Ulam-Hyers stability of tuberculosis and COVID-19 co-infection model under Atangana-Baleanu fractal-fractional operator

Abstract

The intention of this work is to study a mathematical model for fractal-fractional tuberculosis and COVID-19 co-infection under the Atangana-Baleanu fractal-fractional operator. Firstly, we formulate the tuberculosis and COVID-19 co-infection model by considering the tuberculosis recovery individuals, the COVID-19 recovery individuals, and both disease recovery compartment in the proposed model. The fixed point approach is utilized to explore the existence and uniqueness of the solution in the suggested model. The stability analysis related to solve the Ulam-Hyers stability is also investigated. This paper is based on Lagrange's interpolation polynomial in the numerical scheme, which is validated through a specific case with a comparative numerical analysis for different values of the fractional and fractal orders.

Figure 1

TB and COVID-19 co-infection model showing the compartments.

Figure 2

Susceptible to both TB and COVID-19 and time variations for varying values of $k_1$ and $k_2$.

Figure 3

Latent level TB infected people and time variations for varying values of $k_1$ and $k_2$.

Figure 4

Active level TB infected people and time variations for varying values of $k_1$ and $k_2$.

Figure 5

Recovered from TB infected people and time variations for varying values of $k_1$ and $k_2$.

Figure 6

Infected with COVID-19 after recovering from TB people and time variations for varying values of $k_1$ and $k_2$.

Figure 7

Asymptomatic COVID-19 infected people and time variations for varying values of $k_1$ and $k_2$.

Figure 8

Symptomatic COVID-19 infected people and time variations for varying values of $k_1$ and $k_2$.

Figure 9

Recovered from COVID-19 infected people and time variations for varying values of $k_1$ and $k_2$.

Figure 10

Infected with TB after recovering from COVID-19 people and time variations for varying values of $k_1$ and $k_2$.

Figure 11

Both latent TB and COVID-19 co-infected people and time variations for varying values of $k_1$ and $k_2$.

Figure 12

Both active TB and COVID-19 co-infected people and time variations for varying values of $k_1$ and $k_2$.

Figure 13

Both TB and COVID-19 recovered people and time variations for varying values of $k_1$ and $k_2$.

Figure 14

Comparative study of $I^T$ and time variations with $k_1=k_2=0.95$ for varying values of infection rate ${\rho }_1$.

Figure 15

Comparative study of $R^T$ and time variations with $k_1=k_2=0.95$ for varying values of infection rate ${\rho }_1$.

Figure 16

Comparative study of $I^c$ and time variations with $k_1=k_2=0.95$ for varying values of infection rate ${\rho }_2$.

Figure 17

Comparative study of $R^c$ and time variations with $k_1=k_2=0.95$ for varying values of infection rate ${\rho }_2$.

Figure 18

Comparative study of TB and COVID-19 co-infection population density and $k_1=k_2=0.95$ against time.

Figure 19

Comparative study of simulated and real data in COVID-19 infected people and time variations for varying of $k_1$ and $k_2$.