In Cameroon, there is a paucity of empirical research examining the impact of cement and marble works on respiratory health. However, the available data suggest a correlation between exposure to emissions from cement works and an increase in respiratory diseases in the surrounding areas. The Figuil Cement and Marble Works, situated in the northern region of Cameroon, represents a significant source of employment and economic development for the region. However, the industrial activities of this cement plant, in particular the production of cement and marble, result in the generation of considerable emissions of atmospheric pollutants, notably fine particles (PM2.5 and PM10), sulphur oxides (SO₂), and nitrogen oxides (NOx) . These pollutants, which are known to have adverse effects on human health, are of particular concern to the local population residing in the vicinity of the cement plant.

The issue of air pollution in this region has given rise to growing concerns about its impact on public health, particularly in relation to the spread of respiratory diseases. The prevalence of respiratory diseases, including asthma, chronic bronchitis and lung infections, is exacerbated by continuous exposure to elevated levels of air pollutants. Such illnesses can disseminate rapidly throughout populations, particularly in regions where access to healthcare is constrained. A substantial body of research has demonstrated a clear correlation between exposure to air pollution and an elevated prevalence of respiratory illnesses. For example, the World Health Organisation (WHO) has highlighted that fine particles are particularly capable of penetrating deep into the lungs, causing inflammation and exacerbating conditions such as asthma and chronic obstructive pulmonary disease (COPD). In the Figuil locality, studies on air quality are limited, but local reports and testimonies from residents indicate an increased prevalence of respiratory diseases, particularly in children and the elderly, who are more vulnerable to the effects of pollution . The SEIR (Susceptible-Exposed -Infected-Recovered) epidemiological model is a widely used tool for understanding the spread of infectious diseases within populations. Extensions of this model have been employed to examine the dynamics of respiratory diseases in settings where pollution exerts a catalytic influence. Recent research has incorporated environmental factors, such as air quality, into epidemiological models with the objective of improving understanding of the influence of air pollutants on the spread of respiratory diseases . These models simulate the impact of fluctuations in pollution on infection and recovery rates. A number of studies have demonstrated that populations residing in close proximity to industrial areas are at an elevated risk of adverse health outcomes due to the sustained inhalation of toxic pollutants. Children, in particular, are more susceptible to developing chronic respiratory diseases due to their still-developing immune systems . Conversely, the elderly, due to immunosenescence, are more likely to develop respiratory infections and encounter serious complications. To improve air quality and reduce the incidence of respiratory diseases, targeted interventions are required. These include the adoption of cleaner industrial technologies, the introduction of advanced systems for filtering and capturing pollutants at source to reassure air quality monitoring, and raising awareness among local populations of the dangers of pollution. Prevention models incorporating epidemiological and environmental data can be used to develop more effective public health strategies to limit the spread of respiratory diseases in high-risk areas such as Figuil.

The prevalence of respiratory diseases in the vicinity of the Figuil cement and marble works can be attributed to a multitude of factors associated with industrial pollution. The activities of the cement and marble works result in the generation of fine particles and other airborne pollutants, which have an adverse impact on the respiratory health of the surrounding population. Modelling the respiratory diseases associated with air pollution around the Figuil Cement and Marble Works is of paramount importance for the comprehension and mitigation of the health impacts of this source of pollution. In the conventional approach to modelling, a number of age compartments are included in order to reflect the differing susceptibility and responses to pollutants exhibited by different age groups. Nevertheless, a streamlined methodology can offer a comprehensive overview while remaining effective for management and prevention objectives. In order to streamline the analysis while capturing the essential dynamics of respiratory disease transmission, we propose reducing the initial model to two main compartments: children and the elderly. Children are particularly vulnerable due to their developing immune systems and potentially increased exposure to pollutants. Their susceptibility to developing chronic respiratory diseases justifies their separation into a separate compartment. The elderly, due to the natural deterioration of the immune system and lung function, are also very sensitive to the effects of pollution. Including them in a separate compartment makes it easier to understand and address the specific needs of this population.

