Mathematical tools produce new COVID-19 transmission dynamics

Dr Faranak

The education sector, which includes universities, institutes and colleges, has undoubtedly been impacted by the pandemic. With the Movement Control Order implemented in March last year, the industry heavily executed remote learning. Research sectors in universities were massively affected as well.

Experimental data collection and computer simulations for research projects had to be put on hold as laboratories were closed, causing a delay in findings. However, the contribution of research and technology to control and stop the spread of COVID-19 is praiseworthy.

Dr Faranak Rabiei from the School of Engineering, Monash University Malaysia, and postgraduate student Zeeshan Ali took the opportunity to collaborate with scientists from international institutes in applied and computational mathematics to produce a new mathematical modelling of COVID-19's transmission dynamics.

"The existing biological models of COVID-19 is based on classical systems of ordinary differential equations with integer-order. They have been widely studied and developed in several pieces of research. Recently, mathematical models of Coronavirus with fractional order have been considered by many researchers. It is proven that mathematical models designed with the help of fractional calculus (FC) tools are more accurate and stable in comparison to integer-order calculus due to its degree of freedom," Dr Faranak stated.

FC tools are dependent on historical and current states. Thus, they have memory properties. "We know that there are complexities and false information, which makes it hard to provide a suitable mathematical model for COVID-19 with classical order differentiation. The nonlocal operators in FC are ideal because they can capture non-localities and some memory effects. Moreover, the latest developed fractal-fractional order tool provides the relationship between fractal calculus and FC," she added.

The proposed operators in fractal-fractional order mathematical models are more suitable when there is a complex behaviour. The new calculus, named fractal-fractional calculus, is perhaps more efficient than fractal calculus and FC.

With the help of fractal-fractional calculus as a powerful mathematical tool, a novel fractal-fractional model of coronavirus involving different compartments (suspension, infection, recovery, death, etc.) was proposed. "The theoretical parts of the research, such as proof of existence, uniqueness and stability of the solution of proposed models, were discussed in detail," Dr Faranak shared.

Towards the last segments of the research, numerical techniques were developed to analyse and study the transmission dynamics of proposed models of COVID-19 in three regions - Malaysia, Wuhan and Pakistan. The team was able to obtain simulated results for used compartments in the proposed models. The comparison of research simulations with actual disease outbreak data proved the accuracy and efficiency of this study.

Figure 1: Real confirmed cases per day vs simulated confirmed case per day from published research article.

"We concluded that disease transmission in society was reduced by adopting preventive measures (restrictions on individual movement). Although the pandemic has a significant impact on our research, applying the new mathematical model of COVID-19 makes it possible to accurately predict and simulate disease compartments' transmission dynamics. In a nutshell, mathematics is an essential tool that has a high impact on research development in many areas of science and technology," Dr Faranak said.

The outcome of this research highlights the importance of the implementation of suitable strategies and restrictions to control the spread of disease in countries. The presented model's simulated results demonstrate that movement control order significantly impacts the transmission dynamics of disease outbreak in the three regions studied. The fractal-fractional calculus tools may be used as powerful tools to understand and predict the mentioned disease's global dynamics in other countries.