# Determining COVID-19 Dynamics Using Physics Informed Neural Networks

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## Abstract

**:**

## 1. Introduction and Background

#### 1.1. Introduction

#### 1.2. Background

**Definition**

**1.**

**Definition**

**2.**

**Theorem**

**1.**

**Proof.**

## 2. Review of Studies on Physics Informed Neural Networks

#### 2.1. Physics Informed Deep Learning for Traffic State Estimation

#### 2.2. Neural Network Aided Quarantine Control Model Estimation of Global COVID-19 Spread

#### 2.3. Identification and Prediction of Time-Varying Parameters of COVID-19 Model: A Data-Driven Deep Learning Approach

## 3. Problem Formulation and Methodology

#### 3.1. The Neural Network

#### 3.1.1. Residual of Model’s Equations

#### 3.1.2. The Loss Function

#### 3.2. Basic Model Properties

#### 3.2.1. Basic Reproduction Number

#### 3.2.2. SIRD Model Analysis

#### 3.3. Simulation Using Mathematica Generated Data

#### PINNs Model of Mathematica Results

#### 3.4. PINNs Simulations of Alabama State Data

#### 3.5. PINNs Simulation of a Model Using 170 Data Points

#### 3.6. PINNs Simulation of a Model Using All Available Data Points at the Time (576 Data Points)

#### 3.7. PINNs Simulation Forecasting 30 Days

#### 3.8. Deep Learning Sensitivity Analysis

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Moriarty, L.F.; Plucinski, M.M.; Marston, B.J.; Kurbatova, E.V.; Knust, B.; Murray, E.L.; Pesik, N.; Rose, D.; Fitter, D.; Kobayashi, M.; et al. Public health responses to COVID-19 outbreaks on cruise ships worldwide, February-March 2020. Morb. Mortal. Wkly. Rep.
**2020**, 69, 347–352. [Google Scholar] [CrossRef] [PubMed] - Lu, H.; Stratton, C.W.; Tang, Y.W. Outbreak of pneumonia of unknown etiology in Wuhan, China: The mystery and the miracle. J. Med Virol.
**2020**, 92, 401–402. [Google Scholar] [CrossRef] [Green Version] - Chen, Y.C.; Lu, P.E.; Chang, C.S.; Liu, T.H. A time-dependent SIR model for COVID-19 with undetectable infected persons. IEEE Trans. Netw. Sci. Eng.
**2020**, 7, 3279–3294. [Google Scholar] [CrossRef] - Guo, Y.R.; Cao, Q.D.; Hong, Z.S.; Tan, Y.Y.; Chen, S.D.; Jin, H.J.; Tan, K.S.; Wang, D.Y.; Yan, Y. The origin, transmission and clinical therapies on coronavirus disease 2019 (COVID-19) outbreak-an update on the status. Mil. Med. Res.
**2020**, 7, 1–10. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Bloomgarden, Z.T. Diabetes and COVID-19. J. Diabetes
**2020**, 12, 347–348. [Google Scholar] [CrossRef] [Green Version] - Dandekar, R.; Barbastathis, G. Neural Network aided quarantine control model estimation of global Covid-19 spread. arXiv
**2020**, arXiv:2004.02752. [Google Scholar] - Lauer, S.A.; Grantz, K.H.; Bi, Q.; Jones, F.K.; Zheng, Q.; Meredith, H.R.; Azman, A.S.; Reich, N.G.; Lessler, J. The incubation period of coronavirus disease 2019 (COVID-19) from publicly reported confirmed cases: Estimation and application. Ann. Intern. Med.
**2020**, 172, 577–582. [Google Scholar] [CrossRef] [Green Version] - Long, J.; Khaliq, A.Q.M.; Furati, K.M. Identification and prediction of time-varying parameters of COVID-19 model: A data-driven deep learning approach. Int. J. Comput. Math.
**2021**, 1–19. [Google Scholar] [CrossRef] - Boone, L.; Haugh, D.; Pain, N.; Salins, V. 2 tackling the fallout from COVID-19. In Economics in the Time of COVID-19; CEPR Press: London, UK, 2020; p. 37. [Google Scholar]
- Nyabadza, F.; Chirove, F.; Chukwu, W.; Visaya, M. Modelling the potential impact of social distancing on the covid-19 epidemic in south africa. medRxiv
**2020**, 2020, 5379278. [Google Scholar] [CrossRef] - Peng, L.; Yang, W.; Zhang, D.; Zhuge, C.; Hong, L. Epidemic analysis of covid-19 in china by dynamical modeling. arXiv
**2020**, arXiv:2002.06563. [Google Scholar] - Eikenberry, S.; Mancuso, M.; Iboi, E.; Phan, T.; Eikenberry, K.; Kuang, Y.; Kostelich, E.; Gumel, A. To mask or not to mask: Modeling the potential for face mask use by the general public to curtail the covid-19 pandemic. Infect. Dis. Model.
**2020**, 5, 293–308. [Google Scholar] [CrossRef] - Tang, B.; Xia, F.; Tang, S.; Bragazzi, N.L.; Li, Q.; Sun, X.; Liang, J.; Xiao, Y.; Wu, J. The effectiveness of quarantine and isolation determine the trend of the covid-19 epidemics in the final phase of the current outbreak in china. Int. J. Infect. Dis.
