Infection Rate Dynamics of Covid-19: Modeling With Differential Equations and Neural Networks

In December 2019, the Coronavirus disease (COVID-19) caused an outbreak in China, and subsequently, has become a major, global public health issue. An understanding of the disease dynamics can be accomplished by using mathematical modeling. As an illustrative example, here, the SIR (Susceptible-Infectious-Recovered) compartmental model is used to study the spread of infectious disease.  The SIR model is comprised of a system of differential equations, which includes specific time-varying parameters. Through the use of Machine Learning techniques and specifically Physics-Informed Neural Networks (PINNs), one can identify the time-dependent parameters to describe the COVID-19 infection dynamics at a county, state, or country level. With PINNs, one exploits the governing (physical) laws in mathematical form and thus introduces information, due to which the machine learning methods are no longer just dependent on the training datasets. By integrating the underlying physics behind a dynamical system (e.g., infectious disease physics in the current case) with Machine Learning techniques, one can design Deep Learning methods that can be used to model and efficiently study complex phenomena. In this work, the SIR differential equations are assimilated in the Neural Networks during training, and they serve to bring physical knowledge with the data. Here, the authors first construct a discrete SIR simulation model. By numerically integrating the differential equations, data is generated for the populations belonging in each compartment. For simplification, the recovery rate is considered to be constant. However, the authors acknowledge that the infection rate is a time-varying parameter, which needs to be modeled to produce credible SIR infection data. In order to extract a function that describes the infection rate with respect to time, a PINN is eventually used. The architecture of the network includes one input node, four hidden layers, and one output node, while in the activation functions, the authors use the sigmoid and the hyperbolic tangent functions. The network is fed and trained with the infection data associated with the original SIR model. After making use of the day to day infection data, which are available from different established sources, the authors’ goal is to minimize the loss function. The algorithms for the above machine learning techniques are being executed in Python by using the TensorFlow library along with the ADAM optimization tool.  In summary, the outbreak of COVID-19 has strongly impacted today’s demographics and has affected every aspect of human life as one knows it. Studying the disease dynamics at a local or country level offers one insight into how social distancing, testing, and individual hygiene practices have been affecting the spread of the coronavirus. Notably, the basic reproduction number, which is the ratio of the infection rate to the recovery rate, is crucial in terms of informing one about the transmission potential of the disease in a region. To this end, as a first step, through this research, the use of the Physics-Informed Neural Network has been illustrated for extraction of the time-dependent infection parameters that can better reflect the contagiousness of the COVID-19 in different parts of the world.  In future research, other extended compartment models are to be considered in place of the SIR model.