The diagram presents a simplified epidemiological model that elucidates the transmission of respiratory disease among the local population in the vicinity of the Figuil cement and marble works. The model captures the dynamics of respiratory disease by considering different transmission scenarios within the population, with a particular focus on critical ages. The model does not account for vertical transmission and is composed of two age classes: individuals aged 0 to 15 (class ${\mathrm{S}}_1$) and individuals aged over 36 (class ${\mathrm{S}}_2$). The transition between the different compartments is governed by the rates described below.

The terms ${\mathrm{S}}_1$ and ${\mathrm{S}}_2$ are used to represent the compartments of susceptible individuals, that is to say, those who are healthy but who may potentially become infected. The term ${\mathrm{S}}_1$ refers to the age group between 0 and 15 years old, while ${\mathrm{S}}_2$ refers to individuals aged over 36. The presence of air pollution has been demonstrated to increase the susceptibility of individuals to disease, as well as the severity of infection. This is reflected by its influence on the transmission and progression rates (${\mathrm{p}}_{\mathrm{i}}\mathrm{\Lambda }$) in each ${\mathrm{S}}_{\mathrm{i}}$ class. The two groups may become exposed to the disease via the transmission rates ${\mathrm{\beta }}_{1,\mathrm{ }1}$, ${\mathrm{\beta }}_{1,\mathrm{ }2\mathrm{ }},$ ${\mathrm{\beta }}_{2,\mathrm{ }1}$ and ${\mathrm{\beta }}_{2,\mathrm{ }2\mathrm{ }}$, which determine the probability of a susceptible individual becoming exposed after contact with an infectious person.

The compartments ${\mathrm{E}}_{\mathrm{I}}$ and ${\mathrm{E}}_{\mathrm{C}}$ represent those of exposed individuals who are not yet infectious. An individual's transition from the susceptible state to the exposed state is contingent upon their interaction with infectious individuals and their transmission rate β. Similarly, the transition from ${\mathrm{S}}_1$ to ${\mathrm{E}}_{\mathrm{I}}$ and from ${\mathrm{S}}_2$ to ${\mathrm{E}}_{\mathrm{C}}$ is also dependent upon the level of exposure to the contaminant (polluted air).

I represents the compartment of individuals who have been infected and are capable of transmitting the disease. Individuals in compartment ${\mathrm{E}}_{\mathrm{I}}$ can evolve towards compartment I at a rate determined by $\mathrm{\gamma }$, which marks the beginning of the period during which they can infect other individuals.

The letter C represents the compartment of individuals who have developed a chronic form of the disease. It is possible that some individuals, particularly those in the ${\mathrm{E}}_{\mathrm{C}}$ compartment, may progress to a chronic state at a rate $\mathrm{\delta }$. These individuals are no longer infectious, but their state of health deteriorates over the long term.

R represents the compartment of individuals who have recovered from the disease. Recovery may occur from the infectious compartment (I) at a rate designated as ${\mathrm{\gamma }}_1$, or from the chronic compartment (C) at a rate designated as ${\mathrm{\gamma }}_2$.

The model parameters are as follows:

${\mathrm{p}}_{\mathrm{i}}\mathrm{\Lambda }$ represents the proportion of the population exposed to pollution (contaminated air) from the cement plant recruited in each class ${\mathrm{S}}_{\mathrm{i}}$.

The transmission rates of the disease between the different age groups, ${\mathrm{\beta }}_{1,\mathrm{ }1}$, ${\mathrm{\beta }}_{1,\mathrm{ }2\mathrm{ }},$ ${\mathrm{\beta }}_{2,\mathrm{ }1}$ and ${\mathrm{\beta }}_{2,\mathrm{ }2\mathrm{ }}$, are represented by these variables.

The natural or disease-related mortality rate of individuals in class i is represented by ${\mathrm{\mu }}_{\mathrm{i}}$.

The rates of progression of exposure to the chronic and infectious states are represented by $\mathrm{\delta }$ and $\mathrm{\gamma }$.

The rates of recovery of infected individuals and individuals in the chronic phase to the recovered compartment R are represented by ${\mathrm{\gamma }}_1$ and ${\mathrm{\gamma }}_2$.

