This section focuses on the mathematical properties of the baseline TB model given by Eqs. (1) to (6) to gain a deeper understanding of the behavior of the model. The analysis focuses on key aspects such as the positivity of solutions, invariant region, equilibrium points, basic reproduction number, and sensitivity analysis.

Positivity of the solutions

A population is biologically meaningful and well defined if all model solutions are non-negative for all \(t\ge t_{0}\). To ensure the epidemiological relevance of the TB model defined by Eqs. (1) to (6), it is essential to demonstrate the positivity of state variables \(S(t),E_{1}(t),E_{2}(t), I(t),R(t)\), and P(t) at all times \(t>t_{0}\).

Theorem 1

If the initial data \(S(t_{0})\ge 0\), \(E_{1}(t_{0})\ge 0\), \(E_{2}(t_{0})\ge 0\), \(I(t_{0})\ge 0\), \(R(t_{0})\ge 0\), \(P(t_{0})\ge 0\), then the solutions \(S(t),E_{1}(t),E_{2}(t), I(t),R(t)\), and P(t) of the system of model (1) to (6) are non-negative \(\forall t\ge t_{0}\).

Proof

To prove this theorem, we use the approach in [22] and [23]. Consider equation (1):

$$\begin{aligned} \dfrac{dS}{dt} = \Lambda + \gamma R – \left[ \dfrac{\beta }{N}\left( I+\varepsilon _{1}P\right) +\mu \right] S \end{aligned}$$

(8)

Omitting the first two terms, equation (8) can be rewritten as:

$$\begin{aligned} \dfrac{dS(t)}{dt}+f(t)S(t)\ge 0. \end{aligned}$$

(9)

where \(f(t)=\left[ \dfrac{\beta }{N}\left( I+\varepsilon _{1}P\right) +\mu \right]\). The inequality (9) is in the form of a separable first-order and first-degree ordinary differential equation (ODE). Solving for S(t) yields

$$\begin{aligned} \int \limits _{S(t_{0})}^{S(t)}\dfrac{dS(t)}{S(t)}+ \int \limits _{t_{0}}^{t} f(t)dt\ge 0 \end{aligned}$$

(10)

Further simplification leads to:

$$\begin{aligned} S(t)\ge S(t_{0})\exp \left( -\int \limits _{t_{0}}^{t} f(t)dt\right) \end{aligned}$$

(11)

Since \(S(t_{0})\ge 0\) and \(\exp (\cdot )>0\), it implies that

$$\begin{aligned} S(t)\ge 0. \end{aligned}$$

(12)

This shows that the solution S(t) is nonnegative for all \(t \ge t_{0}\). Doing the same approach to the remaining equations in the baseline TB model, it can be simply shown that \(E_{1}(t)\ge 0\), \(E_{2}(t)\ge 0\), \(I(t)\ge 0\), \(R(t)\ge 0\), and \(P(t)\ge 0\) for all \(t\ge t_{0}\). Therefore, all solutions of the baseline TB model (1) to (6) maintain positivity for any non-negative initial conditions. This completes the proof of Theorem 1.\(\square\)

Invariant region

The invariant region describes the domain in which all solutions to the proposed baseline TB model given by Eqs. (1) to (6) are of biological and mathematical importance. All model parameters are non-negative for all \(t\ge t_{0}\). In addition, the solutions with positive initial data remain non-negative with all \(t\ge t_{0}\) and are bounded.

Theorem 2

The set of state variables S(t), \(E_{1}(t)\), \(E_{2}(t)\), I(t), R(t), and P(t) of the baseline TB model Eqs. (1) to (6) is confined in a positive feasible region \(\Phi\).

