Modeling dynamics and stability analysis of pneumonia disease infection with parameters uncertainties control

1. Introduction

Pneumonia is a condition associated with difficulty in breathing due to respiratory illness in the lungs. The disease affects over 2 million children under the age of five, with the majority of cases occurring among the low-income countries [26, 32]. It is a pulmonary infection caused by bacteria, viruses, fungi, or other pathogens. Streptococcus pneumoniae, commonly known as pneumococcus, is the most prevalent cause of bacterial pneumonia [35]. It is characterized by inflammation of the lungs’ air sacs (alveoli), which are filled with fluid or pus, leading to breathing difficulty [18]. According to Tilahun [42] and Oluwatobi and Erinle-Ibrahim [31], pneumococcus transmits via oropharyngeal micro-aspiration and inhalation of aerosols with bacteria or viruses, especially in children and adults who carry the bacteria in their throats with no disease symptoms [9]. It can also be transmitted through airborne droplets from an infected person’s cough or sneeze. According to the World Health Organization (WHO) [44], pneumonia is also associated with high fever and fast breathing, and eventually the infected person becomes extremely sick [17].

Furthermore, smoking history and passive smoking, malnutrition, crowded living conditions, lack of exclusive breastfeeding, indoor air pollution, heart disease, alcoholism and drug misuse, acidosis, diabetes, and antecedent viral infection are all risk factors for the spread of pneumonia disease [16,26]. Adequate treatment, vaccination, effective screening and diagnosis, and environmental control measures for other diseases can all help to prevent pneumonia [37]. In both children and adults, vaccination is the most effective strategy to avoid bacterial and viral pneumonia. Early detection, vaccination, and treatment at the community or primary healthcare facility level, as well as for expectant moms, can prevent newborns from contracting pneumonia [16,17].

Mathematical models are regarded as essential instruments for a better understanding of the epidemiological stability of disease infections in a host, their patterns of transmission in a population, and the development of related, necessary control plans about the disease [8,18,23,26]. Several studies have been done about disease infection spread and control measures, including the following.

Ngari et al. [29] and Oluwatobi and Erinle-Ibrahim [31] developed a susceptible $S(t)$, infectious $I(t)$, and treatment $T(t)$ compartmental model for pneumonia disease transmission. From their work, it is observed that implementing screening and treatment is better than eliminating disease transmission from the community [27]. Also, Temime et al. [41] constructed a model with compartments of the susceptible–infected–treated–vaccinated–susceptible type, which take into account vaccination and the distribution of resistance levels across individuals. The model examines how changes in pneumococcal colonization and resistance over time are influenced by conjugate vaccination in environments with high and widespread antibiotic resistance. Hence, due to the effects of immunization, this may not last over the long term. Therefore, the results suggested that the vaccine alone is insufficient to regulate the selection for resistance in S. pneumoniae.

Another study with susceptible S, carrier C, infected I, and recovered R compartments was proposed by Alya et al. [1] and Kizito and Tumwiine [16] for studying the transmission of pneumonia infection, incorporating treatment, vaccination, and hospitalization as control strategies. Again, Otieno and Paul [33] developed a similar model for pneumonia disease infection transmission. Their research showed that the disease will endure if the basic reproduction number is greater than unity and will be eliminated otherwise from the community, and this is when treatment and vaccination interventions are administered accordingly. Teklu and Kotola [40] built a mathematical model evaluating the effects of vaccination, treatment, and other preventative measures like better hygiene conditions, avoiding close contact with infected people, and avoiding exposure to cigarette smoke. Again, Kassa and Murthy [14] formulated a similar model, and from both studies, stability of equilibrium points and basic reproduction numbers were determined and examined, and their results indicate that vaccination and protection against the pneumonia infection play a significant role in the transmission rate of the infection.

Moreover, Ossaiugbo and Okposo [32] created a model with susceptible $S(t)$, exposed $E(t)$, infectious $I(t)$, and recovered $R(t)$ compartments for pneumonia disease transmission. In their work, analysis of both equilibrium points was performed, and analytically, it was found that the endemic equilibrium point is globally asymptomatic stable when $R_0>1$, while the disease-free equilibrium point is locally asymptomatic stable for $R_0<1$. Moreover, sensitivity tests and findings showed that the most sensitive parameters are the transmission rate and the pace at which an individual is transferring from being exposed to infection, and that treatment and natural immunity revealed to have an impact in reducing pneumonia infection.