The differential equations that govern the movement of individuals between the different health states (susceptible, exposed, infectious, chronic, recovered) in the epidemiological model for the transmission of respiratory diseases around the Figuil cement plant are given by equation (1).

In order to demonstrate that our epidemiological model permits positive and bounded solutions, we will analyse the differential equations of the system and examine their behaviour over time. In particular, we aim to show that the populations in each compartment (${\mathrm{S}}_1, {\mathrm{S}}_2, {\mathrm{E}}_{\mathrm{I}}, {\mathrm{E}}_{\mathrm{C}}, \mathrm{I}, \mathrm{C},\mathrm{ R}$) remain positive and do not exceed certain limits. This implies that the populations in each compartment never become negative. To verify this, we will analyse each equation.

Compartiment ${\mathrm{S}}_1\left(\mathrm{or} {\mathrm{S}}_2\right)$:

The equation of ${\mathrm{S}}_1$ is:

It is assumed that the initial condition, ${\mathrm{S}}_1$(0), is positive, that is to say, ${\mathrm{S}}_1$(0)$\ge 0$. This implies that the population in compartment ${\mathrm{S}}_1$ is positive at the outset.

The term ${\mathrm{p}}_1\mathrm{\Lambda }$ represents a positive constant, denoting the inflow into compartment ${\mathrm{S}}_1$. The terms ${\mathrm{\mu }}_{{\mathrm{S}}_1}{\mathrm{S}}_1$ and $\left({\mathrm{\beta }}_{1,1}\mathrm{I}+{\mathrm{\beta }}_{1,2}\mathrm{I}\right){\mathrm{S}}_1$are negative terms that serve to reduce the population of ${\mathrm{S}}_1$.

In the event that ${\mathrm{S}}_1$ reaches zero, the equation $\dot{{\mathrm{ S}}_1}={\mathrm{p}}_1\mathrm{\Lambda }>0$ indicates that ${\mathrm{S}}_1$ cannot remain at zero and will begin to increase due to the positive inflow ${\mathrm{p}}_1\mathrm{\Lambda }$. It can therefore be concluded that ${\mathrm{S}}_1\left(\mathrm{t}\right)$ will remain positive.

If ${\mathrm{S}}_1$ is positive, the differential equation

$\dot{{\mathrm{S}}_1}={\mathrm{p}}_1\mathrm{\Lambda }-{\mathrm{\mu }}_{{\mathrm{S}}_1}{\mathrm{S}}_1-\left({\mathrm{\beta }}_{1,1}\mathrm{I}+{\mathrm{\beta }}_{1,2}\mathrm{I}\right){\mathrm{S}}_1$ is a form of the Riccati-type relation, which often admits bounded solutions under certain conditions. It is necessary to ensure that ${\mathrm{S}}_1\left(\mathrm{t}\right)$ remains positive; this can be achieved by verifying that the loss rate $\left({\mathrm{\mu }}_{{\mathrm{S}}_1}+{\mathrm{\beta }}_{1,1}\mathrm{I}+{\mathrm{\beta }}_{1,2}\mathrm{I}\right){\mathrm{S}}_1$ is never greater than the input ${\mathrm{p}}_1\mathrm{\Lambda }$. In other words, the objective is to demonstrate that ${\mathrm{p}}_1\mathrm{\Lambda }$ is greater than or equal to $\left({\mathrm{\mu }}_{{\mathrm{S}}_1}+{\mathrm{\beta }}_{1,1}\mathrm{I}+{\mathrm{\beta }}_{1,2}\mathrm{I}\right){\mathrm{S}}_1\left(\mathrm{t}\right)$. This is true as long as ${\mathrm{S}}_1\left(\mathrm{t}\right)$ is sufficiently small, that is to say, ${\mathrm{S}}_1\left(\mathrm{t}\right)\le \frac{{\mathrm{p}}_1\mathrm{\Lambda }}{\left({\mathrm{\mu }}_{{\mathrm{S}}_1}+{\mathrm{\beta }}_{1,1}\mathrm{I}+{\mathrm{\beta }}_{1,2}\mathrm{I}\right)}$.

