#### Volume 56, pp. 1-27, 2022.

## Estimating the time-dependent contact rate of SIR and SEIR models in mathematical epidemiology using physics-informed neural networks

Viktor Grimm, Alexander Heinlein, Axel Klawonn, Martin Lanser, and Janine Weber

### Abstract

The course of an epidemic can often be successfully described mathematically using compartment models. These models result in a system of ordinary differential equations. Two well-known examples are the $SIR$ and the $SEIR$ models. The transition rates between the different compartments are defined by certain parameters that are specific for the respective virus. Often, these parameters are known from the literature or can be determined using statistics. However, the contact rate or the related effective reproduction number are in general not constant in time and thus cannot easily be determined. Here, a new machine learning approach based on physics-informed neural networks is presented that can learn the contact rate from given data for the dynamical systems given by the $SIR$ and $SEIR$ models. The new method generalizes an already known approach for the identification of constant parameters to the variable or time-dependent case. After introducing the new method, it is tested for synthetic data generated by the numerical solution of $SIR$ and $SEIR$ models. The case of exact and perturbed data is considered. In all cases, the contact rate can be learned very satisfactorily. Finally, the $SEIR$ model in combination with physics-informed neural networks is used to learn the contact rate for COVID-19 data given by the course of the epidemic in Germany. The simulation of the number of infected individuals over the course of the epidemic, using the learned contact rate, shows a very promising accordance with the data.

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### Key words

machine learning, physics-informed neural networks, SIR model, SEIR model, epidemic modeling, parameter estimation, COVID-19, SARS-CoV-2, scientific machine learning

### AMS subject classifications

65L09, 68T07, 68T09, 92C60, 92D30

### ETNA articles which cite this article

Vol. 56 (2022), pp. 235-255 Matthias Eichinger, Alexander Heinlein, and Axel Klawonn: Surrogate convolutional neural network models for steady computational fluid dynamics simulations |

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