Modeling the transmission dynamics of delayed pneumonia-like diseases with a sensitivity of parameters

Abstract

Pneumonia is a highly transmitted disease in children. According to the World Health Organization (WHO), the most affected regions include South Asia and sub-Saharan Africa. 15% deaths of children are due to pneumonia. In 2017, 0.88 million children were killed under the age of five years. An analysis of pneumonia disease is performed with the help of a delayed mathematical modelling technique. The epidemiological system contemplates subpopulations of susceptible, carriers, infected and recovered individuals, along with nonlinear interactions between the members of those subpopulations. The positivity and the boundedness of the ongoing problem for nonnegative initial data are thoroughly proved. The system possesses pneumonia-free and pneumonia existing equilibrium points, whose stability is studied rigorously. Moreover, the numerical simulations confirm the validity of these theoretical results.

Keywords: Delayed model; Numerical simulations; Pneumonia disease; Stability analysis.

Figure 1

Flow chart of the pneumonia disease model

Figure 2

Subpopulations plots concerning the time of system (1)–(4) at the pneumonia-free equilibrium of the model when $\tau =0$

Figure 3

Time plots at the pneumonia existing equilibrium when $\tau =0$

Figure 4

Subpopulations plots concerning the time of the system at the pneumonia existing equilibrium with practical uses of different values of delay tactics like $\tau =0.1,0.2,0.5,1$ respectively

Figure 5

Simulation of the reproduction number with different values of the delay parameter

Figure 6

Subpopulation of the infected class concerning time with different delay values

Figure 7

Time plots of the system in two-dimensional way when $\tau =0$

Figure 8

Time plots of the system in two-dimensional way when $\tau =1$