Analysis of Pneumonia Model via Efficient Computing Techniques

: Pneumonia is a highly transmissible disease in children. According to the World Health Organization (WHO), the most affected regions include south Asia and sub-Saharan Africa. Worldwide, 15% of pediatric deaths can be attributed to pneumonia. Computing techniques have a significant role in science, engineering, and many other fields. In this study, we focused on the efficiency of numerical techniques via computer programs. We studied the dynamics of the pneumonia-like infections of epidemic models using numerical techniques. We discuss two types of analysis: dynamical and numerical. The dynamical analysis included positivity, boundedness, local stability, reproduction number, and equilibria of the model. We also discuss well-known computing techniques including Euler, Runge Kutta, and non-standard finite difference (NSFD) for the model. The non-standard finite difference (NSFD) technique shows convergence to the true equilibrium points of the model for any time step size. However, Euler and Runge Kutta do not work well over large time intervals. Computing techniques are the suitable tool for cross-checking the theoretical analysis of the model.

some of them are harmful, causing such issues as brain damage and hearing problems. Meningitis is the most severe disease caused by pneumococcal pneumonia, and it is more common in children who are less than five years old and it can cause long-term disease in individuals over 50 years old. Bacteria are a main and major cause of pneumococcal disease and blood-borne infection. About 1% of children under five years old with this infection die. The chance of death from pneumococcal pneumonia is also higher among the elderly. About 5% of people with pneumonia die, but the ratio is higher among the elderly. Pneumococcal pneumonia can be asymptomatic if there are no bacteria or cold weather during that period. Pneumococcal pneumonia can cause swelling of the throat, necessitating ear tubes in some children. Symptoms of pneumococcal pneumonia can include greenish, yellow, or bloody liquid produced during coughing, weakness, profuse sweating, difficulty breathing, severe headache, and severe chest pain. Symptoms tend to worsen when the patient is hungry or exhausted. In 2014, Mochan et al. [1] dynamically described the interhost immune response to bacterial pneumonia infection in murine strains in a simple ordinary differential equation model. In 2014, Drusano et al. [2] reported the effects of granulocytes in the eradication of bacterial pathogens, and there was no antimicrobial therapy involved in this work. In 2015, Ndelwa et al. [3] produced a dynamic mathematical model for the transmission of pneumonia with screening and medication and analyzed it to assess transmission and effects. In 2015, Kosasih et al. [4] analyzed a mathematical model of cough sounds using waveletbased crackle detection work for rapid diagnosis of bacterial pneumonia in children. In 2016, Cesar et al. [5] mathematically estimated fine particulate matter in a model and evaluated medications for pneumonia and asthma among children. In 2016, Marchello et al. [6] listed atypical bacterial pathogens as the main causes of such lower respiratory diseases as coughs, bronchitis, and CAP. In 2017, Cheng et al. mathematically and dynamically evaluated an IAV-SP model. A quantitative risk-assessment framework was established to improve respiratory health due to COPD [7]. In 2017, Kosasih et al. [8] provided a simple mathematical model showing the analysis of measurements for clinical diagnosis of pneumonia among children. In 2017, Tilahun et al. proposed a deterministic nonlinear mathematical model and analyzed optical control strategies for bacterial pneumonia. Results are shown graphically [9]. In 2018, Raj et al. [10] analyzed the classification of asthma and pneumonia based upon mathematical features of cough sounds among poorer segments of the population. In 2018, Kizito et al. presented a mathematical model that shows the control of pneumonia spread by bacteria. It also gave the dynamics of treatment and formulation of vaccines [11]. In 2018, Mbabazi et al. [12] investigated a nonlinear mathematical model that modeled intra-host co-infection influenza A virus and pneumonia. In 2018, Tilahun et al. [13] proposed a co-infection model for pneumoniatyphoid and mathematically analyzed their characteristic relationship for the development of medical strategies. In 2019, Tilahun et al. described a model of pneumonia-meningitis co-infection with the help of ordinary differential equations and theorems. It explained different techniques for disease clearance [14]. In 2020, Naveed et al. [15] reported a dynamic analysis of coronavirus while assessing the sensitivity of model parameters. In 2019, Kosasih et al. [16] explained the main cause of pneumonia affecting children in early childhood in poor regions of the world. In 2019, Tilahun et al. [17] analyzed a co-infection mathematical model for the bacterial disease of pneumonia and meningitis. In 2019, Mbabazi et al. [18] proposed a mathematical model of pneumococcal pneumonia with time delays and performed Hopf-bifurcation analysis. In 2020, Otoo et al. [19] analyzed a model of pneumonia spread by bacteria. The analysis determined the effects of vaccination on control of this disease. In 2020, Zephaniah et al. [20] presented the dynamics of a mathematical model of pneumonia, showing the result graphically. In 2019, Raza et al. [21] described the stochastic dynamics of gonorrhea-like infections. In 2020, Jung et al. [22] demonstrated the observations using different clinical tests and showed the cause of disease, a novel pathogen. Many mathematical models are studied with different techniques, as shown in previous works [23][24][25][26][27]. Well-known mathematical models can be investigated with the help of efficient techniques [28][29][30][31][32][33][34][35][36][37][38][39]. The rest of the paper is organized as follows. In Sections 2-4, we investigate the dynamic analysis of the model. Section 5 explains the well-known computer methods used on this model. The last two sections present the results, discussion, and conclusion.

