Optimal Control Strategies for a Mathematical Model of Pneumonia Infection

Abstract

In this study, we formulate and analyze a deterministic mathematical model describing the transmission dynamics of pneumonia. A comprehensive stability analysis is conducted for both the disease-free and endemic equilibrium points. The disease-free equilibrium is locally and globally asymptotically stable when the basic reproduction number R₀ < 1, while the endemic equilibrium is locally and globally asymptotically stable when R₀ > 1. To evaluate effective intervention strategies, an optimal control problem is formulated by introducing time-dependent control variables representing awareness campaigns, screening of carriers, and treatment of infected individuals. Applying Pontryagin’s Maximum Principle, the simulation results confirm the effectiveness of the proposed control strategies in reducing the number of infections and mitigating the overall disease burden. The findings offer valuable insights into the control of pneumonia and highlight the potential impact of strategic public health interventions.

1. Introduction

Pneumonia is an acute respiratory disease that poses a major threat to public health worldwide, causing millions of deaths annually, especially among vulnerable groups such as children and the elderly. As an increasing global health challenge, there is a need for effective means to understand and control the spread of this disease [1,2,3,4,5,6,7,8].

Mathematical modeling serves as an indispensable tool for deciphering the epidemiological stability and transmission patterns of infectious diseases, aiding in the formulation of essential control strategies. Such models are instrumental in forecasting the spread of the disease and assessing the efficacy of various interventions, including vaccination and treatment programs. Research indicates that mathematical models integrating vaccination and treatment interventions have helped to reduce pneumonia infection rates [2,9].

In this study, we developed a comprehensive epidemiological model to analyze the spread of pneumonia within a population. The formulation of the present model builds upon the foundational work presented in [9], where we introduced a novel approach to modeling the incidence rate of infections.

Traditionally, epidemic models have employed a bilinear incidence rate, given by $\beta SI$ [10,11,12,13,14,15], where S represents the number of susceptible individuals and I represents the number of infectious individuals. This model assumes direct proportionality between the product of susceptible and infectious, which simplifies the dynamics of disease spread but may not fully capture the complexities of real-world transmission dynamics.

To address the limitations of the bilinear incidence rate, we adopt a saturated incidence rate in our model, expressed as ${\displaystyle \frac{\beta SI}{1+mI}}$. This formulation, which was initially introduced by Capasso and Serio [16], incorporates a saturation term that accounts for the nonlinear effects observed in real-world transmission scenarios, such as behavioral changes and resource limitations as the infection prevalence increases. This form of incidence rate captures the saturation effect, where the transmission rate does not increase indefinitely with the number of infectious individuals. The modified model provides a more realistic framework for understanding the spread of pneumonia in populations where contact rates are not strictly proportional to the number of infectious individuals.

The remainder of this paper is structured as follows: In Section 2, we formulate the mathematical model. Section 3 is devoted to the analysis of the local and global stability of the disease-free and endemic equilibrium points. In Section 4, we investigate the occurrence of forward bifurcation. Section 5 presents the formulation and analysis of the optimal control problem. Numerical simulations illustrating the model dynamics and control strategies are provided in Section 6. Finally, in Section 7, we present a cost-effectiveness analysis of the implemented control strategies.

2. Model Formulation

In the study conducted by Mumbu [9], a mathematical model of pneumonia was formulated using a bilinear incidence term of the form $\beta SI$. In this work, we generalize the model proposed by Mumbu by replacing the bilinear incidence with a saturated incidence rate of the form ${\displaystyle \frac{\beta SI}{1+mI}}$, which better reflects the nonlinear transmission dynamics in the presence of contact inhibition or resource limitations. The model compartments include susceptible $S\left(t\right)$, vaccinated $V\left(t\right)$, carrier $C\left(t\right)$, treated $T\left(t\right)$, infected $I\left(t\right)$, and recovered $R\left(t\right)$ individuals. The dynamics are governed by the following set of nonlinear differential equations:

$$\begin{array}{cc}\hfill {\displaystyle \frac{dS}{dt}}& =\Lambda +\gamma R-{\displaystyle \frac{\beta SI}{1+mI}}-\rho S-\mu S,\hfill \\ \hfill {\displaystyle \frac{dV}{dt}}& =\rho S+\sigma R-\mu V,\hfill \\ \hfill {\displaystyle \frac{dC}{dt}}& ={\displaystyle \frac{\beta SI}{1+mI}}-(\mu +\delta +ϵ)C,\hfill \\ \hfill {\displaystyle \frac{dT}{dt}}& =\delta C+\tau I-(\mu +\theta )T,\hfill \\ \hfill {\displaystyle \frac{dI}{dt}}& =ϵC-(\mu +\tau +\alpha +\kappa )I,\hfill \\ \hfill {\displaystyle \frac{dR}{dt}}& =\theta T+\kappa I-(\mu +\sigma +\gamma )R.\hfill \end{array}$$

(1)

The parameters used in the proposed model are detailed in

Table 1. Description of the parameters of the model.

Parameter	Description	Value/Month	Sources
$\Lambda$	Recruitment rate	10.09	[2]
$\epsilon$	Latent-to-infection transfer rate	0.01096	[17]
$\delta$	Treatment of disease carriers	0.04	[9]
$\beta$	Pneumonia infection rate	0.0287	[2]
m	Saturation parameter	0.5	[2]
$\kappa$	Natural recovery rate of infectious individuals	0.0115	[18]
$\theta$	Treatment recovery rate	0.02	[2]
$\alpha$	Disease-induced death rate	0.36	[9]
$\gamma$	Immunity waning rate	0.00095	[9]
$\mu$	Human mortality rate	0.0002	[19]
$\rho$	Vaccination rate of susceptible individuals	0.0621	[9]
$\sigma$	Vaccination rate of recovered individuals	0.36	[20]
$\tau$	Treatment rate of infectious individuals	0.07	[9]

, and the model flowchart is given in Figure 1.

The positivity and boundedness of the solutions to the proposed model can be established using standard techniques. Since the proofs are similar to those presented in [9], they are omitted here for brevity.

The basic reproduction number, $R_0$, is a key epidemiological metric representing the average number of secondary infections produced by a single infected individual in a completely susceptible population without any intervention [21,22,23]. The basic reproduction number $R_0$ can be calculated using the next-generation operator method, as referenced by Driessche et al. [24].

$$R_0={\displaystyle \frac{\beta \Lambda ϵ}{(\mu +\rho )(\mu +\delta +ϵ)(\mu +\tau +\alpha +\kappa )}}.$$

(2)

This value of $R_0$ provides a threshold for the spread of the disease.