Recently, it has become a challenging factor to achieve the desired optimal control solutions in most epidemic models due to parameter uncertainties [4,28]. Studies like Moradi et al. [24] analyzed the effects of drug usage with uncertainties for cancer chemotherapy treatment, while Cao and Kan [4] discussed COVID-19 control with parameter uncertainties by using treatment and media campaign interventions. Further, Nematollahi et al. [28] proposed an adaptive control technique for containing tuberculosis disease spread, considering parameter uncertainties associated with endogenous reactivation and exogenous reinfection. Therefore, this work proposes the method of adaptive sliding mode control (ASMC) with a closed-loop control system to tackle the pneumonia disease with parameter uncertainties. The technique is very useful for handling parameter’s uncertainties through accurately tracking the disease carriers and infectious agents for the easy and quick containment of pneumonia infection spread from the human population in the presence of vaccination and treatment interventions. Additionally, in this work, no specific case study is involved. Rather, it develops and employs a general study methodology that can be applied wherever relevant.

2. Pneumonia Dynamics Model Flow Chart Formulation

2.1. Model assumptions

The pneumonia model, which consists of six compartments, is formulated based on the following assumptions:

(1)	The model assumes that there is an equal possibility for everyone in the population to become infected with the virus if they are exposed to it.
(2)	All recovered people are assumed to have cleared the bacteria or virus from their bodies and hence do not spread the sickness.
(3)	An extra dose of vaccine is provided to newborns, which will produce optimal levels of response.
(4)	It is assumed that all treated and recovered individuals from the illness get immunized after finishing their dosages, while all the vaccinated children or adults do not advance to the susceptible group due to the booster of vaccine dosages.

2.2. Model variables and parameters

Mathematical model formulated comprises six compartments based on various parameters as follows: $S(t)$ represents susceptible individuals who are at risk of receiving pneumonia infection at time t once become exposed to the disease carriers $C(t)$ with $\pi$ per capita recruitment rate into the susceptible population. $V(t)$ represents vaccinated individuals at time t from susceptible and recovered groups at the rate $\rho$ and $\sigma$, respectively. $C(t)$ is the carrier group for which the force of transmission or infection of susceptible individuals becomes exposed to the disease through $\beta$ and which will carry the pneumonia bacteria before have been able to develop its infection to group $I(t)$ through the rate $\epsilon$ where an infected individual(s) is now capable of spreading the infection rate to other individuals. $T(t)$ is the treated group of carriers and infected individuals at the rate of $\delta$ and $\tau$, respectively. $R(t)$ is the recovered individual through the rate $\theta$ from treatment of pneumonia and $\kappa$ from those who have been infected but gained immunity naturally, however their immunity will wane and hence join susceptible group $S(t)$ through the rate $\gamma$ while others will be revaccinated at the rate $\sigma$ after been recovered. All individuals undergo natural death at the rate $\mu$ and induced disease death rate at $\alpha$.

The summarized descriptions of the model variables and parameters are given in Tables 1 and 2, respectively.

Therefore, based on the assumptions above, and with respect to variables and parameters, we established the schematic flow chart (Fig. 1), with a system of nonlinear differential equations for pneumonia disease infection transmission dynamics in the human population as follows :

2.3. Properties of the pneumonia model dynamic system

2.3.1. Pneumonia invariant region solution

In biological model population dynamics, it is necessary to establish its related boundedness and positive invariance for its meaning and sense [21,25].

Theorem 1. The solution set $S(t)$, $V(t)$, $C(t)$, $T(t)$, $I(t)$, $R(t)$ of the pneumonia model system of equations in (2.1) is contained in the nonnegative feasible region $\mathrm{\Omega }$ [49].