Proof

Suppose that \(\Phi =\left( S(t),E_{1}(t),E_{2}(t),I(t),R(t)\right) \in \mathbb {R}_{+}^{5}\) for all time \(t\ge t_{0}\). The total human population at time t is given by the equation:

$$\begin{aligned} N(t) = S(t) + E_{1}(t) + E_{2}(t) + I(t) + R(t) \end{aligned}$$

(13)

Differentiating both sides with respect to t yields

$$\begin{aligned} \dfrac{dN}{dt}=\dfrac{dS}{dt}+ \dfrac{dE_{1}}{dt}+\dfrac{dE_{2}}{dt}+\dfrac{dI}{dt}+\dfrac{dR}{dt}. \end{aligned}$$

(14)

Substituting the model Eqs. (1) to (5) into equation (14) and simplifying yields

$$\begin{aligned} \dfrac{dN}{dt} = \Lambda – \mu N -\delta I. \end{aligned}$$

(15)

Omitting the last term, equation (15) can be rewritten as:

$$\begin{aligned} \dfrac{dN}{dt} \le \Lambda – \mu N. \end{aligned}$$

(16)

The inequality (16) is in the form of a separable ODE. Solving for N(t) yields

$$\begin{aligned} \int \limits _{N(t_{0})}^{N(t)}\dfrac{dN(t)}{\Lambda -\mu N(t)}\le \int \limits _{t_{0}}^{t} dt. \end{aligned}$$

(17)

Further simplification leads to:

$$\begin{aligned} N(t)\le \dfrac{\Lambda }{\mu }+\left( N(t_{0})-\dfrac{\Lambda }{\mu }\right) e^{-\mu (t-t_{0})} \end{aligned}$$

(18)

When \(t=t_{0}\), then \(N=N(t_{0})\). When \(t\rightarrow \infty\), then \(N(t)\rightarrow \dfrac{\Lambda }{\mu }\). Finally, the feasible region is

$$\Phi =\left\{ \left( S(t),E_{1}(t),E_{2}(t), I(t),R(t)\right) \in \mathbb {R}_{+}^{5}: N(t_{0})\le N(t)\le \dfrac{\Lambda }{\mu }\right\}$$

The domain \(\Phi\) is positive invariant under the flow induced by the equations of the baseline TB model (1) to (6). Therefore, all feasible solutions of the model enter the feasible region \(\Phi\), hence the proposed Mpox model is well posed and is both epidemiologically and mathematically meaningful and we consider to generate the analysis. This implies that the proposed baseline TB model is positive invariant in the region \(\Phi\), for non-negative initial conditions in the region \(\Phi\). This completes the proof.\(\square\)

Existence of equilibrium points

The equilibrium points of the system given by Eqs. (1) to (6) are obtained by equating the right-hand sides to zero and solving for the state variables, i.e.,

$$\begin{aligned} \Lambda – \dfrac{\beta }{N}\left( I+\varepsilon _{1}P\right) S + \gamma R – \mu S & =0\end{aligned}$$

(19)

$$\begin{aligned} q\dfrac{\beta }{N}\left( I+\varepsilon _{1}P\right) S + \varepsilon _{2}\dfrac{\beta }{N}\left( I+\varepsilon _{1}P\right) R – (\sigma +\alpha \kappa +\mu )E_{1} & =0\end{aligned}$$

(20)

$$\begin{aligned} (1-q)\dfrac{\beta }{N}\left( I+\varepsilon _{1}P\right) S + \sigma E_{1} – (\kappa +\mu )E_{2} & =0\end{aligned}$$

(21)

$$\begin{aligned} \alpha \kappa E_{1} +\kappa E_{2} – (r+\mu +\delta )I & =0\end{aligned}$$

(22)

$$\begin{aligned} rI – \varepsilon _{2}\dfrac{\beta }{N}\left( I+\varepsilon _{1}P\right) R – (\gamma +\mu )R & =0\end{aligned}$$

(23)

$$\begin{aligned} \eta I – \mu _{p}P=0 \end{aligned}$$

(24)

Identifying the equilibrium points helps us understand the potential long-term behavior of the dynamical system. There are two types of equilibrium points for the baseline TB model: TB-free equilibrium point(s), and endemic equilibrium point(s). These are discussed below.

TB-free equilibrium points (DFE)

The point where the disease is absent from the population is referred to as a disease-free equilibrium (DFE). This point is obtained by setting to zero both the exposed and infectious populations, as well as the pathogen population. Thus, by substituting \(E_{1}=E_{2}=I=P=0\) into Eqs. (19, 20, 21, 22, 23) to (24), yields the point \(\text {DFE}\left( S^{*},0,0,0,0,0\right)\), where

$$\begin{aligned} S^{*} =\dfrac{\Lambda }{\mu } \end{aligned}$$

(25)

The DFE point will be used to calculate the basic reproduction number (\(R_{0}\)) for the baseline TB model, as illustrated in Basic reproduction number (\(R_{0}\)) section.