3. Stability Analysis of the Pneumonia Model

3.1. Steady States of the Model

The pneumonia-free equilibrium point $E_0$ of (1) is

$$E_0=\left({\displaystyle \frac{\Lambda }{\mu +\rho }},{\displaystyle \frac{\rho \Lambda }{\mu (\mu +\rho )}},0,0,0,0\right).$$

(3)

In investigating disease dynamics within a population, the presence of an endemic equilibrium point signifies that the disease has become a persistent feature within a population compartment. Let us denote this point, where the pneumonia disease is a dominant factor at equilibrium, as $E^{*}$. The endemic equilibrium is given by

$$\begin{array}{cc}\hfill S^{*}& ={\displaystyle \frac{q_2{q}_4(1+m{I}^{*})}{ϵ\beta }},\hfill \\ \hfill V^{*}& ={\displaystyle \frac{\beta \sigma I^{*}\left(k{q}_3ϵ+\delta \theta q_4+\theta \tau ϵ\right)+(1+m{I}^{*})\rho q_2{q}_3{q}_4{q}_5}{ϵ\beta \mu q_3{q}_5}},\hfill \\ \hfill C^{*}& ={\displaystyle \frac{q_4}{ϵ}}{I}^{*},\hfill \\ \hfill T^{*}& ={\displaystyle \frac{\left(\delta q_4+\tau ϵ\right)}{ϵq_3}}{I}^{*},\hfill \\ \hfill R^{*}& ={\displaystyle \frac{\left(\kappa q_3ϵ+\delta \theta q_4+\theta \tau ϵ\right)}{ϵq_3{q}_5}}{I}^{*},\hfill \\ \hfill I^{*}& ={\displaystyle \frac{q_1{q}_2{q}_3{q}_4{q}_5\left(R_0-1\right)}{\Psi }},\hfill \end{array}$$

(4)

where

$$\begin{array}{cc}\hfill q_1& =\rho +\mu ,\hfill \\ \hfill q_2& =\mu +\delta +ϵ,\hfill \\ \hfill q_3& =\mu +\theta ,\hfill \\ \hfill q_4& =\mu +\tau +\alpha +\kappa ,\hfill \\ \hfill q_5& =\mu +\sigma +\gamma .\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill \Psi & =\beta \gamma \mu \left[(\delta +\theta +\mu )(\kappa +\mu +\tau )+ϵ(\theta +\mu +\tau )\right]\hfill \\ \hfill & +\beta (\theta +\mu )(\mu +\sigma )(\delta +\mu +ϵ)(\kappa +\mu +\tau )\hfill \\ \hfill & +\alpha [\beta \left(\gamma \delta \mu +\gamma (\theta +\mu )(\mu +ϵ)+(\theta +\mu )(\mu +\sigma )(\delta +\mu +ϵ)\right)\hfill \\ \hfill & +m(\theta +\mu )(\mu +\rho )(\gamma +\mu +\sigma )(\delta +\mu +ϵ)]\hfill \\ \hfill & +m(\theta +\mu )(\mu +\rho )(\gamma +\mu +\sigma )(\delta +\mu +ϵ)(\kappa +\mu +\tau ).\hfill \end{array}$$

In the following subsections, we investigate the local asymptotic stability (LAS) and global asymptotic stability (GAS) of the equilibrium points of the pneumonia model.

Theorem 1.

If the basic reproduction number $R_0<1$, then the pneumonia-free equilibrium point $E_0$ is (LAS), while it is unstable for $R_0>1$.

Proof. 

To establish the local stability of the pneumonia-free equilibrium point $E_0$, we compute the Jacobian matrix of the pneumonia model (Equation (1)) at $E_0$, where

$$J\left(E_0\right)=\left(\begin{array}{cccccc}-q_1& 0& 0& 0& -{\displaystyle \frac{\beta \Lambda }{q_1}}& \gamma \\ \rho & -\mu & 0& 0& 0& \sigma \\ 0& 0& -q_2& 0& {\displaystyle \frac{\beta \Lambda }{q_1}}& 0\\ 0& 0& \delta & -q_3& \tau & 0\\ 0& 0& \epsilon & 0& -q_4& 0\\ 0& 0& 0& \theta & \kappa & -q_5\end{array}\right).$$

(6)

The first four eigenvalues are

$${\lambda }_1=-q_1,{\lambda }_2=-\mu ,{\lambda }_3=-q_3,{\lambda }_4=-q_5.$$

The remaining eigenvalues, ${\lambda }_5$ and ${\lambda }_6$, are obtained from the characteristic equation

$${\lambda }^2+(q_2+q_4)\lambda +q_2{q}_4\left(1-R_0\right)=0.$$

(7)

According to the Routh–Hurwitz criterion, Equation (7) possesses strictly negative eigenvalues if and only if $R_0<1$ is satisfied. The fulfillment of this criterion guarantees that the system is asymptotically stable; conversely, its violation leads to instability. □

Theorem 2.

The endemic point of the pneumonia system (Equation (1)) is LAS if $R_0>1$.

Proof. 

The Jacobian matrix around $E^{*}$ is as follows:

$$J\left(E^{*}\right)=\left(\begin{array}{cccccc}{\displaystyle -\frac{\beta I^{*}}{1+m{I}^{*}}-q_1}& 0& 0& 0& -{\displaystyle \frac{\beta S^{*}}{{(1+m{I}^{*})}^2}}& \gamma \\ \rho & -\mu & 0& 0& 0& \sigma \\ {\displaystyle \frac{\beta I^{*}}{1+m{I}^{*}}}& 0& -q_2& 0& {\displaystyle \frac{\beta S^{*}}{{(1+m{I}^{*})}^2}}& 0\\ 0& 0& \delta & -q_3& \tau & 0\\ 0& 0& ϵ& 0& -q_4& 0\\ 0& 0& 0& \theta & \kappa & -q_5\end{array}\right).$$

(8)

The first eigenvalue ${\lambda }_1$ of $J\left(F^{*}\right)$ is negative, with ${\lambda }_1=-\mu <0$. The characteristic equation computes the other five eigenvalues, which are

$${\lambda }^5+c_1{\lambda }^4+c_2{\lambda }^3+c_3{\lambda }^2+c_4\lambda +c_5=0,$$

(9)

where

$$\begin{array}{cc}\hfill c_1=& q_1+q_2+q_3+q_4+q_5+\Phi ,\hfill \\ \hfill c_2=& q_2\left(q_1+q_3+q_5+\Phi +{\displaystyle \frac{m{q}_4\Phi }{\beta }}\right)+q_4\left(q_1+\Phi \right)+q_5\left(q_1+q_4+\Phi \right)+q_3\left(q_1+q_4+q_5+\Phi \right),\hfill \\ \hfill c_3=& q_5\left({\displaystyle \frac{m{q}_2{q}_4\Phi }{\beta }}+q_2\Phi +q_4\Phi \right)+q_3\left({\displaystyle \frac{m{q}_2{q}_4\Phi }{\beta }}+q_4\Phi +q_5\left(q_4+\Phi \right)+q_2\left(q_5+\Phi \right)\right)\hfill \\ \hfill & +q_1\left({\displaystyle \frac{m{q}_2{q}_4\Phi }{\beta }}+q_3{q}_4+\left(q_3+q_4\right)q_5+q_2\left(q_3+q_5\right)\right)+q_2{q}_4\Phi ,\hfill \\ \hfill c_4=& q_5\left[q_3\left({\displaystyle \frac{m{q}_2{q}_2\Phi }{\beta }}+q_2\Phi +q_4\Phi \right)+q_1\left({\displaystyle \frac{m{q}_2{q}_4\Phi }{\beta }}+q_2{q}_3+q_3{q}_4\right)+q_2{q}_4\Phi \right]\hfill \\ \hfill & +q_3\left({\displaystyle \frac{m{q}_1{q}_2{q}_4\Phi }{\beta }}+q_2{q}_4\Phi \right),\hfill \\ \hfill c_5=& ϵq_1{q}_3{q}_5\Omega \left(R_0-1\right),\hfill \end{array}$$

$$\Omega ={\displaystyle \frac{q_2{q}_4}{ϵ(m{I}^{*}+1)}},\Phi ={\displaystyle \frac{\beta I^{*}}{m{I}^{*}+1}}.$$