Proof. Let the feasible region as $\mathrm{\Omega }=\{S(t),V(t),C(t),T(t),I(t),R(t)\}\in {ℝ}_{+}^6$ for all $t\ge 0$ with total of human population $N(t)$ for any time $(t)$ as

By differentiating Eq. (2.2) with respect to time t, we lead to By substituting Eq. (2.1) in (2.3), we get By computing and simplifying in (2.4), it leads to Combining Eqs. (2.2) and (2.5), we get By employing integrating factor and initial conditions, we reach By implying that as $t\to \infty$, it is simplified to Hence, the system solutions contained in $\mathrm{\Omega }$ are indeed positive invariant. This concludes that every solution set in (2.1) enters the region $\mathrm{\Omega }$ [13].  □

2.3.2. Positivity solutions of the pneumonia model

In any biological population, it is meaningful and well defined whenever all solution sets of the model are nonnegative for $\forall t\ge 0$ [13, 19]. This deduces the following theorem.

Theorem 2. If the given initial data set $\{\{[S(0),V(0),C(0),T(0),I(0),R(0)]\ge 0\}\in \mathrm{\Omega }\}$, then the pneumonia solution set $\{S(t),V(t),C(t),T(t),I(t),R(t)\}$ of the system in (2.1) is positive for $\forall t\ge 0.$

Proof. To prove for positivity existence of the model in (2.1) such that all compartments are positive, and thus, from susceptible group, we have

Explicitly, Eq. (2.8) can be written as By using variables’ separation technique, integrating and applying initial conditions in (2.9), it gives Therefore, Eq. (2.10) indicates that, as $t\to \infty$, we get $S(t)\ge 0$.

Through the same procedures, other classes will be proven for their positive solutions. Hence, this deduces that all systems of pneumonia equations in (2.1) possess positive solutions for $\forall t\ge 0$, such that $S(t)\ge 0,V(t)\ge 0,C(t)\ge 0,T(t)\ge 0,I(t)\ge 0,R(t)\ge 0$. □

3. Steady States of the Model

3.1. Disease-free equilibrium point

By solving equations in (2.1) with respect to its variables, the disease-free equilibrium point $E^0$ is found as

3.2. Basic reproduction number, $R_0$

The basic reproduction number $R_0$ refers to the average number of secondary infected individuals produced from an initial single primary infected individual in its life-time duration when introduced into the host population in the absence of intervention [25]. When $R_0<1$, biologically, the disease is weak enough to invade the population, unlike for $R_0>1,$ where it dominates and lasts longer [47,48]. Castillo-Chavez, Feng, and Huang approach as in Martcheva [21] was employed to compute $R_0$ for which the system of equations is classified into non-infected, carrier, and infectious groups.

Let

Referring the system of Eq. (3.2), the variable C at disease-free equilibrium point $E^0$ is computed which gives From the same Eq. (3.2), appropriate substitution for solving variable I leads to If we evaluate at $E^0$ such that $I=0$, by taking derivative of Jacobian matrix with respect to I, we get By definition [21], $A=M-D$, where M defines a matrix and D is a diagonal matrix as well such that $M\ge 0$, $D>0$.

Thus, from Eq. (3.5), it is clear that

By considering the study of [21], $R_0$ is the spectral radius defined by the matrix $M{D}^{-1}$, which is

Thus, we get basic reproduction number as

3.3. The pneumonia existence and uniqueness of the equilibrium point

The endemic equilibrium point exists when the pneumonia disease is found in the population compartments. Let $E^{\ast }$ define pneumonia disease dominating the equilibrium point, then the following theorem is equivalent [39].

Theorem 3. The system (2.1) has a unique positive endemic equilibrium point, $E^{\ast }=[S^{\ast },V^{\ast },C^{\ast },T^{\ast },I^{\ast },R^{\ast }]$ if and only if the basic reproduction number $R_0>1$.

Proof. To prove the existence of an endemic equilibrium point, we employ the technique proposed by Soulaimani and Kaddar [39]. Suppose that $R_0>1$, and $(S^{\ast },V^{\ast },C^{\ast },T^{\ast },I^{\ast },R^{\ast })$ is a nonnegative endemic equilibrium point of the system (2.1), which can be written as

The corresponding solution in terms of $I^{\ast }$ from (3.7) is as where Let $z(S,I)$ be a continuously and differentiable function on ${ℝ}_{+}^{⊭}$ such that $z(S,0)=z(0,I)=0$ for all $S\ge 0$, $I\ge 0$ with the hypotheses that [39]

H1:	$\frac{\partial z}{\partial S}(S,I)>0$ and $\frac{\partial z}{\partial I}(S,I)>0$ for all $(S,I)\ge 0$.
H2:	$\frac{\partial z}{\partial I}(S,I)$ is a monotone decreasing function of $I>0$ with any fixed $S\ge 0$ and that ${lim}_{I\to 0^{+}}\frac{z(S,I)}{I}$ is a continuous increasing monotonic function on $S\ge 0$.

