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.

Pneumonia is a disease of the lungs that can cause minor to severe illness in people of different ages. The swelling of the lungs that occurs during pneumonia is most commonly caused by infection with bacteria or molds. There are also a few noninfectious types of pneumonia. These are caused by inhaling contaminated materials into the lungs. Most pneumococcal poisons are insignificant, but 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. [

For any arbitrary time

We consider all parameters positive and show that the solution is bounded in

So, S

So,

There are two steady states of

where

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

_{b} =

Be not be negative and

The above discussion is about the matrix

_{b} =

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

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:

The system (1–4) is described under Euler technique, as follows:

The system (1–4) is described under Runge Kutta technique, as follows:

Stage 1:

Stage 2:

Stage 3:

Stage 4:

Final stage:

The system (1–4) is described under NSFD technique, as follows:

The Jacobian matrix is defined as

After that, by assuming the values of disease-free equilibrium

The given Jacobian is

The eigenvalues of the Jacobian matrix are

In this section, we investigate the computing results for the said model with the help of computer software and the scientific literature presented in

Parameters | Values |
---|---|

Λ | 0.5 |

𝜔 | 0.1124 |

𝜃 | 0.563 |

𝜂 | 0.00641 |

𝛽 | 0.515 |

𝜇 | 0.5 |

𝜋 | 0.7096 |

𝜎 | 0.53 |

𝛿 | 2 (DFE) 2.5 (EE) |

𝜏 | 0.641 |

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

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.

We thank LetPub (