(10)

According to the Routh–Hurwitz criterion, all roots of the characteristic (Equation (9)) have negative real parts if the following conditions hold:

$$c_1>0,c_5>0,c_1{c}_2-c_3>0,c_3(c_1{c}_2-c_3)-c_1(c_1{c}_4-c_5)>0,$$

$$(c_1{c}_2-c_3)(c_3{c}_4-c_2{c}_5)-{(c_1{c}_4-c_5)}^2>0.$$

The condition $c_5>0$ is satisfied if and only if $R_0>1$. Moreover, it can be demonstrated analytically, by substituting from Equations (5) and (10), that all other Routh–Hurwitz conditions are satisfied. □

3.2. Global Stability of the Pneumonia-Free Equilibrium

The following theorem is devoted to proving the global asymptotic stability of the pneumonia-free equilibrium point.

Theorem 3.

If the basic reproduction number $R_0<1$, the pneumonia-free equilibrium, $E_0$, is globally asymptotically stable (GAS).

Proof. 

To analyze the global stability of the pneumonia-free equilibrium point $E_0$ in the pneumonia infection model (Equation (1)), we use the Lyapunov function approach [25,26]. The Lyapunov function is defined as

$${\Psi }_1=C+{\displaystyle \frac{q_2}{ϵ}}I.$$

The derivative of ${\Psi }_1$ with respect to time is given by

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{\Psi }_1}{dt}}& ={\displaystyle \frac{\beta SI}{1+mI}}-q_{2}C+{\displaystyle \frac{q_2}{ϵ}}\left(ϵC-q_{4}I\right)\hfill \\ \hfill & \le {\displaystyle \frac{\beta \Lambda I}{q_1}}-{\displaystyle \frac{q_2{q}_{4}I}{ϵ}}\hfill \\ \hfill & \le {\displaystyle \frac{q_2{q}_{4}I}{ϵ}}\left(R_0-1\right).\hfill \end{array}$$

Then, according to LaSalle’s invariance principle, the pneumonia-free equilibrium, $E_0$, is globally asymptotically stable once $R_0<1$. □

The following theorem establishes the global stability of the pneumonia endemic equilibrium point $E^{*}$ at $m=0$ and $\gamma =0$.

Theorem 4.

If the basic reproduction number $R_0>1$, the endemic equilibrium point, $E^{*}$, is globally asymptotically stable (GAS).

Proof. 

In order to analyze the global stability of the pneumonia coexistence equilibrium point $E^{*}$, we define the Lyapunov function as follows:

$$\begin{array}{cc}\hfill {\Psi }_2=& \left[S-S^{*}-S^{*}ln\left({\displaystyle \frac{S}{S^{*}}}\right)\right]+\left[C-C^{*}-C^{*}ln\left({\displaystyle \frac{C}{C^{*}}}\right)\right]\hfill \\ \hfill & +{\displaystyle \frac{\beta S^{*}{I}^{*}}{\epsilon C^{*}}}\left[I-I^{*}-I^{*}ln\left({\displaystyle \frac{I}{I^{*}}}\right)\right].\hfill \end{array}$$

(11)

Using the pneumonia system (Equation (1)) at the endemic equilibrium point $E^{*}$, one obtains

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{\Psi }_2}{dt}}=& \left(1-{\displaystyle \frac{S^{*}}{S}}\right)\left[\beta S^{*}{I}^{*}+q_1{S}^{*}-\beta SI-q_{1}S\right]\hfill \\ \hfill & +\left(1-{\displaystyle \frac{C^{*}}{C}}\right)\left[\beta SI-{\displaystyle \frac{\beta S^{*}{I}^{*}C}{C^{*}}}\right]+{\displaystyle \frac{\beta S^{*}{I}^{*}}{\epsilon C^{*}}}\left(1-{\displaystyle \frac{I^{*}}{I}}\right)\left[\epsilon C-{\displaystyle \frac{\epsilon C^{*}I}{I^{*}}}\right]\hfill \\ \hfill & =\left(1-{\displaystyle \frac{S^{*}}{S}}\right)\left[-(\mu +\rho )\left(S-S^{*}\right)+\beta S^{*}{I}^{*}\left(1-{\displaystyle \frac{SI}{S^{*}{I}^{*}}}\right)\right]\hfill \\ \hfill & +\left(1-{\displaystyle \frac{C^{*}}{C}}\right)\beta S^{*}{I}^{*}\left[{\displaystyle \frac{SI}{S^{*}{I}^{*}}}-{\displaystyle \frac{C}{C^{*}}}\right]+{\displaystyle \frac{\beta S^{*}{I}^{*}}{\epsilon C^{*}}}\left(1-{\displaystyle \frac{I^{*}}{I}}\right)\epsilon C^{*}\left[{\displaystyle \frac{C}{C^{*}}}-{\displaystyle \frac{I}{I^{*}}}\right]\hfill \\ \hfill & =-q_1{\displaystyle \frac{(S-S^{*})(S-S^{*})}{S}}+\beta S^{*}{I}^{*}\left(1-{\displaystyle \frac{S^{*}}{S}}\right)\left(1-{\displaystyle \frac{SI}{S^{*}{I}^{*}}}\right)\hfill \\ \hfill & +\beta S^{*}{I}^{*}\left(1-{\displaystyle \frac{C^{*}}{C}}\right)\left[{\displaystyle \frac{SI}{S^{*}{I}^{*}}}-{\displaystyle \frac{C}{C^{*}}}\right]+K_1\epsilon C^{*}\left(1-{\displaystyle \frac{I^{*}}{I}}\right)\left[{\displaystyle \frac{C}{C^{*}}}-{\displaystyle \frac{I}{I^{*}}}\right]\hfill \\ & =-(\mu +\rho ){\displaystyle \frac{{(S-S^{*})}^2}{S}}+\beta S^{*}{I}^{*}\left(1-{\displaystyle \frac{S^{*}}{S}}-{\displaystyle \frac{SI}{S^{*}{I}^{*}}}+{\displaystyle \frac{I}{I^{*}}}\right)\hfill \\ \hfill & +\beta S^{*}{I}^{*}\left({\displaystyle \frac{SI}{S^{*}{I}^{*}}}-{\displaystyle \frac{C^{*}}{C}}-{\displaystyle \frac{SI{C}^{*}}{S^{*}{I}^{*}C}}+1\right)+\beta S^{*}{I}^{*}\left({\displaystyle \frac{C}{C^{*}}}-{\displaystyle \frac{I}{I^{*}}}-{\displaystyle \frac{I^{*}C}{I{C}^{*}}}+1\right)\hfill \\ \hfill & =-(\mu +\rho ){\displaystyle \frac{{(S-S^{*})}^2}{S}}\hfill \\ \hfill & +\beta S^{*}{I}^{*}\left[1-{\displaystyle \frac{S^{*}}{S}}-{\displaystyle \frac{SI}{S^{*}{I}^{*}}}+{\displaystyle \frac{I}{I^{*}}}+{\displaystyle \frac{SI}{S^{*}{I}^{*}}}-{\displaystyle \frac{C^{*}}{C}}-{\displaystyle \frac{SI{C}^{*}}{S^{*}{I}^{*}C}}+1+{\displaystyle \frac{C}{C^{*}}}-{\displaystyle \frac{I}{I^{*}}}-{\displaystyle \frac{I^{*}C}{I{C}^{*}}}+1\right]\hfill \\ \hfill & =-(\mu +\rho ){\displaystyle \frac{{(S-S^{*})}^2}{S}}+\beta S^{*}{I}^{*}\left[3-{\displaystyle \frac{S^{*}}{S}}-{\displaystyle \frac{SI{C}^{*}}{S^{*}{I}^{*}C}}-{\displaystyle \frac{I^{*}C}{I{C}^{*}}}\right].\hfill \end{array}$$