Then, from the system in (2.1), we define the function $f(I)$ mapping from ${ℝ}^{+}$ to $ℝ$ as

After relevant substitution, it follows that

Using the hypotheses $H_1$–$H_2$ in (2.1) and (3.7), f is a strictly decreasing monotone function on ${ℝ}^{+}$ at $E^0$ such that

By re-arranging the above and use Eq. (3.6), we get It is equivalent to It implies that if and only if $R_0>1$. Therefore, there exists a unique $I^{\ast }$ such that $f(I^{\ast })>0$ provided that $R_0>1$. Hence, the system at $E^{\ast }$ has a unique nonnegative endemic state [39]. This completes the proof of the theorem. □

4. Stability Analysis of Pneumonia Model

4.1. Local stability of non-existence of pneumonia disease equilibrium $E_0$

In the absence of pneumonia disease infection in the population, the following theorem holds.

Theorem 4. If the basic reproduction number $R_0<1$, then the pneumonia disease-free equilibrium point $E_0$ is locally asymptotically stable $($LAS$),$ while unstable for $R_0>1$ in the region $\mathrm{\Omega }.$

Proof. To prove the local stability of the disease-free equilibrium point $E_0$, we employed the eigenvalues technique as used in [15, 50]. The system in (2.1) is linearized to get Jacobian matrix $J(E_0)$ as

From (4.1), through row and column reductions [20], easily we get the corresponding eigenvalues ${\lambda }_1=-(\mu +\rho )$, ${\lambda }_2=-\mu$, ${\lambda }_4=-(\mu +\theta )$, and ${\lambda }_6=-(\mu +\sigma +\gamma )$.

Then, the remaining eigenvalues, ${\lambda }_3$ and ${\lambda }_5$, will be obtained from the reduced Jacobian matrix that

for which (4.2) is simplified to a second-order characteristics polynomial equation where By Routh–Hurwitz criterion, the system in (4.3) will have definite negative eigenvalues whenever both coefficients $b_1$ and $b_2$ are strictly positive.

Thus, clearly $b_1>0$ and $b_2>0$ if and only if $[(\mu +\delta +\epsilon )(\mu +\alpha +\kappa +\tau )-\beta \epsilon S]>0$. By considering Eq. (3.6), we get $(1-R_0)>0$, $\Rightarrow R_0<1$.

Thus, all eigenvalues (roots) of $J(E_0)$ in (4.1) are real, distinct, and negative if and only if $R_0<1$. In this state, the system is asymptotically stable, otherwise will be unstable [50]. Therefore, this completes the proof of the theorem. □

4.2. Global stability analysis of non-existence pneumonia equilibrium

In this section, the following theorem is relevant.

Theorem 5. If basic reproduction number $R_0\le 1$, then the non-existence pneumonia disease equilibrium point $E_0$ is globally asymptotically stable (GAS), and unstable whenever $R_0>1$ in the region $\mathrm{\Omega }.$

Proof. To determine the global stability of non-existence pneumonia disease equilibrium, $E_0$, the following technique was employed [15, 21]. Let us define the function as

where $Z_n$ and $Z_i$ present vectors from non-spreading and spreading disease infection compartments, respectively, and $Z_{(E_0)}$ of the same length as $Z_n$ is the disease-free equilibrium point such that $Z_i={(C,T,I)}^T$, $Z_n={(S,V,R)}^T$.

Correspondingly, from (4.4) we have

and To prove the global stability of disease-free equilibrium point, we should find that, $P_2$ is the Metzler matrix such that $P_2(Z_{i,j})\ge 0$ for all $i\ne j$ contains positive all off-diagonal elements, and P matrix has real and negative eigenvalues definitely.

