An Accurate Three-Step Hybrid Block Method Via Optimization Approach for Solving Mathematical Model of Continuous Fever

Emergence of novel infectious diseases and the resurgence of already known ones and its variants elicit significant concern in our contemporary world. Thus, it is very crucial to utilize all available resources to monitor and control their spread. Most of the epidemiological models developed to study and analyze the characteristics of diseases produced system of differential equations that are coupled in nature, which has become a challenge to researchers to find exact solutions. This work proposes an accurate three-step hybrid block method through optimization approach for solving mathematical models of continuous fever. The techniques of interpolation and collocation were applied to a power series polynomial for the derivation of the method using a three-parameter approximation of the hybrid points. The hybrid points were obtained by minimizing the local truncation error of the main method. The discrete schemes were produced as by-products of the continuous scheme and used to simultaneously solve mathematical models of continuous fever in block mode. The analysis of the basic properties of the method revealed that the schemes are self-starting, convergent, and A-stable. In addition, the analysis of the order of accuracy of the method showed that there is a gain of one order of accuracy in the main scheme where the optimization was carried out. Thereby, enhancing the accuracy of the whole method. The accuracy of the method was ascertained using three numerical examples. Comparison of the numerical results of the new method with those of the existing methods revealed that the newly developed method compares favorably with the existing hybrid block methods. Hence, the new method should be employed for the numerical solution of initial value problems of ordinary differential equations to obtain more accurate results.