Applying the arithmetic mean–geometric mean inequality, we conclude that ${\displaystyle \frac{d{\Psi }_2}{dt}}$ is negative definite. Thus, the Lyapunov function is decreasing along the trajectories of the system, except at the equilibrium point itself, where it is zero. This implies that the endemic equilibrium is globally asymptotically stable under the condition $R_0>1$. □

4. Forward Bifurcation Analysis

In this section, we invoke the forward bifurcation of the pneumonia system (Equation (1)). To this end, we apply the center manifold theory [27] to derive sufficient conditions under which $E^{*}$ remains locally asymptotically stable whenever the basic reproduction number exceeds one, i.e., $R_0>1$. To facilitate the analysis, we introduce a change in notation for the state variables: let $x_1=S$, $x_2=V$, $x_3=C$, $x_4=T$, $x_5=I$, and $x_6=R$. These variables are then assembled into the state vector $x={(x_1,x_2,x_3,x_4,x_5,x_6)}^{\top }$. Using this notation, the system of differential equations governing the dynamics can be expressed as

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{x}_1}{dt}}=f_1& =\Lambda +\gamma x_6-{\displaystyle \frac{\beta x_1{x}_5}{1+m{x}_5}}-q_1{x}_1,\hfill \\ \hfill {\displaystyle \frac{d{x}_2}{dt}}=f_2& =\rho x_1+\sigma x_6-\mu x_2,\hfill \\ \hfill {\displaystyle \frac{d{x}_3}{dt}}=f_3& ={\displaystyle \frac{\beta x_1{x}_5}{1+m{x}_5}}-q_2{x}_3,\hfill \\ \hfill {\displaystyle \frac{d{x}_4}{dt}}=f_4& =\delta x_3+\tau x_5-q_3{x}_4,\hfill \\ \hfill {\displaystyle \frac{d{x}_5}{dt}}=f_5& =\epsilon x_3-q_4{x}_5,\hfill \\ \hfill {\displaystyle \frac{d{x}_6}{dt}}=f_6& =\theta x_4+\kappa x_5-q_5{x}_6.\hfill \end{array}$$

(12)

To apply the center manifold theory, it is essential to first identify the right eigenvector $w=(w_1,w_2,$$w_3,w_4,w_5,w_6{)}^{\top }$ associated with the zero eigenvalue of the Jacobian matrix J. This eigenvector plays a central role in characterizing the local behavior of the system near the equilibrium point. To determine w, we solve the following homogeneous linear system:

$$\left(\begin{array}{cccccc}-q_1& 0& 0& 0& -{\displaystyle \frac{\beta \Lambda }{q_1}}& \gamma \\ \rho & -\mu & 0& 0& 0& \sigma \\ 0& 0& -q_2& 0& {\displaystyle \frac{\beta \Lambda }{q_1}}& 0\\ 0& 0& \delta & -q_3& \tau & 0\\ 0& 0& \epsilon & 0& -q_4& 0\\ 0& 0& 0& \theta & \kappa & -q_5\end{array}\right)\left(\begin{array}{c}{w}_1\\ w_2\\ w_3\\ w_4\\ w_5\\ w_6\end{array}\right)=\left(\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 0\end{array}\right).$$

Thus, all components of the eigenvector w are explicitly determined in terms of the model parameters as

$$\begin{array}{cc}\hfill {\omega }_1& ={\displaystyle \frac{1}{q_1^2}}\left({\displaystyle \frac{\gamma q_1\left(ϵ\theta \tau +kϵq_3+\delta \theta q_4\right)}{ϵq_3{q}_5}}-\beta \Lambda \right),\hfill \\ \hfill {\omega }_2& ={\displaystyle \frac{1}{\mu q_1^2}}\left({\displaystyle \frac{q_1\left(\gamma \rho +\sigma q_1\right)\left(ϵ\theta \tau +kϵq_3+\delta \theta q_4\right)}{ϵq_3{q}_5}}-\beta \Lambda \rho \right),\hfill \\ \hfill {\omega }_3& ={\displaystyle \frac{q_4}{ϵ}},\hfill \\ \hfill {\omega }_4& ={\displaystyle \frac{ϵ\tau +\delta q_4}{ϵq_3}},\hfill \\ \hfill {\omega }_5& =1,\hfill \\ \hfill {\omega }_6& ={\displaystyle \frac{ϵ\theta \tau +kϵq_3+\delta \theta q_4}{ϵq_3{q}_5}}.\hfill \end{array}$$

We now proceed to determine the left eigenvector $v=(v_1,v_2,v_3,v_4,v_5,v_6)$ associated with the zero eigenvalue of the Jacobian matrix $J_{\mathrm{DFE}}$. This vector satisfies the homogeneous linear system $v·J_{\mathrm{DFE}}=0$. Thus, the left eigenvector v is fully determined as

$$v=\left(0,0,{\displaystyle \frac{\epsilon }{\mu +\delta +\epsilon }},0,1,0\right).$$

With both the left and right eigenvectors now known, we proceed to compute the second-order coefficients $M_1$ and $M_2$ that arise in the center manifold expansion [27]. These coefficients are given by

$$M_1=\sum _{i,j,k=1}^6{v}_k{w}_i{w}_j{\displaystyle \frac{{\partial }^2{f}_k}{\partial x_i\partial x_j}},M_2=\sum _{i,k=1}^6{v}_k{w}_i{\displaystyle \frac{{\partial }^2{f}_k}{\partial x_i\partial \beta }}.$$

(13)

Thus,

$$\begin{array}{cc}\hfill M_1& ={\displaystyle \frac{2\beta \Lambda \left(\beta +m{q}_1\right)}{q_1^2}}\left({\displaystyle \frac{\gamma q_1\left(\delta \theta q_4+3q_3ϵ+\theta \tau ϵ\right)}{\Lambda q_3{q}_5ϵ\left(\beta +m{q}_1\right)}}-1\right),\hfill \\ \hfill M_2& ={\displaystyle \frac{\Lambda ϵ}{q_1{q}_2}}.\hfill \end{array}$$

(14)

Consequently, given that the coefficient $M_2$ is positive, it is deduced from Castillo-Chavez and Song’s theorem, as referenced in [27], that the pneumonia infection model outlined in Equation (1) will experience a forward bifurcation if the coefficient $M_1$, as defined by Equation (14), is negative. Specifically, this happens when the following inequality is satisfied:

$${\displaystyle \frac{\gamma q_1\left(\delta \theta q_4+3q_3ϵ+\theta \tau ϵ\right)}{\Lambda q_3{q}_5ϵ\left(\beta +m{q}_1\right)}}<1.$$

5. Optimal Control Strategies

To identify the most effective intervention strategies for eradicating pneumonia within a specified time frame, we extend the baseline model (Equation (1)) by incorporating three time-dependent control functions. Each control represents a distinct public health measure:

(i)

$u_1$ is an awareness campaign.