Then, by using (4.5) and (4.6) in (4.4), we obtain

By taking partial derivatives of the non-transmitting and transmitting compartments of $J(E_0)$ in (4.1) with respect to (4.7), the following corresponding matrices are obtained : and From the results (4.8) and (4.9), all matrices $(P,P_1$, and $P_2)$ are $3\times 3$. Furthermore, we see that the eigenvalues of matrix P are real, distinct, and negative (i.e. $-(\mu +\rho ),-\mu$, and $-(\mu +\sigma +\gamma ))$. Also, $P_2$ is the Metzler matrix such that its leading diagonal elements are negative definitely while the off-diagonal entries are positive [15]. Hence, the theorem. □

4.3. Analysis for global stability of pneumonia existence equilibrium point $E_{\ast }$

The endemic state defines the dominance of the pneumonia disease infection in the population which is equivalent to the following theorem [50].

Theorem 6. If the basic reproduction number $R_0>1$, then the pneumonia existing equilibrium point is GAS and entered in the region $\mathrm{\Omega }.$

Proof. To establish the global stability analysis of pneumonia persistence disease equilibrium point defined in (2.1), the logarithmic Lyapunov function approach was employed [15, 21]. Let the function be defined as

Then, it follows that By taking derivative in Eq. (4.10) with time t corresponding to Eq. (2.1), we get By substituting Eq. (2.1) into (12), we get From the system in (2.1), the corresponding endemic equilibrium point can be written as Then, by utilizing Eq. (4.14) into (4.13) we obtain Through simplification and re-arrangement in (4.15), it gives By setting $V=V^{\ast }$, $C=C^{\ast }$, $T=T^{\ast }$, $I=I^{\ast }$ and $R=R^{\ast }$ in (4.16), we get the reduced function G as Hence, Eq. (4.17) is a strictly Lyapunov function that $\frac{dG}{dt}<0$. However, if $S=S^{\ast }$, then (4.17) is reduced to $\frac{dG}{dt}\le 0$. Therefore, under LaSalle’ principle, the system behaves global asymptotic stability, and hence concludes the proof of the theorem [15]. □

5. Sensitivity Analysis and Numerical Solutions

5.1. Parameters’ sensitivity analysis

Here, the normalized forward sensitivity indices of all differentiable parameters p with respect to $R_0$ were computed, and which generally defined by

where p represents any parameter in the basic reproduction number $R_0$. For example, the sensitivity index of $R_0$ corresponding to the parameter $\alpha$ is given as ${\mathrm{{\rm Y}}}_p^{R_0}=\frac{\partial R_0}{\partial \alpha }\times \frac{\alpha }{R_0}=-0.815033$ after employing parameters’ values found in Table 4. Other indices are found in a similar way and the corresponding results are indicated in Table 3.

5.1.1. Interpretation of the parameter sensitivity indices

Figure 2 represents the sensitivity indices profile of $R_0$ with respect to model parameters found in $R_0$. The sensitivity indices with negative signs imply the rate that parameters contribute in decreasing the value of $R_0$ while with positive signs indicate that when parameter values are increased, the value of $R_0$ increases as well. For example, the parameters $\beta$ and $\pi$ have the most positive indices, which implies more contribution increase of the value of $R_0$ followed by $\epsilon$. On the other hand, the parameters $\delta$, $\rho$, and $\alpha$ have large negative indices implies more significance contribution in decreasing the value of $R_0$. Biological implication is that, the more the negative values, the more the diminishing pneumonia infection represented by $R_0$, and conversely for positive which eliminate efficacy and hence increases the value of $R_0$.

6. Parameters’ Uncertainties Sliding Mode Control for Pneumonia Disease Infection

Epidemic models obviously contain uncertainty parameters resulted from recording or manipulating data for disease infections rate, estimating parameter values or unknown cases, Huynh et al. [11] and Fang et al. [5]. Due to this undoubtful challenge, this work established ASMC technique to handle the situation with highly performance of managing uncertainties for both linear and nonlinear systems [6,43,7]. The control structure aimed to minimize the number of carrier and infectious individuals possibly zero by tracking through predefined reference trajectory [3,2,45,46]. Generally, the ASMC is governed under the following model equation [12] :

where $e(t)$ is the tracking error between the reference and output and $\omega$ is a turning parameter or sliding surface constant gain, which helps to shape $\varphi (t)$. It is selected by the design which determines the performance of the system on the sliding surface $\varphi (t)$.

