## 3 Results and Discussion
Its maximum errors were \(max\left|x_{i}^{exact}-x_{i}^{Atdams-Bashforth}\right|=0.007965958\), \(max\left|x_{i}^{exact}-x_{i}^{Atdams-Bashforth}\right|=0.03635441\), \(max\left|x_{i}^{exact}-x_{i}^{Atdams-Bashforth}\right|=0.01277912\) for the age categories 65-74, 75-89, and 90+, respectively, significantly higher, especially in the 75-89 age category, indicating a lower accuracy of the method. The backward Euler method had the lowest orders of accuracy \(max\left|x_{i}^{exact}-x_{i}^{BackumEuler}\right|=0.005310639\), \(max\left|x_{i}^{exact}-x_{i}^{BackumEuler}\right|=0.02423627\), and \(max\left|x_{i}^{exact}-x_{i}^{BackumEuler}\right|=0.008519411\) for the age groups 65-74, 75-89, and 90+, respectively. Although this method has a simple implementation, its high uncertainty makes it less favorable for our application, especially in age groups with large data variation. Based on the obtained data, the fourth-order Runge-Kutta method (\(O(\Delta x^{4})\)) was the optimal choice for the numerical solution of the considered problem, as it improved the accuracy of the numerical solution by providing the smallest errors and the best agreement with the analytical calculations. \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Solution Methods** & **Order of Method** & **Age Category** &
\begin{tabular}{l} **Maximum Error with** \\ **Exact Solution** \\ \end{tabular} \\ \hline \multirow{3}{*}{Runge–Kutta} & \multirow{3}{*}{\(O\left(\Delta x^{4}\right)\)} & 65–74 & 0.00002677413 \\ \cline{3-3}  & & 75–89 & 0.0004091325 \\ \cline{3-3}  & & 90+ & 0.00004571414 \\ \hline \multirow{3}{*}{Adams–Bashforth} & \multirow{3}{*}{\(O\left(\Delta x^{2}\right)\)} & 65–74 & 0.007965958 \\ \cline{3-3}  & & 75–89 & 0.03635441 \\ \cline{3-3}  & & 90+ & 0.01277912 \\ \hline \multirow{3}{*}{Backward Euler} & \multirow{3}{*}{\(O\left(\Delta x\right)\)} & 65–74 & 0.005310639 \\ \cline{3-3}  & & 75–89 & 0.02423627 \\ \cline{1-1} \cline{3-3}  & & 90+ & 0.008519411 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Values of accuracy orders of numerical methods. Next, let us consider our problem using the already selected fourth-order Runge-Kutta method. Figure 8 presents data illustrating changes in the ageing process for the three age groups: 65-74 years old, 75-89 years old, and 90+ years old. The abscissa axis (horizontal axis) denotes time, and the ordinate axis (vertical axis) shows the changes in aging according to Figures 2-4. In the 65-74 age group, moderate fluctuations in indicators with relatively low amplitude were observed, which may indicate a more stable aging process in that interval. In that group, the influence of bad habits, such as alcohol consumption and smoking, on the development of postinfarction cardiosclerosis (correlation coefficient \(b_{1}\)), increased blood pressure, as well as changes in body weight, which may accelerate the aging process, was noted. In the 75-89 age category, the amplitude of the fluctuations was higher than in 65-74-year-olds but lower than in persons over 90 years of age, indicating the intermediate nature of changes associated with aging. For that group, we can assume that the aging process is associated with the level of \(a_{2}\) interaction with the increase in blood pressure, the appearance of the \(c_{2}\) correlation coefficient reflecting the risk of developing chronic heart failure, as well as the \(d_{2}\) coefficient characterizing the probability of coronary heart disease. The analysis of the graphs showed that the peaks in each age category may correspond to the time intervals during which the ageing processes were most intensive. The group of 65-74-year-olds showed the most stable dynamics of changes, probably due to the influence of bad habits, which may indicate a more predictable nature of ageing at that age. On the contrary, the group of 90+-year-olds showed the greatest fluctuations, which may indicate a decrease in the predictability of the ageing process or its increased sensitivity to various factors, such as the respondent's gender, level of education, and smoking.