(ii)

$u_2$ is a screening for carriers.

(iii)

$u_3$ is a treatment for the infected.

Introducing these controls into Equation (1) yields the following optimal control system:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{dS}{dt}}=\Lambda +\gamma R-{\displaystyle \frac{\beta \left(1-u_1\right)SI}{1+mI}}-(\mu +\rho )S,\hfill \\ \hfill & {\displaystyle \frac{dV}{dt}}=\rho S+\sigma R-\mu V,\hfill \\ \hfill & {\displaystyle \frac{dC}{dt}}={\displaystyle \frac{\beta \left(1-u_1\right)SI}{1+mI}}-\left(\mu +\delta +\epsilon +u_2\right)C,\hfill \\ \hfill & {\displaystyle \frac{dT}{dt}}=\left(u_2+\delta \right)C+\tau I-(\mu +\theta )T,\hfill \\ \hfill & {\displaystyle \frac{dI}{dt}}=\epsilon C-\left(\mu +\tau +\alpha +\kappa +u_3\right)I,\hfill \\ \hfill & {\displaystyle \frac{dR}{dt}}=\theta T+\left(\kappa +u_3\right)I-(\mu +\sigma +\gamma )R.\hfill \end{array}$$

(15)

To determine the optimal levels of the controls $u_1\left(t\right)$, $u_2\left(t\right)$, and $u_3\left(t\right)$, we define the objective function J as

$$J={\int }_0^T\left[A_{1}C\left(t\right)+A_{2}I\left(t\right)+{\displaystyle \frac{1}{2}}\sum _{i=1}^3{B}_i{u}_i^2\left(t\right)\right]dt.$$

(16)

The positive constants $A_1$ and $A_2$ quantify the societal cost per carrier and per infected, respectively [28], while $B_i>0$ measures the cost-efficiency of control $u_i$. The integrated model reflects two goals: reducing the epidemiological burden, which is represented by the linear terms $A_{1}C\left(t\right)+A_{2}I\left(t\right)$, and limiting the economic cost of the interventions, which is captured by the quadratic terms ${\displaystyle \frac{1}{2}}{B}_i{u}_i^2\left(t\right)$. The objective is to minimize the cost function J over all admissible controls, which represents a trade-off between the costs associated with the disease burden and the costs of the control interventions.

The Hamiltonian and Optimality System

The Hamiltonian function H is defined as

$$\begin{array}{cc}\hfill \mathit{H}=& A_{1}C+A_{2}I+{\displaystyle \frac{1}{2}}\sum _{i=1}^3{B}_i{u}_i^2\left(t\right)+{\lambda }_1\left(t\right){\displaystyle \frac{dS\left(t\right)}{dt}}+{\lambda }_2\left(t\right){\displaystyle \frac{dV\left(t\right)}{dt}}\hfill \\ \hfill & +{\lambda }_3\left(t\right){\displaystyle \frac{dC\left(t\right)}{dt}}+{\lambda }_4\left(t\right){\displaystyle \frac{dT\left(t\right)}{dt}}+{\lambda }_5\left(t\right){\displaystyle \frac{dI\left(t\right)}{dt}}+{\lambda }_6\left(t\right){\displaystyle \frac{dR\left(t\right)}{dt}}.\hfill \end{array}$$

(17)

The functions ${\lambda }_i\left(t\right)$ for $i=1,2,3,4,5,6$ are adjoint variables [29]. The existence of optimal control pairs is established using the theoretical framework provided by Fleming and Rishel [30]. We now apply the necessary optimality conditions to the Hamiltonian function (Equation (17)). Then, there exist adjoint variables (${\lambda }_1\left(t\right),{\lambda }_2\left(t\right),\dots ,{\lambda }_6\left(t\right)$) satisfying the following adjoint system:

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{\lambda }_1}{dt}}& ={\displaystyle \frac{(1-u_1\left(t\right))\beta I({\lambda }_1-{\lambda }_3)}{1+mI}}+\rho ({\lambda }_1-{\lambda }_2)+\mu {\lambda }_1,\hfill \\ \hfill {\displaystyle \frac{d{\lambda }_2}{dt}}& =\mu {\lambda }_2,\hfill \\ \hfill {\displaystyle \frac{d{\lambda }_3}{dt}}& =-A_1-{\lambda }_4(u_2\left(t\right)+\delta )+(\mu +\delta +\epsilon +u_2\left(t\right)){\lambda }_3-\epsilon {\lambda }_5,\hfill \\ \hfill {\displaystyle \frac{d{\lambda }_4}{dt}}& =-\theta {\lambda }_6+(\mu +\theta ){\lambda }_4,\hfill \\ \hfill {\displaystyle \frac{d{\lambda }_5}{dt}}& =-A_2+{\displaystyle \frac{(1-u_1\left(t\right))\beta S({\lambda }_1-{\lambda }_3)}{1+mI}}-\tau {\lambda }_4\hfill \\ \hfill & +(\mu +\tau +\alpha +\kappa +u_3\left(t\right)){\lambda }_5-(\kappa +u_3\left(t\right)){\lambda }_6,\hfill \\ \hfill {\displaystyle \frac{d{\lambda }_6}{dt}}& =-\gamma {\lambda }_1-\sigma {\lambda }_2+(\mu +\sigma +\gamma ){\lambda }_6.\hfill \end{array}$$

(18)

The adjoint variables satisfy the transversality conditions ${\lambda }_i\left(t_f\right)=0$ for $i=1,2,\dots ,6$. Furthermore, the optimal control triplet $(u_1^{*},u_2^{*},u_3^{*})$ is characterized by the following conditions:

$$\begin{array}{cc}\hfill u_1^{*}\left(t\right)& =max\left\{min\left\{{\displaystyle \frac{\beta S\left(t\right)I\left(t\right)({\lambda }_3\left(t\right)-{\lambda }_1\left(t\right))}{B_1(1+mI\left(t\right))}},1\right\},0\right\},\hfill \\ \hfill u_2^{*}\left(t\right)& =max\left\{min\left\{{\displaystyle \frac{C\left(t\right)({\lambda }_3\left(t\right)-{\lambda }_4\left(t\right))}{B_2}},1\right\},0\right\},\hfill \\ \hfill u_3^{*}\left(t\right)& =max\left\{min\left\{{\displaystyle \frac{I\left(t\right)({\lambda }_5\left(t\right)-{\lambda }_6\left(t\right))}{B_3}},1\right\},0\right\}.\hfill \end{array}$$

(19)

6. Numerical Simulations

In this section, we describe the numerical simulations of the pneumonia infection model (Equation (1)) that were conducted to validate the analytical results using the parameter values listed in Table 1 (unless otherwise specified). These simulations aim to evaluate the potential impact of pneumonia control measures on the spread of the disease within the considered community. Figure 2a demonstrates that the endemic equilibrium is globally asymptotically stable when $R_0$ exceeds unity ($R_0=2.3614>1$), consistent with the conditions outlined in the theoretical stability criteria. These findings indicate that, under certain parameter regimes, the disease persists in the population despite intervention efforts, highlighting the importance of parameter sensitivity in disease control strategies.