Basic reproduction number (\(R_{0}\))

The basic reproduction number (\(R_{0}\)) in TB modeling is a key indicator of disease transmission. To compute \(R_{0}\) for the baseline TB model, the next-generation matrix technique is used [23]. The model equations are split to form two matrices, F and V. The matrix F contains infections terms while matrix V contains a negation of all other transition terms. Considering only the “diseased” states (E1, E2, I and P), the model equations can be written in the general form as:

$$\begin{aligned} \dfrac{d{\textbf {X}}}{dt}=F({\textbf {X}})-V({\textbf {X}}), \end{aligned}$$

(26)

where \(F({\textbf {X}})\) and \(V({\textbf {X}})\) are column vectors given by

$$\begin{aligned} F({\textbf {X}}) = \begin{bmatrix} q\dfrac{\beta }{N}\left( I+\varepsilon _{1}P\right) S + \varepsilon _{2}\dfrac{\beta }{N}\left( I+\varepsilon _{1}P\right) R\\ (1-q)\dfrac{\beta }{N}\left( I+\varepsilon _{1}P\right) S\\ 0\\ 0 \end{bmatrix} \end{aligned}$$

(27)

$$\begin{aligned} V({\textbf {X}}) = \begin{bmatrix} (\sigma +\alpha \kappa +\mu )E_{1}\\ -\sigma E_{1} + (\kappa +\mu )E_{2}\\ -\alpha \kappa E_{1} -\kappa E_{2} + (r+\mu +\delta )I\\ -\eta I + \mu _{p}P \end{bmatrix} \end{aligned}$$

(28)

Let the Jacobian matrices \({\textbf {f}}\) and \({\textbf {v}}\) be defined as:

$$\begin{aligned} {\textbf {f}}=\dfrac{\partial F}{\partial (E1,E2,I,P)}=\begin{bmatrix}\dfrac{\partial F}{\partial E1}&\dfrac{\partial F}{\partial E2}&\dfrac{\partial F}{\partial I}&\dfrac{\partial F}{\partial P} \end{bmatrix} \end{aligned}$$

(29)

$$\begin{aligned} {\textbf {v}}=\dfrac{\partial V}{\partial (E1,E2,I,P)}=\begin{bmatrix}\dfrac{\partial V}{\partial E1}&\dfrac{\partial V}{\partial E2}&\dfrac{\partial V}{\partial I}&\dfrac{\partial V}{\partial P} \end{bmatrix} \end{aligned}$$

(30)

Evaluating \({\textbf {f}}\) and \({\textbf {v}}\) at equilibrium point DFE yields

$$\begin{aligned} {\textbf {f}} = \begin{bmatrix} 0& 0& q\beta & q\beta \varepsilon _{1}\\ 0& 0& (1-q)\beta & (1-q)\beta \varepsilon _{1}\\ 0& 0& 0& 0\\ 0& 0& 0& 0 \end{bmatrix} \end{aligned}$$

(31)

$$\begin{aligned} {\textbf {v}} = \begin{bmatrix} (\sigma +\alpha \kappa +\mu )& 0& 0& 0\\ -\sigma & (\kappa +\mu )& 0& 0\\ -\alpha \kappa & -\kappa & (r+\mu +\delta )& 0\\ 0& 0& -\eta & \mu _{p} \end{bmatrix} \end{aligned}$$

(32)