Now, from Eq. (2.1), the control function $u(t)$ is introduced corresponding to the carrier and infected population groups such that the system becomes

Thus, we formulated the adaptive sliding mode control (ASMC) flow structure involving pneumonia compartments as presented in Fig. 11. From (6.2) our intended compartments to deal with for controlling the pneumonia infection spread in the community are The objective is to contain the carrier $C_r$ and infected $I_r$ individuals as the major contributors and transmitters of the disease pathogens spread by accurately tracking a predefined reference trajectories (i.e. $C_r,I_r\to 0)$ [28]. This is done when tracking error $e(t)$ converges to zero for all $t\ge 0$ such that where $\tilde{C}$ and $Ĩ$ are the corresponding tracking errors of carrier and infected individuals, while $C_r$ and $I_r$ are the corresponding desired values from the actual of carrier C, and infected I.

Taking derivative in (6.1) with respect to time t becomes

Using characterization control $u_1(t)$ with respect to carrier individuals compartment in (6.3), (6.7) can be written in the form of Then by combining expressions in (6.5) and (6.8), it gives Appropriate substitution of (6.3) into (6.9), followed by simplification and rearrangement, ends up with In a similar way, controlling $u_2(t)$ with respect to infected individuals in (6.4) will be found as To achieve the desired design of adaptive control laws, Eqs. (6.10) and (6.11) are composed in the form of matrix X and vector z containing the corresponding state variables and parameters of the system such that and where $X_1,X_2$ are functions of the system state variables $(S,V,C,T,I,R)$, and $z_1,z_2$ are vectors contain uncertain parameters of the system such that Moreover, since the actual system parameters $z_1$ and $z_2$ contain uncertainties as proposed in Nematollahi et al. [28], then the ASMC control laws are therefore designed using the estimated model parameters ${ẑ}_1$ and ${ẑ}_2$ with the corresponding adaptive laws that handle the discrepancies between the actual and the estimated parameters in (6.12) and (6.13) as and where ${ẑ}_1$ and ${ẑ}_2$ represent vectors of the estimated parameters for which ${ẑ}_1={[\widehat{\beta }\widehat{\mu }\widehat{\delta }\widehat{\epsilon }]}^t$ and ${ẑ}_2={[\widehat{\epsilon }\widehat{\mu }\widehat{\alpha }\widehat{\tau }\widehat{\kappa }]}^t$, while $\widehat{\mathrm{\Gamma }}\mathrm{\text{sgn}}$ is a non-continuous term included in the control law.

In order to compensate any mismatch between the estimated and the actual parameters, the adaptive gains ${\widehat{\mathrm{\Gamma }}}_1$ and ${\widehat{\mathrm{\Gamma }}}_2$ are included to achieve the intended control objectives. These gains are updated via the adaptation laws which defined as [36]

The nonnegative of constants $r_1$ and $r_2$ implies the rate at which the adaptive gains are updated with respect to the tracking error of $\tilde{C}$ and $Ĩ$ with initial values ${\widehat{\mathrm{\Gamma }}}_1(0)$ and ${\widehat{\mathrm{\Gamma }}}_2(0)$, respectively.

6.1. Closed-loop pneumonia control system

In this section, the closed-loop control system of pneumonia is developed by using ASMC designed in the previous section. Equations (6.16) and (6.17) then are substituted in (6.3) and (6.4), respectively, which give

Thus, the closed-loop system is established as and such that where $\frac{d{\tilde{C}}_r}{dt}$, $\frac{d{Ĩ}_r}{dt}$ are derivatives of tracking errors, ${\tilde{z}}_1,{\tilde{z}}_2$ are bounded model parameters estimation error.