In contrast, Figure 2b indicates that the disease-free equilibrium is (LAS) and also (GAS) under these conditions, aligning with expectations when both the infection rate is low and the treatment rate is high. In this scenario, the basic reproduction number is $R_0=0.7080<1$, supporting the conclusion that the infection would die out over time, which is consistent with epidemic control objectives.

The results underscore that parameter selection critically influences both the existence and stability of different equilibrium states. Understanding these relationships is essential for developing effective disease management policies. The combination of theoretical analysis and numerical simulations provides a comprehensive picture of the model’s behavior, facilitating potential applications to real-world scenarios and informing control measures for pneumonia outbreaks.

Figure 3 depicts the impact of the saturation coefficient m on the progression of pneumonia within the population over 100 months. Three distinct scenarios are illustrated, corresponding to m values of $0.1$, $0.2$, and $0.5$, with each curve representing the population of affected individuals over time under these conditions. The results demonstrate that at the lowest saturation level of $m=0.1$, the model predicts a rapid increase in cases, reaching a higher peak before gradually declining. As the saturation coefficient increases to $0.2$ and $0.5$, the initial rate of infection spread diminishes, leading to lower maximum populations of affected individuals.

This suggests that higher saturation levels tend to moderate the initial outbreak, resulting in less intense surges in infections. Over the simulation duration, all trajectories tend to stabilize at relatively low levels, indicating a decline in disease prevalence. The findings suggest that the saturation coefficient m, which influences the nonlinear transmission dynamics—potentially representing factors such as decreased contact between individuals due to awareness of infection, changes in community behavior—plays a crucial role in controlling epidemic intensity. Increasing m effectively dampens the initial outbreak’s severity, highlighting its potential role in strategic disease management and intervention planning.

As shown in Figure 4, the bifurcation diagram reveals a bifurcation point at the critical value $\beta =0.031838$. This result is in perfect agreement with the theoretical threshold derived from the bifurcation analysis in Section 4.

In this study, three types of controls were considered: the first control (awareness campaigns ($u_1$)), the second control (screening for carriers ($u_2$)), and the third control (treatment for infected people ($u_3$)). We discuss four control strategies below to compare their effectiveness.

6.1. ${\mathrm{STR}}_1\left(\mathrm{A}\right)$: Implementing Screening for Carriers $u_2$ and Treatment for the Infected $u_3$ Only

The numerical simulations demonstrate that the presence of these control interventions $(u_1,u_2,$ and $u_3)$ significantly minimizes the number of carriers and infectious individuals. In contrast, their absence leads to a rapid increase in carrier numbers.

Figure 5a–c collectively illustrate the dynamic effects of combined control efforts on the progression of pneumonia infection within different epidemiological compartments over 30 months. Specifically, Figure 5a portrays the trajectory of the carrier population, demonstrating that the implementation of screening for carriers ($u_2$) and active treatment of infected individuals ($u_3$) (with $u_2$ and $u_3$ in effect, where $u_1=0,u_2\ne 0$, and $u_3\ne 0$) leads to a significant decline in carrier numbers compared to the baseline scenario, where no controls are applied (with all control measures inactive, $u_1=u_2=u_3=0$). This suggests that early detection and intervention effectively curtail the reservoir of infectious carriers, thereby reducing subsequent transmission.

Figure 5b focuses on the infected population (I), revealing that the coordinated application of screening and treatment strategies ($u_1=0,u_2\ne 0$, and $u_3\ne 0$) results in a substantial decrease in actively infected cases. This decline underscores the role of timely diagnosis and treatment in accelerating recovery rates and limiting disease spread within the community.

The control profile associated with $u_2$, as illustrated in Figure 5c, remains at its maximum intensity for nearly $14.37$ months, demonstrating an extended period during which screening efforts are intensely applied. Conversely, the control corresponding to $u_3$ sustains its maximum level for approximately $12.12$ months before experiencing a decline, reflecting a strategic relaxation in treatment efforts as the epidemic progresses towards the control or stabilization. This observed pattern highlights the importance of maintaining a prolonged high-level screening initiative, coupled with a phased reduction in treatment interventions, to optimize disease management over the course of the intervention period.

6.2. ${\mathrm{STR}}_2\left(\mathrm{B}\right)$: Implementing Awareness Campaigns ($u_1$) and Treatment for the Infected ($u_3$) Only

Figure 6a–c illustrate the effects of combined control strategies on key epidemiological compartments over a specified time frame, providing insight into the mechanisms through which interventions influence disease dynamics. In particular, Figure 6a highlights the impact on the carrier population; the data indicate that implementing awareness campaigns ($u_1$) alongside treatment of infected individuals ($u_3$) results in a notable decline in the number of carriers. This reduction reflects the effectiveness of awareness campaigns raising community awareness about the transmission and intervention in diminishing the reservoir of infectious individuals capable of propagating the disease.

Figure 6b presents the temporal trend of the infected class, showing that the combined application of awareness campaigns and treatment significantly curtails active infections. This indicates that early awareness campaigns should be launched ($u_1$), followed by prompt treatment ($u_3$), which plays a critical role in alleviating the burden of the disease and facilitating faster recovery within the population.

In Figure 6c, the control profiles $u_1$ (awareness campaigns) and $u_3$ (treatment for infected individuals) are maintained at their maximum levels for nearly the entire duration of the intervention period. This indicates that continuous high-intensity efforts in these areas are considered most effective in mitigating disease progression and transmission within the modeled scenario.

6.3. ${\mathrm{STR}}_3\left(\mathrm{C}\right)$: Implementing Awareness Campaigns ($u_1$) and Screening for Carriers ($u_2$) Only

Figure 7a–c illustrate the temporal effects of targeted control interventions on specific disease compartments, emphasizing the roles of awareness campaigns ($u_1$), carrier screening ($u_2$), and treatment of infected individuals ($u_3$) in mitigating disease progression over a 30-month period.

In Figure 7a, the variability in the carrier population (C) under uncontrolled versus controlled scenarios highlights the significant impact of screening and awareness efforts. The implementation of $u_1$ and $u_2$ results in a steady decrease in carrier numbers, which is attributable to early detection and behavioral modifications stemming from increased public knowledge. This decline demonstrates that proactive screening facilitates timely identification, enabling carriers to take appropriate health measures, thus reducing their contribution to ongoing transmission.

Figure 7b depicts the trajectory of the infected class (I), contrasting the absence and presence of interventions involving awareness campaigns, screening, and treatment. The controlled scenario shows a marked reduction in the number of actively infected individuals, indicating that timely treatment, combined with awareness and screening efforts, effectively curtails disease propagation. These measures help not only in managing individual cases but also in interrupting transmission chains within the population.

According to Figure 7c, the control profile $u_1$ (awareness campaigns) is maintained at its maximum level throughout nearly the entire period ($28.36$ months), highlighting the importance of sustained awareness efforts. In contrast, the control $u_2$ (carrier screening) remains at its upper bound for the first $9.35$ months, after which it decreases and rises temporarily to less than $0.1$, then decreases again, indicating a strategic intensity reduction in screening activities after initial intensive intervention.