Formulation of Pneumonia Model
For any arbitrary time t, the parameters and variables of pneumonia disease described as follows: S(t): represents the susceptible, who is at risk of acquiring infection pneumonia, C(t): represents the carrier individuals carrying the pneumonia bacteria and can transfer the infection, I(t) : represents the infective individuals that are capable of transmitting the infection to individuals at risk, R(t): represents the individuals who have been recovered after the treatment of Pneumonia, μ: represents the natural mortality rate of individuals per capita, : represents the recruitment rate into susceptible population per capita, θ: represents the proportion of susceptible individuals who joins the carriers, σ : represents the disease induced mortality rate birth rate of human population per capita, β: represents the recovery rate of carriers per capita, α: represents the infection force of susceptible individuals, τ : represents the recovery rate of individuals who are infected of Pneumonia per capita, π: represents the rate of developing symptoms by carriers, η: represents the rate of treated individuals becoming susceptible, γ : represents the rate of susceptible individuals getting vaccinated, ω: represents the rate of treated individuals having vaccinated, ω: represents the coefficient of transmission for the carrier subgroup, δ: represents the rate of transmission, p: represents the probability that shows a contact is efficient enough to cause infection, k: represents the rate of contact. The governing equations of the model are as follows: (1)

Fundamental Properties of Model
We consider all parameters positive and show that the solution is bounded in = {(S, C, I, R) 4 So, S ≥ 0 similarly shows that for Eqs. (2)-(4)
Proof: Firstly, adding the Eqs. (1)-(4) as follows: where N 0 is the initial condition of N, This show that 0 ≤ N ≤ μ and N = S + C + I + R, then all variable is bounded in .

Reproduction Number of Pneumonia Model
The next-generation matrix method is presented for the system (1-4). We calculate two types of matrices like transmission and transition after assuming the disease-free equilibrium as follows: The spectral radius of the model is denoted by

Local Stability
Theorem: The disease-free equilibrium of model (1-4) is locally asymptotically stable if the reproduction number is less than one and unstable if it is greater than one.
Proof: To prove the local asymptotically stable disease-free equilibrium, we take the Jacobian matrix of SCIR Model of pneumonia model at disease-free equilibrium. To show that trace is less than zero and a determinant greater than zero.
where, K a = (μ + β + π), Be not be negative and δ[K a (1 − θ) + θ(ωK b + π)] is positive and also K a K b > 0 and we note that determinant (J) also positive, which is −μ(−K a K b (μ + η)) > 0, thus we have The above discussion is about the matrix J, a trace is less than zero and a determinant greater than zero. So, the disease-free equilibrium point is locally asymptotically stable if R 0 < 1.
Theorem: If the reproduction number is greater than one, then the endemic equilibrium of the model Eqs. (1)-(4) is locally asymptotically stable in .
Proof: The Jacobian matrix at endemic equilibrium is as follows: By using Routh Hurwitz method for order 4 th as follows: ⎡ The endemic equilibrium is locally asymptotically stable for the reproduction number greater than one if

Computing Techniques
In this section, we present the well-known techniques like Euler, Runge Kutta, and non-standard finite difference for the system (1-4) as follows:

Euler Technique
The system (1-4) is described under Euler technique, as follows: where the time step is represented by h.

Runge-Kutta Technique
The system (1-4) is described under Runge Kutta technique, as follows: Stage 1: Stage 2: Stage 3: Stage 4: Final stage: where the time step is represented by h.

Non-standard Finite Difference Technique
The system (1-4) is described under NSFD technique, as follows: where the time step is represented by h.

Convergence Analysis
Theorem: The computing technique of the proposed system (10-13) is stable for any n ≥ 0 if the absolute eigenvalues are less than one [40].

Computing Results
In this section, we investigate the computing results for the said model with the help of computer software and the scientific literature presented in Tab. 1 as follows:

Results and Discussion
We present the solution to the system (1-4) via Matlab ordinary differential equations-45 at disease-free and endemic equilibria of the model in Figs. 1a and 1b. Also, the solutions of the system (5-8) via the Euler method at different time step sizes are in Figs. 2a and 2b. The solution of the system (9) via the Runge Kutta method at different time step sizes is in Figs. 3a and 3b. In the same, we plot the solutions of the system (10-13) via the NSFD method in Figs. 4a and 4b. In Figs. 5a-5d, the comparison section shows the investigation of computer methods such as Euler and Runge Kutta with NSFD approximations. Here, we observe that Euler and Runge Kutta show negativity and unboundedness and violate the dynamical properties of the model. However, our proposed numerical approximation is reliable, inexpensive, independent of the time step, and an efficient computational method.

Conclusion
We here investigated analyses of pneumonia infections via well-known computing techniques. Computer results of epidemic models are an authentic tool to cross-check the dynamical analysis of the model. For the sake of computational analysis, Euler, Runge Kutta, and the non-standard finite difference techniques (NSFD) are presented. Throughout the analysis, we observe that Euler and Runge Kutta are time-dependent techniques. Even when we increase the duration of the time step, these techniques violate such dynamic properties as positivity, boundedness, and dynamical consistency. However, NSFD is always convergent and independent of the size of the time step. These things could be observed from the comparison section. This idea could be extended to different types of disease modeling.