Therefore, when ${\mathrm{S}}_1\left(\mathrm{t}\right)$ is positive but small, ${\mathrm{S}}_1\left(\mathrm{t}\right)$ remains positive as long as the terms $\left({\mathrm{\mu }}_{{\mathrm{S}}_1}+{\mathrm{\beta }}_{1,1}\mathrm{I}+{\mathrm{\beta }}_{1,2}\mathrm{I}\right)$ do not exceed ${\mathrm{p}}_1\mathrm{\Lambda }$. It can be similarly reasoned that ${\mathrm{S}}_2\left(\mathrm{t}\right)$ also remains positive.

Compartment ${\mathrm{E}}_{\mathrm{I}}$ (or ${\mathrm{E}}_{\mathrm{C}}$):

For the sake of argument, we may assume that ${\mathrm{E}}_{\mathrm{I}}$ can become negative, that is to say, ${\mathrm{E}}_{\mathrm{I}}$ < 0. Substituting this into the equation $\dot{{\mathrm{E}}_{\mathrm{I}}}=\left({\mathrm{\beta }}_{1,1}\mathrm{I}+{\mathrm{\beta }}_{1,2}\mathrm{I}\right){\mathrm{S}}_1-\left({\mathrm{\mu }}_{{\mathrm{E}}_{\mathrm{I}}}+\mathrm{\gamma }\right){\mathrm{E}}_{\mathrm{I}}$, we obtain: Substituting ${\mathrm{E}}_{\mathrm{I}}<0$ into the equation yields: The temporal derivative of ${\mathrm{E}}_{\mathrm{I}}$ is given by the following equation:

$\dot{{\mathrm{E}}_{\mathrm{I}}}=\left({\mathrm{\beta }}_{1,1}\mathrm{I}+{\mathrm{\beta }}_{1,2}\mathrm{I}\right){\mathrm{S}}_1-\left({\mathrm{\mu }}_{{\mathrm{E}}_{\mathrm{I}}}+\mathrm{\gamma }\right){\mathrm{E}}_{\mathrm{I}}$ The variable ${\mathrm{E}}_{\mathrm{I}}$ is therefore defined as:

Given that ${\mathrm{E}}_{\mathrm{I}}$ is less than zero and that ${\mathrm{\mu }}_{{\mathrm{E}}_{\mathrm{I}}}+\mathrm{\gamma }$ are positive (representing the death rate and progression rate), the term ${\mathrm{\mu }}_{{\mathrm{E}}_{\mathrm{I}}}+\mathrm{\gamma }$ Thus, ${\mathrm{E}}_{\mathrm{I}}$ is positive. It can be seen that the right-hand side of the equation is: The equation can be rearranged as follows:

$\left({\mathrm{\beta }}_{1,1}\mathrm{I}+{\mathrm{\beta }}_{1,2}\mathrm{I}\right){\mathrm{S}}_1-\left({\mathrm{\mu }}_{{\mathrm{E}}_{\mathrm{I}}}+\mathrm{\gamma }\right){\mathrm{E}}_{\mathrm{I}}$ It can be demonstrated that ${\mathrm{E}}_{\mathrm{I}}$ will be greater than: $\left({\mathrm{\beta }}_{1,1}\mathrm{I}+{\mathrm{\beta }}_{1,2}\mathrm{I}\right){\mathrm{S}}_1$, given that the value of $-\left({\mathrm{\mu }}_{{\mathrm{E}}_{\mathrm{I}}}+\mathrm{\gamma }\right){\mathrm{E}}_{\mathrm{I}}$ is less than zero. The value of ${\mathrm{E}}_{\mathrm{I}}$ is positive. Consequently, the term $-\left({\mathrm{\mu }}_{{\mathrm{E}}_{\mathrm{I}}}+\mathrm{\gamma }\right){\mathrm{E}}_{\mathrm{I}}$ will exert a positive influence on $\dot{{\mathrm{E}}_{\mathrm{I}}}$. If ${\mathrm{E}}_{\mathrm{I}}$ were initially negative, the temporal derivative (${\mathrm{E}}_{\mathrm{I}}$) would be positive, indicating an increase in ${\mathrm{E}}_{\mathrm{I}}$ towards positive values. This is inconsistent with the assumption that ${\mathrm{E}}_{\mathrm{I}}$ could remain negative. Therefore, it can be concluded that ${\mathrm{E}}_{\mathrm{I}}$ must be positive or zero. It can therefore be concluded that, by applying a similar line of reasoning to that used for ${\mathrm{E}}_{\mathrm{I}}$, ${\mathrm{E}}_{\mathrm{C}}$ must remain positive or zero.