6.4. ${\mathrm{STR}}_4\left(\mathrm{D}\right)$: Implementing Awareness Campaigns ($u_1$), Screening for Carriers ($u_2$), and Treatment for the Infected ($u_3$) Measures

Figure 8a shows the comparison of carrier individuals over time, with the uncontrolled scenario ($u_1=u_2=u_3=0$) exhibiting an initial increase and then a gradual decline and the controlled scenario ($u_1\ne 0,u_2\ne 0,$ and $u_3\ne 0$) showing a rapid decrease in carrier cases within the first few months, approaching near-zero levels by around 5 months. This illustrates the effectiveness of the control strategies in reducing carrier prevalence.

In Figure 8b, the uncontrolled scenario ($u_1=u_2=u_3=0$) shows a rapid decline in infection cases initially, then gradually levels off, reaching low levels after approximately 10 months. In contrast, the controlled scenario ($u_1\ne 0,u_2\ne 0,$ and $u_3\ne 0$) exhibits a faster reduction, with the number of infected cases decreasing sharply and approaching zero within about 5 months. This demonstrates the enhanced effectiveness of the control measures in significantly accelerating the reduction of infection prevalence over the 30 months.

As depicted in Figure 8c, the control profile $u_1$ is maintained at its maximum level for a duration that extends nearly to $9.5$ months. In the case of the control $u_2$, it remains at the upper bound for $9.4$ months before experiencing a decline after the $9.4$-month mark. Similarly, the control $u_3$ stays at the upper bound for a period of 7 months, after which it decreases.

7. Cost-Effectiveness Analysis

Eradicating or controlling infectious diseases in a community often demands substantial time and financial resources, necessitating a cost–benefit analysis. This section focuses on identifying cost-effective interventions for combating the infection of pneumonia through numerical simulations of an optimal control system.

To determine which of these controls was best, the best intervention techniques were applied to reduce disease spread, and their impacts on health and the economy were evaluated [31]. Four basic performance measures were used.

First, we conducted an efficiency analysis (EI). The efficiency index (EI) is calculated using the following equation:

$$\mathrm{EI}=\left(1-{\displaystyle \frac{A_c}{A_0}}\right)\times 100,$$

(20)

where

• $A_c$: total recorded infections resulting from the control application;
• $A_0$: total recorded infections without any control application.

The total number of infected individuals within the time period $[0, 30]$ is expressed by the following equation [31,32]:

$$A={\int }_0^{30}I\left(t\right)dt.$$

(21)

Second, we used the infection averted ratio (IAR). The percentage of avoided infection is defined as follows:

$$\mathrm{IAR}={\displaystyle \frac{\mathrm{Number}\mathrm{of}\mathrm{infection}\mathrm{averted}}{\mathrm{Number}\mathrm{of}\mathrm{recovered}}}.$$

(22)

The number of infections averted is calculated by subtracting the number of infections with the control from the number without the control. The strategy that takes the highest percentage is the most cost-effective [33].

Next, we used the average cost-effectiveness ratio (ACER) and the incremental cost-effectiveness ratio (ICER).

The average cost-effectiveness ratio (ACER) is calculated as follows:

$$\mathrm{ACER}={\displaystyle \frac{\mathrm{Total}\mathrm{cost}\mathrm{produced}\mathrm{by}\mathrm{the}\mathrm{intervention}}{\mathrm{Total}\mathrm{number}\mathrm{of}\mathrm{infection}\mathrm{averted}}}.$$

(23)

The cost of the intervention was calculated using the objective function [34].

For every extra health result, the incremental cost-effectiveness ratio (ICER) is the additional cost. The number of controls implemented is assumed to be directly proportional to the costs of the various control initiatives. In order to incrementally compare two or more competing intervention strategies, one intervention is contrasted with the next less successful one [34].

$$\mathrm{ICER}={\displaystyle \frac{\mathrm{Difference}\mathrm{in}\mathrm{infection}\mathrm{averted}\mathrm{costs}\mathrm{in}{\mathrm{STR}}_1\mathrm{and}{\mathrm{STR}}_2}{\mathrm{Difference}\mathrm{in}\mathrm{the}\mathrm{total}\mathrm{number}\mathrm{of}\mathrm{infection}\mathrm{averted}\mathrm{in}{\mathrm{STR}}_1\mathrm{and}{\mathrm{STR}}_2}}.$$

(24)

The numerator represents the difference in the costs of preventing the disease or preventing cases, as well as the costs of preventing productivity losses and others. In contrast, it includes the difference in health outcomes achieved, which may include the total number of infections avoided or the number of exposed individuals protected from carriers or actual infected cases [34].

We compared the effectiveness of four control strategies. The low values for both the ICER and ACER indicate that the strategy is the most economically efficient in terms of cost compared to health benefits, as each represents the cost of achieving one unit of health effectiveness. Therefore, the strategy with the lowest ICER is the most resource-efficient option. The IAR measures the strategy’s effectiveness in reducing infections compared to recovery outcomes. The higher the IAR, the more effective the strategy is in terms of health impact compared to cost and, therefore, the more feasible it is. The EI measures the percentage reduction in the cumulative number of infections resulting from implementing the strategy compared to the reference case without intervention. Higher values indicate greater effectiveness in reducing the disease burden, which enhances the credibility of this strategy as an effective health option. By combining these indicators, it becomes clear that the strategy that combines the highest EI, the highest IAR, and the lowest ACER and ICER is the most integrated strategy in terms of health and economic efficiency.

The separation of control strategies into double and triple scenarios was undertaken to facilitate the use of the incremental cost-effectiveness-ratio analysis, enabling a comprehensive assessment of their cost efficiency. Specifically, as illustrated in

Table 2. Baseline cost-effectiveness comparison of strategies A–C.

Strategies	Total Infectious Averted	Total Cost (USD)	ICER	ACER	IAR	${\mathit{A}}_{\mathit{c}}$	EI
${\mathrm{STR}}_1\left(\mathrm{A}\right)$	691.5940	0.1383	0.0002	0.0002	0.4730	209.2899	76.7684%
${\mathrm{STR}}_2\left(\mathrm{B}\right)$	671.2128	0.2999	−0.0079	0.0004	0.5586	229.6710	74.5060%
${\mathrm{STR}}_3\left(\mathrm{C}\right)$	218.4699	0.1940	0.0002	0.0009	0.2927	682.4140	24.2506%

, the evaluation of cost-effectiveness for the double scenario involves ranking the control strategies based primarily on the increasing order of the total number of infectious individuals averted. This approach allows for an organized comparison of strategies by their effectiveness in reducing infection cases relative to their associated costs, aiding policymakers in identifying the most economically efficient options for disease control.

