On mathematical modelling of measles disease via collocation approach

Abstract

Measles, a highly contagious viral disease, spreads primarily through respiratory droplets and can result in severe complications, often proving fatal, especially in children. In this article, we propose an algorithm to solve a system of fractional nonlinear equations that model the measles disease. We employ a fractional approach by using the Caputo operator and validate the model's by applying the Schauder and Banach fixed-point theory. The fractional derivatives, which constitute an essential part of the model can be treated precisely by using the Broyden and Haar wavelet collocation methods (HWCM). Furthermore, we evaluate the system's stability by implementing the Ulam-Hyers approach. The model takes into account multiple factors that influence virus transmission, and the HWCM offers an effective and precise solution for understanding insights into transmission dynamics through the use of fractional derivatives. We present the graphical results, which offer a comprehensive and invaluable perspective on how various parameters and fractional orders influence the behaviours of these compartments within the model. The study emphasizes the importance of modern techniques in understanding measles outbreaks, suggesting the methodology's applicability to various mathematical models. Simulations conducted by using MATLAB R2022a software demonstrate practical implementation, with the potential for extension to higher degrees with minor modifications. The simulation's findings clearly show the efficiency of the proposed approach and its application to further extend the field of mathematical modelling for infectious illnesses.

Keywords: Haar wavelet; Ulam-Hyers stability; fixed point theory; fractional SEIR modeling; numerical analysis.

Figure 1.. Flow chart.

Figure 2.. Graphical presentation of sensitivity index

Figure 3.. Susceptible population by HWCM at $\chi=1$.

Figure 4.. Plot of exposed peoples by HWCM at $\chi=1$.

Figure 5.. Plot of infected people by HWCM at $\chi=1$.

Figure 6.. Plot of recovered people by HWCM at $\chi=1$.

Figure 7.. Plot of $\mathbb{S}(\tau)$ for various values of $\chi$.

Figure 8.. Plot of $\mathbb{E}(\tau)$ for various values of $\chi$.

Figure 9.. Plot of infected population $\mathbb{I}(\tau)$ for various values of $\chi$.

Figure 10.. Plot of $\mathbb{R}(\tau)$ for various values of $\chi$.

Figure 11.. Dynamics of the SEIR system at $\chi=1$.