Compartment I (or C):

For the sake of argument, let us assume that I can become negative, that is to say, I<0. This leads to the equation $\dot{\mathrm{I}}={\mathrm{\gamma E}}_{\mathrm{I}}-\left({\mathrm{\mu }}_{\mathrm{I}}+{\mathrm{\gamma }}_1\right)\mathrm{I}$.

Given that I is less than zero and the term $\left({\mathrm{\mu }}_{\mathrm{I}}+{\mathrm{\gamma }}_1\right)\mathrm{I}$ is positive, it follows that the latter will add a positive term to $\mathrm{\gamma E}$. Therefore, we can conclude that $\dot{\mathrm{I}}$ will be positive. If the derivative of I with respect to time, $\mathrm{İ}$ is positive while I is less than zero, it follows that I will increase towards positive values. This is in contradiction with the initial assumption that I can remain negative. It follows that I must be positive or zero. The same line of reasoning can be applied to C, which also guarantees the positivity of the solutions.

Compartment R:

It is assumed that R can become negative, that is to say, $\mathrm{R}<0$. Given that $\dot{\mathrm{R}}={\mathrm{\gamma }}_1\mathrm{I}+{\mathrm{\gamma }}_2\mathrm{C}-{\mathrm{\mu }}_{\mathrm{R}}\mathrm{R}$, the term ${\mathrm{\gamma }}_1\mathrm{I}+{\mathrm{\gamma }}_2\mathrm{C}$ is positive. If R is negative, then the term $-{\mathrm{\mu }}_{\mathrm{R}}\mathrm{R}$ is positive, given that $-{\mathrm{\mu }}_{\mathrm{R}}\mathrm{R}$ is positive and R is negative. Therefore, when R is less than zero, the term $-{\mathrm{\mu }}_{\mathrm{R}}\mathrm{R}$ will be positive and increase the rate of change of R (i.e. $\dot{\mathrm{R}}$). If R were negative, then $\dot{\mathrm{R}}$ would be positive. This implies that R will increase towards positive values due to the term $-{\mathrm{\mu }}_{\mathrm{R}}\mathrm{R}$ being positive. Consequently, R cannot remain negative; it must be positive or zero.

The objective of this demonstration is to prove that the solutions are bounded. This is to be achieved by establishing the existence of positive constants ${\mathrm{M}}_{{\mathrm{S}}_1}$, ${\mathrm{M}}_{{\mathrm{S}}_2}$, ${\mathrm{M}}_{{\mathrm{E}}_{\mathrm{I}}}$, ${\mathrm{M}}_{{\mathrm{E}}_{\mathrm{C}}}$, ${\mathrm{M}}_{ \mathrm{I}}$, ${\mathrm{M}}_{\mathrm{C}}$, ${\mathrm{M}}_{\mathrm{R}}$ such that: It is also required that ${\mathrm{S}}_1\left(\mathrm{t}\right)\le {\mathrm{M}}_{{\mathrm{S}}_1}, {\mathrm{S}}_2\left(\mathrm{t}\right)\le {\mathrm{M}}_{{\mathrm{S}}_2},{\mathrm{E}}_{\mathrm{I}}\left(\mathrm{t}\right)\le {\mathrm{M}}_{{\mathrm{E}}_{\mathrm{I}}}, {\mathrm{E}}_{\mathrm{C}}\left(\mathrm{t}\right)\le {\mathrm{M}}_{{\mathrm{E}}_{\mathrm{C}}}, \mathrm{I}\left(\mathrm{t}\right)\le {\mathrm{M}}_{\mathrm{I}}, \mathrm{C}\left(\mathrm{t}\right)\le {\mathrm{M}}_{\mathrm{C}}, \mathrm{R}\left(\mathrm{t}\right)\le {\mathrm{M}}_{\mathrm{R}}.$

The total sum N(t) of individuals in all compartments at a given time t is given by the following equation:

The total derivative can now be calculated.