The ICER is computed as

$$\begin{array}{cc}\hfill \mathrm{ICER}& ={\mathrm{STR}}_1\left(\mathrm{A}\right)={\displaystyle \frac{0.1383}{691.5940}}=0.0002,\hfill \\ \hfill \mathrm{ICER}& ={\mathrm{STR}}_2\left(\mathrm{B}\right)={\displaystyle \frac{0.2999-0.1383}{671.2128-691.5940}}=-0.0079,\hfill \\ \hfill \mathrm{ICER}& ={\mathrm{STR}}_3\left(\mathrm{C}\right)={\displaystyle \frac{0.1940-0.2999}{218.4699-671.2128}}=0.0002.\hfill \end{array}$$

In Table 2, when comparing strategies ${\mathrm{STR}}_1\left(\mathrm{A}\right)$, ${\mathrm{STR}}_2\left(\mathrm{B}\right)$, and ${\mathrm{STR}}_3\left(\mathrm{C}\right)$ based on various evaluation indicators, we find a clear difference in health aspects, with convergence in economic aspects. ${\mathrm{STR}}_1\left(\mathrm{A}\right)$ recorded the highest value for the efficiency index, indicating its effectiveness in reducing the number of cumulative infections compared to ${\mathrm{STR}}_3\left(\mathrm{C}\right)$. In contrast, ${\mathrm{STR}}_2\left(\mathrm{B}\right)$ showed the highest value in terms of the infection avoidance ratio, as it succeeded in reducing the number of infections compared to the number of recoveries to a greater extent. Strategies ${\mathrm{STR}}_3\left(\mathrm{C}\right)$ and ${\mathrm{STR}}_1\left(\mathrm{A}\right)$ are the most balanced from an economic perspective, as they recorded an equal ICER value, while strategy ${\mathrm{STR}}_2\left(\mathrm{B}\right)$ is the economically effective option.

After recomputing the ICER for strategies two and three, the results are summarized in

Table 3. Recomputed ICER and cost-effectiveness for strategies B and C.

Strategies	Total Infectious Averted	Total Cost (USD)	ICER	ACER	IAR	${\mathit{A}}_{\mathit{c}}$	EI
${\mathrm{STR}}_2\left(\mathrm{B}\right)$	671.2128	0.2999	−0.0079	0.0004	0.5586	229.6710	74.5060%
${\mathrm{STR}}_3\left(\mathrm{C}\right)$	218.4699	0.1940	0.0002	0.0009	0.2927	682.4140	24.2506%

. This recalculation provides a more precise comparison of the incremental cost-effectiveness of these strategies relative to the previously analyzed options. By refining the ICER values, the analysis offers clearer insights into which strategy delivers the most cost-effective reduction in infections, supporting informed decision-making for the optimal allocation of resources in disease control efforts.

Table 3 also shows that strategy ${\mathrm{STR}}_2\left(\mathrm{B}\right)$ is more effective and less costly than ${\mathrm{STR}}_3\left(\mathrm{C}\right)$. Strategy ${\mathrm{STR}}_2\left(\mathrm{B}\right)$ is a combination of awareness campaigns and treatment, so strategy ${\mathrm{STR}}_3\left(\mathrm{C}\right)$ will be excluded.

$$\begin{array}{cc}\hfill \mathrm{ICER}& ={\mathrm{STR}}_2\left(\mathrm{B}\right)={\displaystyle \frac{0.2999}{671.2128}}=0.0004,\hfill \\ \hfill \mathrm{ICER}& ={\mathrm{STR}}_3\left(\mathrm{C}\right)={\displaystyle \frac{0.1940}{218.4699}}=0.0009,\hfill \end{array}$$

Table 4. Verification of ICER/ACER for strategies B versus C.

Strategies	Total Infectious Averted	Total Cost (USD)	ICER	ACER	IAR	${\mathit{A}}_{\mathit{c}}$	EI
${\mathrm{STR}}_2\left(\mathrm{B}\right)$	671.2128	0.2999	0.0004	0.0004	0.5586	229.6710	$74.5060\%$
${\mathrm{STR}}_3\left(\mathrm{C}\right)$	218.4699	0.1940	0.0009	0.0009	0.2927	682.4140	$24.2506\%$

shows that the value of strategy ${\mathrm{STR}}_2\left(\mathrm{B}\right)$ is more effective and also less costly than ${\mathrm{STR}}_3\left(\mathrm{C}\right)$; so, ${\mathrm{STR}}_3\left(\mathrm{C}\right)$ will also be excluded.

$$\mathrm{ICER}={\mathrm{STR}}_4\left(\mathrm{D}\right)={\displaystyle \frac{0.1351}{692.8497}}=0.0002,$$

$$\begin{array}{cc}\hfill \mathrm{ICER}& ={\mathrm{STR}}_4\left(\mathrm{D}\right)={\displaystyle \frac{0.1351-0.2999}{692.8497-671.2128}}=-0.0076,\hfill \\ \hfill \mathrm{ICER}& ={\mathrm{STR}}_2\left(\mathrm{B}\right)={\displaystyle \frac{0.2999}{671.2128}}=0.0004,\hfill \end{array}$$

From

Table 5. Pairwise cost-effectiveness comparison: strategy D vs. strategy B.

Strategies	Total Infectious Averted	Total Cost (USD)	ICER	ACER	IAR	${\mathit{A}}_{\mathit{c}}$	EI
${\mathrm{STR}}_4\left(\mathrm{D}\right)$	692.8497	0.1351	−0.0076	0.0002	0.5428	208.0341	76.9078%
${\mathrm{STR}}_2\left(\mathrm{B}\right)$	671.2128	0.2999	0.0004	0.0004	0.5586	229.6710	74.5060%

, the ${\mathrm{STR}}_4\left(\mathrm{D}\right)$ strategy is more economical and more efficient. It is also clear from Table 3 that the ${\mathrm{STR}}_4\left(\mathrm{D}\right)$ strategy is also better than ${\mathrm{STR}}_2\left(\mathrm{B}\right)$. Therefore, the ${\mathrm{STR}}_4\left(\mathrm{D}\right)$ strategy is the best among all strategies.

The intervention strategy that showed the most promise in reducing infections is the ${\mathrm{STR}}_4\left(\mathrm{D}\right)$ strategy, which involves implementing screening for carriers and treatment for infected individuals. According to the analyses presented in Table 5 and

Table 6. Cost-effectiveness for strategy D (screening + treatment).

Strategies	Total Infectious Averted	Total Cost (USD)	ICER	ACER	IAR	${\mathit{A}}_{\mathit{c}}$	EI
${\mathrm{STR}}_4\left(\mathrm{D}\right)$	692.8497	0.1351	0.0002	0.0002	0.5428	208.0341	$76.9078\%$

, ${\mathrm{STR}}_4\left(\mathrm{D}\right)$ achieves the highest number of infections averted $\left(692.8497\right)$ and is identified as the most economical and efficient strategy, being better than other tested strategies. The comprehensive comparison of control strategies confirms that this combined approach provides the greatest reduction in disease spread relative to its cost.

8. Conclusions

In this study, we systematically explored various intervention strategies aimed at controlling the spread of pneumonia within a community. Through detailed numerical simulations and comprehensive cost-effectiveness analyses, it became evident that integrated approaches combining awareness campaigns, carrier screening, and timely treatment yield the most substantial reductions in disease transmission. Of particular note was the strategy involving simultaneous screening and treatment, which consistently demonstrated superior performance in decreasing the number of infectious individuals while remaining economically efficient. The temporal dynamics of intervention efforts underscored the importance of sustaining high-intensity activities, especially in awareness and screening, during critical phases of outbreak management. Importantly, the findings highlight that early and aggressive implementation of combined control measures can significantly curtail the epidemic trajectory, offering valuable insights for policymakers seeking to optimize resource allocation. Overall, the results affirm that a multi-faceted sustained intervention approach not only effectively suppresses disease proliferation but also aligns with cost-effective health strategies, facilitating more resilient and responsive public health policies.