Since the system state variables and model parameter estimation errors $({\tilde{z}}_1,{\tilde{z}}_2)$ are bounded, then the following bounded terms $C{X}_1(S,V,C,T,I,R){\tilde{z}}_1$ and $I{X}_2(S,V,C,T,I,R){\tilde{z}}_2$ can be developed such that for nonnegative constants ${\zeta }_1$ and ${\zeta }_2$, we have [34]

6.2. Stability analysis of the closed-loop pneumonia control system

Indeed, for the system be controllable, it must be stable and attainable [4]. This fact therefore can be proved by analyzing the stability of the system containing carrier and infected individuals by employing a quadratic Lyapunov function defined as [24]

Taking derivative in (6.26) gives By using Eqs. (6.22) and (6.23), Eq. (6.27) becomes However, $\tilde{C}\mathrm{\text{sgn}}(\tilde{C})=|\tilde{C}|$ and $Ĩ\mathrm{\text{sgn}}(Ĩ)=|Ĩ|$, thus by inserting the adaptive laws in (6.18) and (6.19), Eq. (6.28) is simplified to Furthermore, from Barbalat’s lemma [38], the asymptotic stability can be achieved by taking second derivative in (6.29) such that Since carrier and infected individuals dynamics in (6.3) and (6.4) are bounded, then the derivatives of the corresponding classes $\frac{dC}{dt}$ and $\frac{dI}{dt}$ are bounded as well. Similar boundedness can be deduced for $\frac{d{C}_r}{dt}$, $\frac{d{I}_r}{dt}$, $\frac{d\tilde{C}}{dt}$, and $\frac{dĨ}{dt}$, and thus, by referring the Barbalat’s lemma, this eventually proves that $\frac{d^{2}W}{d{t}^2}$ is bounded as $\frac{dW}{dt}\to 0$ as $t\to 0$.

Indeed, the tracking errors $(\tilde{C}\to 0$, $Ĩ\to 0$) also converge, which means carriers and infected population converging toward their reference trajectories $(C\to C_r$, $I\to I_r)$ and this concludes that the closed-loop pneumonia control system is asymptotically stable despite of uncertainty of parameters. This concludes the proof of the theorem.

Theorem 7. For a given closed-loop system in (6.2), the established control variables and adaptive laws can guarantee that: (i) all the variables involved in the closed-loop system are bounded; (ii) the state variables C and I converge to their desired values.

6.3. Numerical simulations and discussion

This section presents the numerical analysis of the parameter values with the support of MATLAB software. All parameter values were given in the rate of months. The initial conditions of human population were estimated as susceptible $S=1000$, vaccinated $V=400$, carrier $C=200$, treated $T=150$, infected $I=300$, and recovered $R=100$.

Figure 12(a) shows the presence of treatment and vaccination of pneumonia disease control, thus the number carriers and infectious which are responsible for transmitting the pneumonia pathogens is sufficiently minimized to 250 and nearly zero, respectively. This is contrary to Fig. 12(b), where there are rapid increase of carriers to about 700 individuals in the absence of any interventions.

Figures 12(c) and 12(d) represent the absence of single pneumonia disease intervention of treatment and vaccination, respectively. In comparison perspective, treatment intervention seems to have more impact on reducing pneumonia disease carriers than vaccination. Their impact for infectious is likely similar (see Figs. 3(c) and 3(d)).

Figures 13(a) and 13(b) represent the impacts of increasing treatment doses to the infected and exposed individuals. In the presence of vaccination, there are high impacts for reducing the carriers, while there is almost no significant change for the absence of vaccination (see Figs. 4(a) and 4(b)), respectively.

Conversely, Figs. 13(c) and 13(d) indicate the role of increasing vaccination doses to the infected and exposed individuals in the absence and presence of treatment, respectively. Vaccination alone has a slight impact on decreasing pneumonia disease carriers (see Fig. 13(c)) compared with the combination of treatment (see Fig. 13(d)). However, they both reduce insignificant infectious.

6.4. Graphical interpretations for parameters uncertainties control

Figure 14(a), during the absence of disease control, indicates that the carrier population is responsible for transmitting the pneumonia pathogens, which will dominate the population for a long duration beyond 30 months, while the infectious will converge to zero after almost 10 months. On the other hand, (b) shows that number of disease carriers will be diminished immediately from the third month approaching to zero after 30 months later. Similarly, the infectious will be reduced in a few months compared with the absence of the disease control.