In essence, the positive and negative terms associated with the transmission of the infection are in equilibrium, resulting in the following equation:

Adjustment for natural variation in mortality:

$\frac{\mathrm{dN}\left(\mathrm{t}\right)}{\mathrm{dt}}=\mathrm{\Lambda }\left({\mathrm{p}}_1+{\mathrm{p}}_2\right)-\mathrm{\mu }\mathrm{N}\left(\mathrm{t}\right)$ in this equation, μ represents a weighted average of natural mortality rates. The general solution to this linear differential equation is as follows:

This shows that N(t) is bounded by $\frac{\mathrm{\Lambda }\left({\mathrm{p}}_1+{\mathrm{p}}_2\right)}{\mathrm{\mu }}$.

It can be demonstrated that a positive constant, ${\mathrm{M}}_{\mathrm{N}}$, exists such that N(t) is bounded by ${\mathrm{M}}_{\mathrm{N}}$ for all t ≥0. This is due to the fact that N(t) is the sum of the compartments ${\mathrm{S}}_1\left(\mathrm{t}\right),{\mathrm{S}}_2\left(\mathrm{t}\right),{\mathrm{E}}_{\mathrm{I}}\left(\mathrm{t}\right)$, and therefore the result follows from the boundedness of these individual compartments. Given that ${\mathrm{E}}_{\mathrm{C}}\left(\mathrm{t}\right),\mathrm{I}\left(\mathrm{t}\right),\mathrm{C}\left(\mathrm{t}\right),\mathrm{R}\left(\mathrm{t}\right)$ and N(t) are bounded, it follows that each compartment ${\mathrm{S}}_1\left(\mathrm{t}\right),{\mathrm{S}}_2\left(\mathrm{t}\right),{\mathrm{E}}_{\mathrm{I}}\left(\mathrm{t}\right),{\mathrm{E}}_{\mathrm{C}}\left(\mathrm{t}\right),\mathrm{I}\left(\mathrm{t}\right),\mathrm{C}\left(\mathrm{t}\right),\mathrm{R}\left(\mathrm{t}\right)$ is also bounded. It follows that there exist positive constants ${\mathrm{M}}_{{\mathrm{S}}_1}$, ${\mathrm{M}}_{{\mathrm{S}}_2}$, ${\mathrm{M}}_{{\mathrm{E}}_{\mathrm{I}}}$, ${\mathrm{M}}_{{\mathrm{E}}_{\mathrm{C}}}$, ${\mathrm{M}}_{\mathrm{I}}$, ${\mathrm{M}}_{\mathrm{C}}$, ${\mathrm{M}}_{\mathrm{R}}$ such that: This implies that the total population size cannot grow indefinitely and is therefore bounded.

The inequalities are as follows:

The given system of differential equations can be expressed as a Cauchy problem.

The Cauchy problem is then formulated as follows: The initial value X(0) is equal to ${\mathrm{X}}_0$, where ${\mathrm{X}}_0$represents the given initial conditions. The function F(X(t)) is infinitely differentiable on ${\mathbb{R}}_{+}^7$, and thus locally Lipschitzian there. The Cauchy-Lipschitz theorem allows us to conclude that there exists a unique maximum solution to the Cauchy problem associated with the differential equation (1) for the initial condition (${\mathrm{t}}_0$, ${\mathrm{X}}_0$) ∈ ${\mathbb{R}}_{+}^7.$ Furthermore, since F(X(t)) is of class ${\mathrm{C}}^{\mathrm{\infty }}$, this solution is also of class ${\mathrm{C}}^{\mathrm{\infty }}$.

