Step forward in epidemiological modeling: Introducing the rate indicator function to capture waves

Abstract

In the last centuries, mathematical models have been used to depict the dynamic evolution of infectious diseases’ spread. The aim is to determine the total numbers of infected, recovered, and susceptible individuals; however, they only represent accumulated values due to the complexities of collected data. Having the graphical representations of these classes as accumulated values, one may not directly be able to predict waves to determine a day-to-day number of newly infected cases as some additional calculations are required. Collected data from real-world situations are represented as day-to-day new infections; they help determine in addition numbers of waves. However, current existing mathematical models cannot be used for wave prediction. While knowing the predicted numbers of waves, policymakers can take adequate measures to control the situation. To solve this problem, we questioned the fact that the rates of infection and recovery are constant and suggested an indicator rate function obtained using experimental data. To see the effect of this function, we considered a simple SIR model, which was modified by introducing rate indicator functions for infected and recovered classes. To include nonlocal behaviors in the mathematical model, different types of differential operators, including classical and fractional derivatives, were used. The models were solved numerically using well-known numerical schemes. The numerical solutions were plotted for different theoretical parameters; the results depict real-world behaviors. To test the efficiency of this new approach, we collected data from South Africa and compared them with our model. Not only our model could predict waves, but also it fits experimental data very accurately. This new approach will open new doors of investigation toward a revolution in epidemiological modeling.