The associated next generation matrix is given by \({\textbf {G}} = {\textbf {f}}*{\textbf {v}}^{-1}\), i.e.,

$$\begin{aligned} {\textbf {G}} = \begin{bmatrix} \dfrac{(\beta \kappa q(\mu _p + \eta \varepsilon _{1})(\sigma + \alpha \kappa + \alpha \mu )}{\mu _p(\kappa + \mu )(\mu + \sigma + \alpha \kappa )(\delta + \mu + r)} & \dfrac{\beta \kappa q(\mu _p + \eta \varepsilon _{1})}{\mu _p(\kappa + \mu )(\delta + \mu + r)} & \dfrac{\beta q(\mu _p + \eta \varepsilon _{1})}{\mu _p(\delta + \mu + r)} & \dfrac{\beta q\varepsilon _{1}}{\mu _p}\\ \\ \dfrac{\beta \kappa (\mu _p + \eta \varepsilon _{1})(1-q)(\sigma + \alpha \kappa + \alpha \mu )}{\mu _p(\kappa + \mu )(\mu + \sigma + \alpha \kappa )(\delta + \mu + r)} & \dfrac{\beta \kappa (\mu _p + \eta \varepsilon _{1})(1-q)}{\mu _p(\kappa + \mu )(\delta + \mu + r)} & \dfrac{\beta (\mu _p + \eta \varepsilon _{1})(1-q)}{\mu _p(\delta + \mu + r)} & \dfrac{\beta \varepsilon _{1}(q – 1)}{\mu _p}\\ \\ 0& 0& 0& 0\\ \\ 0& 0& 0& 0 \end{bmatrix} \end{aligned}$$

(33)

The basic reproduction number, \(R_{0}\), is the spectral radius of the matrix \({\textbf {G}}\), which is the largest eigenvalue of \({\textbf {G}}\) in magnitude. It is given by

$$\begin{aligned} R_{0}=\dfrac{\beta \kappa (\mu _p + \eta \varepsilon _{1})(\mu + \sigma + \alpha \kappa – \mu q + \alpha \mu q)}{\mu _p(\kappa + \mu )(\mu + \sigma + \alpha \kappa )(\delta + \mu + r)} \end{aligned}$$

(34)

If \(R_{0}>1\), it indicates that the TB bacteria will persist in the human population. This can potentially lead to more deaths over time, emphasizing the crucial need for public health intervention measures to combat the spread of bacteria. If \(R_{0}, the TB bacteria is likely to die out of the human population in the near future, although ongoing monitoring and potential preventative actions remain advisable. In several studies, it has been shown that the DFE point is asymptotically stable locally and globally if \(R_{0} and unstable if \(R_{0}> 1\) [22].

Endemic equilibrium points (EE)

This equilibrium represents a steady state solution in which TB disease persists within the population at a stable level. This point is obtained by substituting \(E_{1}\ne 0,E_{2}\ne 0,I\ne 0\) and \(P\ne 0\) into Eqs. (19, 20, 21, 22, 23) to (24). Let the endemic equilibrium point of the baseline TB model be \(\text {EE}=(S^{*},E_{1}^{*},E_{2}^{*},I^{*},R^{*},P^{*})\). For the existence of the endemic equilibrium conditions \(S^{*}>0\), \(E_{1}^{*}>0\), \(E_{2}^{*}>0\), \(I^{*}>0\), \(R^{*}>0\), \(P^{*}>0\) must be satisfied. Thus, we obtain the following solutions:

$$\begin{aligned} S^{*} & = \dfrac{\Lambda +\gamma R^{*}}{\dfrac{\beta }{N^{*}}\left( I^{*}+\varepsilon _{1}P^{*}\right) + \mu },\end{aligned}$$

(35)

$$\begin{aligned} E_{1}^{*} & = \dfrac{q\dfrac{\beta }{N^{*}}\left( I^{*}+\varepsilon _{1}P^{*}\right) S^{*} + \varepsilon _{2}\dfrac{\beta }{N^{*}}\left( I^{*}+\varepsilon _{1}P^{*}\right) R^{*}}{\sigma +\alpha \kappa +\mu },\end{aligned}$$

(36)

$$\begin{aligned} E_{2}^{*} & = \dfrac{(1-q)\dfrac{\beta }{N^{*}}\left( I^{*}+\varepsilon _{1}P^{*}\right) S^{*} + \sigma E_{1}^{*}}{\kappa +\mu }, \end{aligned}$$

(37)

$$\begin{aligned} I^{*} & = \dfrac{\alpha \kappa E_{1}^{*} +\kappa E_{2}^{*}}{r+\mu +\delta },\end{aligned}$$