The calculation of the infection-free equilibrium point (DFE) is as follows: the DFE is reached when ${\mathrm{E}}_{\mathrm{I}}={\mathrm{E}}_{\mathrm{C}}=\mathrm{I}=\mathrm{C}=0$. Assuming that the populations of susceptible ${\mathrm{S}}_{\mathrm{i}}$ are at equilibrium, we can write:: $\left\{\begin{array}{c}0={\mathrm{p}}_1\mathrm{\Lambda }-{\mathrm{\mu }}_{{\mathrm{S}}_1}{\mathrm{S}}_1\\ 0={\mathrm{p}}_2\mathrm{\Lambda }-{\mathrm{\mu }}_{{\mathrm{S}}_2}{\mathrm{S}}_2\end{array}\right.$

The solutions are as follows: $\left({\mathrm{S}}_1^{*},{\mathrm{S}}_2^{*},{\mathrm{E}}_{\mathrm{I}},{\mathrm{E}}_{\mathrm{C}},\mathrm{I},\mathrm{C},\mathrm{R}\right)=\left(\frac{{\mathrm{p}}_1\mathrm{\Lambda }}{{\mathrm{\mu }}_{{\mathrm{S}}_1}}, \frac{{\mathrm{p}}_2\mathrm{\Lambda }}{{\mathrm{\mu }}_{{\mathrm{S}}_2}}, 0, 0, 0, 0, 0\right)$

The basic reproduction number (${\mathrm{R}}_0$) is calculated using the spectrum of the Jacobian matrix evaluated at the DFE. This matrix is formed by linearising the system around the DFE. The new infection rate matrix (F) and the transition rate matrix between infected compartments (V) are given by:

The Jacobians of F and V are evaluated in the absence of infection, whereby ${\mathrm{E}}_{\mathrm{I}}={\mathrm{E}}_{\mathrm{C}}=\mathrm{I}=\mathrm{C}=0$.

The basic reproduction number, denoted by ${\mathrm{R}}_0$, is given by two distinct formulas: ${\mathrm{R}}_0=\mathrm{\rho }\left(-\mathrm{F}{\mathrm{V}}^{-1}\right)$ with $\mathrm{\rho }\left(\mathrm{A}\right)$. In the first formula, ρ is the spectral radius of the matrix $-\mathrm{F}{\mathrm{V}}^{-1}$, which is defined as the dominant eigenvalue. This formula is applicable when ${\mathrm{S}}_{\mathrm{P}}$(A) represents the spectrum of A. In the second formula, A is the matrix whose eigenvalues are the solutions of the characteristic equation:

det$(-\mathrm{F}{\mathrm{V}}^{-1}-\mathrm{\lambda I})=0$.

The matrix $\mathrm{F}{\mathrm{V}}^{-1}$ is given by the equation

The expression for ${\mathrm{R}}_0$ can be approximated by the following approximation:

Theorem 1: In the case of system (1), if ${\mathrm{R}}_0\le \mathrm{ }1$, then the DFE is globally asymptotically stable along the positive orthant ${\mathbb{R}}_{+}^7$. Conversely, if ${\mathrm{R}}_0>1$, the DFE is unstable.

Proof: In order to study the stability of the DFE, it is necessary to analyse the eigenvalues of the Jacobian matrix of the system evaluated at the DFE . If all the eigenvalues have a negative real part, the DFE is locally asymptotically stable. It will be demonstrated that the initial situation arises when ${\mathrm{R}}_0\le 1$. The Jacobian matrix of the system (1) evaluated at the disease-free equilibrium is given by J(0) = F + V. Given that F is non-negative and V is a stable Metzler matrix, it follows that F + V is a regular decomposition of J(0). Therefore, by virtue of , we can conclude that $\mathrm{\rho }\left(-\mathrm{F}{\mathrm{V}}^{-1}\right)$ is equivalent to where α(M) denotes the stability modulus of the matrix M, which is defined as the largest real part of the elements of its spectrum. It follows that the disease-free equilibrium is locally asymptotically stable. This, in turn, implies, in accordance with Hirsch's theorem, that the disease-free equilibrium (in this case, the origin) is globally asymptotically stable if ${\mathrm{R}}_0=\mathrm{\rho }\left(-\mathrm{F}{\mathrm{V}}^{-1}\right)\mathrm{ }<\mathrm{ }1$.

