁, μ_f, and μ_p.

Fig. 5 highlights the most influential parameters, including β, δ, ω, k₁, μ_f, and μ_p, emphasizing their significant role in shaping disease dynamics. An increase in the contact rate (β) and the outflow rates of funding (μ_f) and prevention programs (μ_p) will amplify disease transmission, resulting in a higher number of TB-infected individuals within the community. Conversely, enhancing the inflow rate of prevention programs through increased funding (k₁) and improving the treatment rate (ω) will effectively reduce the spread of infection and lower the overall disease burden in the population.

4.3.1. The impact of varying funding and prevention programs removal rate

Using the parameter values in Table 3, Fig. 6 demonstrates how changes in the funding removal rate (μ_f) and prevention program removal rate (μ_p) influence the number of infectious individuals, i(t), based on the parameter values listed in Table 1. The results show that as μ_f and μ_p increase, the number of TB cases also rises, and conversely, a decrease in these rates leads to fewer cases. Additionally, the analysis reveals that as the number of infectious individuals grows, the endemicity of the disease within the community increases, driven by the increasing number of infections. To effectively control the spread of the disease, it is crucial to reduce the rates at which funding and prevention programs are removed.

Fig. 6.

A numerical simulation demonstrating the impact of the funding removal rate (μ_f) and the prevention program removal rate (μ_p) on disease transmission.

4.4. The impact of varying prevention programs and funding inflow rate on TB dynamics

Fig. 7 using the parameter values presented in Table 3 illustrates the impact of prevention programs and funding on the dynamics of TB. The results show that an increase in k₁ and k₂ leads to higher inflow rates for prevention programs and funding, denoted by α and ρ, respectively, which results in a decrease in the number of infectious cases. On the other hand, when k₁ and k₂ decrease, TB cases increase. The analysis indicates that as the number of infectious individuals decreases, the endemicity of the disease in the community also declines, due to the reduction in infection cases. Therefore, to effectively control the disease within the population, it is essential to increase the inflow rates for funding and prevention programs.

Fig. 7.

A numerical simulation illustrating the effect of the funding inflow rate (ρ) and the prevention program inflow rate (α) on disease transmission.

4.5. Impact of interventions (Funding, prevention programs and treatment) on TB dynamics

As shown in Fig. 8(a) and (b), the number of infected individuals decreases more significantly over time when interventions are applied compared to when no interventions are implemented. These results highlight the effectiveness of control measures in reducing disease prevalence in endemic areas. Additionally, Fig. 8 (a) demonstrates the impact of a single intervention, such as treatment, on the population. While treatment alone helps reduce the number of infectious cases, it proves less effective than combining treatment with other interventions, such as funding and prevention programs, as illustrated in Fig. 8 (b). This integrated approach ultimately results in a substantial decline in infection rates within the community.

Fig. 8.

Numerical simulations demonstrating the impact of interventions, including treatment, prevention programs, and funding, on disease transmission dynamics by using the parameter values given in Table 3.

4.6. The impact of funding, prevention programs, and treatment on ${\mathcal{R}}_0$

This section examines the effects of various control measures on disease transmission. Specifically, the impacts of funding, prevention programs, and treatment interventions are analyzed, with findings presented in Fig. 9.

Fig. 9.

Impact of key parameters on the effective reproduction number ${\mathcal{R}}_0$ for tuberculosis (TB) dynamics. Subfigures (a) and (b) show that as funding increases, reflected in improvements in the growth rate of prevention programs k₁ and the treatment rate ω, the effective reproduction number ${\mathcal{R}}_0$ decreases. Subfigures (c) and (d) illustrate that increasing the contact rate β significantly raises ${\mathcal{R}}_0$. The parameter values presented in Table 3.

Fig. 9(a)–(b) illustrates that as funding increases, leading to improvements in the growth rate of prevention programs ($k_1$) and the treatment rate ω, the effective reproduction number ${\mathcal{R}}_0$ declines. This trend indicates that both enhanced funding and the expansion of robust prevention and treatment efforts can significantly reduce disease transmission. A lower ${\mathcal{R}}_0$ corresponds to reduced endemicity, as fewer secondary infections occur. Therefore, in regions where tuberculosis (TB) remains endemic, it is essential to invest in well-funded and rapidly scaling prevention programs alongside effective treatment strategies to control and eventually eliminate the disease from the population. Fig. 9(c)–(d) further shows that an increase in the contact rate β substantially raises ${\mathcal{R}}_0$, highlighting the role of increased social interaction or transmission opportunities in accelerating the spread of TB. On the other hand, a higher treatment rate ω contributes to reducing secondary infections. However, while treatment plays a crucial role in controlling the disease, it must be supported by patient adherence programs and complementary preventive measures to ensure sustained reductions in transmission.

5. Conclusion

This study presents a mathematical model to analyze the role of funding on the dynamics of tuberculosis (TB), with particular focus on funded-prevention programs. Our results reveal that increases in funding can lead to reductions in the effective reproduction number ${\mathcal{R}}_0$ through enhanced prevention programs, modelled by the parameter k₁ and more efficient treatment modelled by ω. Furthermore, the model highlights conditions under which a backward bifurcation may occur, indicating that reducing ${\mathcal{R}}_0$ below unity may not always guarantee disease elimination unless additional control measures are implemented.

From an application standpoint, these results have significant implications for TB control policies, particularly in low- and middle-income countries where funding is often constrained. The current global situation shows that TB remains one of the top infectious disease killers, with millions of new cases and deaths annually, especially in regions with weak health systems and high HIV prevalence. Despite global targets for TB elimination, many countries are not on track due to underfunded prevention efforts, treatment stockouts, and poor patient follow-up systems.

Our findings suggest that to make meaningful progress toward TB elimination, governments and global health agencies must prioritize sustained investment in both the expansion of prevention programs and the improvement of treatment infrastructure. Specifically, policies that boost the rate at which prevention programs grow and ensure the widespread availability and uptake of treatment can significantly reduce transmission. Furthermore, the model indicates that interventions must also address behavioural and systemic factors that influence contact rates.

While this study provides valuable insights into the impact of funding, prevention, and treatment on TB dynamics, several limitations must be considered. A major drawback was the lack of up-to-date data on funding and prevention programs, particularly regarding inflow and outflow rates of funds. The sensitivities around funding information from governments make it difficult to link the model to funding data. We had to rely on available online data on the number of TB cases. We focused on TB data from South Africa, and through the model fitting, we were able to determine parameters that were then used during the simulations.

Despite the useful results obtained from our model, the model presented in this paper has its shortcomings. The model does not incorporate spatial factors, population mobility, or the behavioural complexities associated with treatment adherence, such as non-compliance or inconsistent care. The research presented in this paper could be improved by considering behavioural factors, such as health-seeking behaviour, stigma, and community engagement, which play a crucial role in the transmission of TB and treatment adherence. Spatial models that account for geographic factors and population mobility can also be developed using the model developed here as a basis to improve the model's applicability, particularly in diverse and large populations. In settings with high transmission, our model reinforces the need for proactive and comprehensive strategies that tackle both the biological and socio-economic drivers of TB.

CRediT authorship contribution statement

V.M. Mbalilo: Writing – review & editing, Writing – original draft. F. Nyabadza: Writing – review & editing, Supervision. S.P. Gatyeni: Writing – review & editing, Supervision.

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Conflict of interest

The authors declare that there is no conflict of interest concerning the publication of this article.

Handling Editor: Dr Yiming Shao