6.4.1. Absence of control intervention in the closed-loop control system of pneumonia disease with presence of uncertainties

Figure 15 shows that the disease carriers C and infectious I were well tracked along the trajectories immediately from the initial time to the defined one at various presence of uncertainties rates like (0%, 1%, 5%, 10%) regardless of the absence of control elements $(u_1=u_2=0)$ in the population group panels (a)–(d), respectively. However, indeed, the carriers will last longer in the population beyond 30 months specified time, while the infected individuals will be diminished soon after five initial months. This means that the pneumonia disease infections will take longer time to be eliminated from the human population.

6.4.2. Impact of control intervention in the closed-loop control system of pneumonia disease to human population in presence of uncertainties

Figure 16 indicates that by decreasing the disease control rates just from $0.5$ to $0.05$ in the closed-loop system (panels (a)–(d)), the disease carriers C will become more stronger to invade the population and hence take longer time to be cleared compared with the infectious I whose duration will sharply decline to zero value soon after 10 to 15 months duration. However, the presence of uncertainties considerably (0%, 1%, 5%, 10%) in panels (a)–(d) in the system did not affect the tracking behavior along the specified carrier and infected trajectories. In this case, we observed that as more control rates decrease the more time, the disease carrier and infectious survive in the human population and vice versa.

6.4.3. Variation of control rates in the closed-loop control system of pneumonia disease to human population in presence of uncertainties

Figure 17 shows that in presence of disease control $(u_1=u_2=0.5)$, the disease carriers C and infectious I were both quickly reduced constantly towards zero digit immediately after early 4 to 5 months. Also, it is found that regardless of variation of uncertainties rates in the system, for instance (0%, 1%, 5%, 10%) in panels (a)–(d) yet, the control system managed to track well along the defined trajectories of carrier and infected population. This implies that the disease carriers and infectious will be cleared immediately from the human population, just in the first 5 months.

7. Conclusion

In this paper, we presented the impacts of treatment and vaccination interventions in the combating of pneumonia disease spread. From Table 3, we see the most positively and negatively influential parameters that contributing to the increase and decrease of pneumonia infections, respectively. The variations and physical behaviors in form of contours and 3D faces of these parameters are presented in Figs. 3–10 with respect to the basic reproduction number $R_0$.

Furthermore, the study discussed the ASMC for controlling the disease carriers C and infectious I through their trajectory tracking to achieve the desired solution. From the findings, it is observed that the pneumonia disease control in the closed-loop control system for the carriers and infectious population is more achieved by managing system uncertainties through successfully tracking their defined trajectories. The presence of control parameter values $(u_1,u_2)$ plays a great role in containing the disease transmitters more efficiently in the human population when more uncertainties are managed.

Specially, the closed-loop control system involving high control rates like ($u_1,u_2=0.5$) and above brings more success in reducing the disease carriers C and infectious I in more shortly time that’s less than 5 months duration regardless of the presence of uncertainties considerably (0–10%) rates in the particular groups. Closed-loop control system with ASMC managed the presence of uncertainties in the system and hence facilitates more and quickly reduction of disease carriers and infectious for easily to be cleared from the human population provided the control rates exists in reasonable amount and its reverse is true (see Fig. 17). Accuracy tracking of carrier and infectious population trajectories admits the efficiency of this method in combating the pneumonia disease infections from the human population (see Figs. 15–17). Managing uncertainties is more efficiency in the disease control compared to its absence (see Figs. 12 and 13).

Therefore, it can be concluded that, by controlling the parameters uncertainties by the use of ASMC in the closed-loop system, the pneumonia disease infections rate can be quickly and easily contained from human population. This work has proved the success control of the disease by using the defined control rates and through successfully tracking of the disease transmitters’ trajectories. Hence, the technique signifies its usefulness in managing uncertainties for the disease control as illustrated and discussed above.

Acknowledgments

This work has been supported by the Mathematics for Sustainable Development (MATH4SDG) project, which is a research and development project running in the period 2021–2026 at Makerere University — Uganda, University of Dar es Salaam — Tanzania, and the University of Bergen — Norway, funded through the NORHED II program under the Norwegian Agency for Development Cooperation (NORAD, Project No. 68105).

Competing Interests

Author declares no conflict of interest regarding this work.

Data Availability

Data supporting the work are included in this report.