Proposition: In the case of system (1), a unique endemic equilibrium point $\left({\mathrm{S}}_1^{*},{\mathrm{S}}_2^{*},{\mathrm{E}}_{\mathrm{I}}^{*},{\mathrm{E}}_{\mathrm{C}}^{*},{\mathrm{I}}^{*},{\mathrm{C}}^{*},{\mathrm{R}}^{*}\right)$ is obtained by solving system (1) with, for example, the substitution method in the positive orthant, provided that ${\mathrm{R}}_0>\mathrm{ }1.$

Theorem 2: (The overall stability of the endemic balance is of paramount importance.) In the event that ${\mathrm{R}}_0$ is greater than 1, the single endemic equilibrium point $\left({\mathrm{S}}_1^{*},{\mathrm{S}}_2^{*},{\mathrm{E}}_{\mathrm{I}}^{*},{\mathrm{E}}_{\mathrm{C}}^{*},{\mathrm{I}}^{*},{\mathrm{C}}^{*},{\mathrm{R}}^{*}\right)$ of system (1) is globally asymptotically stable.

Proof: In order to examine the system's overall stability, we will simplify the model equation by grouping the infectious latents ${\mathrm{E}}_{\mathrm{I}}$ and chronic latents ${\mathrm{E}}_{\mathrm{C}}$ into a single class labelled ${\mathrm{I}}_1$, and the infectious (I) and chronic patients (C) into a second class labelled ${\mathrm{I}}_2$. The resulting simplified model is as follows:

The distinctive endemic equilibrium point $\left({\mathrm{S}}_1^{*},{\mathrm{S}}_2^{*},{\mathrm{I}}_1^{*},{\mathrm{I}}_2^{*},{\mathrm{R}}^{*}\right)$ is defined by the following relationships:

The following Lyapunov candidate function is worthy of consideration:

The derivative of the Lyapunov candidate function V along trajectories of the ordinary differential system (2) is given by the following expression:

By employing the system relations at the endemic equilibrium point of the system (3), we arrive at the following conclusion:

By posing $\mathrm{\alpha }=\frac{{{\mathrm{\beta }}_{1,2}{\mathrm{S}}_1^{*}+\mathrm{\beta }}_{2,2}{\mathrm{S}}_2^{*}}{{\mathrm{\mu }}_{\mathrm{I}}+{\mathrm{\gamma }}_1+{\mathrm{\mu }}_{\mathrm{C}}+{\mathrm{\gamma }}_2}$,

it just ${{\mathrm{\beta }}_{1,2}{\mathrm{S}}_1^{*}+\mathrm{\beta }}_{2,2}{\mathrm{S}}_2^{*}=\mathrm{\alpha }\left({\mathrm{\mu }}_{\mathrm{I}}+{\mathrm{\gamma }}_1+{\mathrm{\mu }}_{\mathrm{C}}+{\mathrm{\gamma }}_2\right)$

In order to establish the relationships set out in (3),

we have: $\left({\mathrm{\beta }}_{1,1}{\mathrm{I}}_1^{*}+{\mathrm{\beta }}_{1,2}{\mathrm{I}}_2^{*}\right){\mathrm{S}}_1^{*}+\left({\mathrm{\beta }}_{2,1}{\mathrm{I}}_1^{*}+{\mathrm{\beta }}_{2,2}{\mathrm{I}}_2^{*}\right){\mathrm{S}}_2^{*}=\left({\mathrm{\mu }}_{{\mathrm{E}}_{\mathrm{I}}}+\mathrm{\gamma }+{\mathrm{\mu }}_{{\mathrm{E}}_{\mathrm{C}}}+\mathrm{\delta }\mathrm{ }\right){\mathrm{I}}_1^{*}$