Estimating Time-Dependent COVID-19 Parameters Using Kolmogorov-Arnold Network and Physics-Informed Neural Network

Abstract

We introduce a novel method for estimating COVID-19 time-varying parameters. These parameters are in the context of an SIRD compartmental differential equations. The time-dependent parameters are the transmission rate $\beta(t)$, recovery rate $\gamma(t)$, and mortality rate $\mu(t)$. The method harnesses the novel Kolmogorov-Arnold Network (KAN), which is a type of artificial neural network. For the KAN in this paper, we learn activation functions that are represented using Fourier series, hence the abbreviation KAN-F. We define three KAN-F functions $\widehat{\beta}$, $\widehat{\gamma}$, $\widehat{\mu}$ that model the true parameters $\beta(t)$, $\gamma(t)$, $\mu(t)$, respectively. We investigate two model architectures for the KAN-F: the first has 8 input variables consisting of $S$, $I$, $R$, $D$, and their numerical gradients at any time $t$, while the second has 4 input variables excluding the numerical gradients. The objective loss function that has to be minimized is subject to Physics-Informed Neural Network (PINN) or Epi-DNN. We estimate the time-dependent parameters of COVID-19 using data from three South-East Asian countries: Indonesia, Singapore, and Malaysia. The time period of choice coincides with the period where SARS-CoV-2 Delta variant (B.1.617.2) was dominant. Using Epi-DNN and KAN-F, we are able to estimate $\beta(t)$, $\gamma(t)$, and $\mu(t)$, with decent accuracy and comparative efficiency. In addition to estimating the rates during the training period, we also predict transmission rates over 30 days during forecasting period. We found that the output of KAN-F over the forecasting period can give good predictions if we scale the output by a factor of 17\% for Indonesia and 30\% for Singapore and Malaysia.