(38)

$$\begin{aligned} R^{*} & = \dfrac{rI^{*}}{\varepsilon _{2}\dfrac{\beta }{N^{*}}\left( I^{*}+\varepsilon _{1}P^{*}\right) + \gamma +\mu },\end{aligned}$$

(39)

$$\begin{aligned} P^{*} & = \dfrac{\eta I^{*}}{\mu _{p}}. \end{aligned}$$

(40)

In this case, the solution exists and is unique. Analyzing the dynamics of EE is important for understanding the long-term behavior of TB disease. From the above conditions, it can be concluded that the endemic equilibrium solution is stable if and only if \(R_{0}> 1\) exhibits persistence of TB transmission in the human population. In several studies, it has been shown that the EE point is asymptotically stable if \(R_{0}> 1\) and unstable if \(R_{0} [22].

Sensitivity analysis

Sensitivity analysis quantifies how a model’s predictions change in response to variations in its parameters. The aim is to identify the model parameters that most significantly impact the basic reproduction number. This helps identify the key factors driving the dynamics of the disease, enabling policymakers to prioritize interventions targeting those areas. In essence, sensitivity analysis provides valuable insight into the complex dynamics of TB transmission, informing decisions on resource allocation and intervention strategies.

The normalized forward sensitivity index of a model parameter (m) with respect to the basic reproduction number (\(R_{0}\)) quantifies the change in \(R_{0}\) resulting from a relative change in m. It is calculate by:

$$\begin{aligned} S=\dfrac{m}{R_{0}}\dfrac{\partial R_{0}}{\partial m} \end{aligned}$$

(41)

A positive value of S indicates that an increase in m will lead to an increase in \(R_{0}\). In contrast, a negative value of S suggests that an increase in m will decrease \(R_{0}\). A higher absolute value of S indicates a greater influence of the parameter m on \(R_{0}\). The parameters in the basic reproduction number given by equation (34) are: \(\Lambda\), \(\beta\), \(\kappa\), r, \(\mu\), and \(\delta\). Using formula (41), the sensitivity indices are given by

$$\begin{aligned} S_{\beta }=\dfrac{\beta }{R_{0}}\dfrac{\partial R_{0}}{\partial \beta }, \quad S_{\kappa }=\dfrac{\kappa }{R_{0}}\dfrac{\partial R_{0}}{\partial \kappa },\quad S_{r}=\dfrac{r}{R_{0}}\dfrac{\partial R_{0}}{\partial r} \end{aligned}$$

(42)

$$\begin{aligned} S_{\mu }=\dfrac{\mu }{R_{0}}\dfrac{\partial R_{0}}{\partial \mu }, \quad S_{\delta }=\dfrac{\delta }{R_{0}}\dfrac{\partial R_{0}}{\partial \delta },\quad S_{\alpha }=\dfrac{\alpha }{R_{0}}\dfrac{\partial R_{0}}{\partial \alpha }, \quad S_{\sigma }=\dfrac{\sigma }{R_{0}}\dfrac{\partial R_{0}}{\partial \sigma } \end{aligned}$$

(43)

$$\begin{aligned} S_{\eta }=\dfrac{\eta }{R_{0}}\dfrac{\partial R_{0}}{\partial \eta }\quad S_{\mu _p}=\dfrac{\mu _p}{R_{0}}\dfrac{\partial R_{0}}{\partial \mu _p}, \quad S_{\varepsilon _{1}}=\dfrac{\varepsilon _{1}}{R_{0}}\dfrac{\partial R_{0}}{\partial \varepsilon _{1}}, \quad S_{q}=\dfrac{q}{R_{0}}\dfrac{\partial R_{0}}{\partial q} \end{aligned}$$

(44)

The partial derivatives were computed using the Symbolic Math Toolbox (syms) in MATLAB. For parameters with positive indices, decreasing their values would reduce the spread of TB bacteria in the human population. In contrast, increasing the values of parameters with negative indices would also reduce the spread of TB bacteria. Parameters with high-sensitivity indices are often prioritized for further investigation or control measures. In particular, sensitivity analysis can be used to assess the potential effectiveness of various control strategies under different scenarios. This helps in comparing the cost effectiveness of different interventions and selecting the most optimal ones. The quantitative results of the sensitivity analysis are presented in Sensitivity indices of model parameters section.