**2020**, 95, 288–293. [Google Scholar] [CrossRef] [PubMed] - Ndairou, F.; Area, I.; Nieto, J.; Torres, D. Mathematical modeling of covid-19 transmission dynamics with a case study of wuhan. Chaos Solitons Fractals
**2020**, 135, 109846. [Google Scholar] [CrossRef] - Anguelov, R.; Banasiak, J.; Bright, C.; Lubuma, J.; Ouifki, R. The big unknown: The asymptomatic spread of covid-19. BIOMATH
**2020**, 9, 2005103. [Google Scholar] [CrossRef] - Jia, J.; Ding, J.; Liu, S.; Liao, G.; Li, J.; Duan, B.; Wang, G.; Zhang, R. Modeling the control of covid-19: Impact of policy interventions and meteorological factors. arXiv
**2020**, arXiv:2003.02985. [Google Scholar] - Kucharski, A.; Russell, T.; Diamond, C.; Liu, Y.; Edmunds, J.; Funk, S.; Eggo, R.; Sun, F.; Jit, M.; Munday, J.; et al. Early dynamics of transmission and control of covid-19: A mathematical modelling study. Lancet Infect. Dis.
**2020**, 20, 553–558. [Google Scholar] [CrossRef] [Green Version] - Liu, Y.; Gayle, A.; Wilder-Smith, A.; Rocklöv, J. The reproductive number of covid-19 is higher compared to sars coronavirus. J. Travel Med.
**2020**, 27, aaa021. [Google Scholar] [CrossRef] [Green Version] - Shayak, B.; Sharma, M.; Rand, R.H.; Singh, A.; Misra, A. Transmission dynamics of covid-19 and impact on public health policy. medRxiv
**2020**. [Google Scholar] [CrossRef] [Green Version] - Sun, T.; Wang, Y. Modeling covid-19 epidemic in heilongjiang province, china. Chaos Solitons Fractals
**2020**, 138, 109949. [Google Scholar] [CrossRef] - Boccaletti, S.; Ditto, W.; Mindlin, G.; Atangana, A. Modeling and forecasting of epidemic spreading: The case of COVID-19 and beyond. Chaos Solitons Fractals
**2020**, 135, 109794. [Google Scholar] [CrossRef] - Castillo, O.; Melin, P. Forecasting of covid-19 time series for countries in the world based on a hybrid approach combining the fractal dimension and fuzzy logic. Chaos Solitons Fractals
**2020**, 140, 110242. [Google Scholar] [CrossRef] - Ivorra, B.; Ferrandez, M.R.; Vela-Perez, M.; Ramosa, A.M. Mathematical modeling of the spread of the coronavirus disease 2019 (COVID-19) taking into account the undetected infections. The case of China. Commun. Nonlinear Sci. Numer. Simul.
**2020**, 88, 105303. [Google Scholar] [CrossRef] [PubMed] - Huang, J.; Agarwal, S. Physics informed deep learning for traffic state estimation. In Proceedings of the 2020 IEEE 23rd International Conference on Intelligent Transportation Systems (ITSC), Rhodes, Greece, 20–23 September 2020; pp. 1–6. [Google Scholar]
- Raissi, M.; Perdikaris, P.; Karniadakis, G.E. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J. Comput. Phys.
**2019**, 378, 686–707. [Google Scholar] [CrossRef] - Jagtap, A.D.; Karniadakis, G.E. Extended Physics-Informed Neural Networks (XPINNs): A Generalized Space-Time Domain Decomposition Based Deep Learning Framework for Nonlinear Partial Differential Equations. Commun. Comput. Phys.
**2020**, 28, 2002–2041. [Google Scholar] - Goodfellow, I.; Bengio, Y.; Courville, A. Deep Learning; MIT Press: Cambridge, UK, 2016; Volume 1. [Google Scholar]
- Barto, A.G.; Dietterich, T.G. Reinforcement learning and its relationship to supervised learning. Handb. Learn. Approx. Dyn. Program.
**2004**, 10, 9780470544785. [Google Scholar] - Li, S.; Ma, B.; Chang, H.; Shan, S.; Chen, X. Continuity-discrimination convolutional neural network for visual object tracking. In Proceedings of the 2018 IEEE International Conference on Multimedia and Expo (ICME), San Diego, CA, USA, 23–27 July 2018; pp. 1–6. [Google Scholar]
- Dey, A. Machine learning algorithms: A review. International J. Comput. Sci. Inf. Technol.
**2016**, 7, 1174–1179. [Google Scholar] - Sra, S.; Nowozin, S.; Wright, S.J. (Eds.) Optimization for Machine Learning; MIT Press: Cambridge, MA, USA, 2012. [Google Scholar]
- Misyris, G.S.; Venzke, A.; Chatzivasileiadis, S. Physics-informed neural networks for power systems. In Proceedings of the 2020 IEEE Power & Energy Society General Meeting (PESGM), Virtual Event, 3–6 August 2020; pp. 1–5. [Google Scholar]
- University of Eswatini. Available online: https://datastudio.google.com/reporting/b847a713-0793-40ce-8196-e37d1cc9d720/page/2a0LB (accessed on 1 October 2021).
- Huang, C.; Wang, Y.; Li, X.; Ren, L.; Zhao, J.; Hu, Y.; Zhang, L.; Fan, G.; Xu, J.; Gu, X.; et al. Clinical features of patients infected with 2019 novel coronavirus in Wuhan, China. Lancet
**2020**, 395, 497–506. [Google Scholar] [CrossRef] [Green Version] - Thabet, S.T.M.; Abdo, M.S.; Shah, K.; Abdeljawadd, T. Study of transmission dynamics of COVID-19 mathematical model under ABC fractional order derivative. Results Phys.
**2020**, 19, 103507. [Google Scholar] [CrossRef] - Inui, S.; Fujikawa, A.; Jitsu, M.; Kunishima, N.; Watanabe, S.; Suzuki, Y.; Umeda, S.; Uwabe, Y. Chest CT findings in cases from the cruise ship diamond princess with Coronavirus disease (COVID-19). Radiol. Cardiothorac. Imaging
**2020**, 2, e200110. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**A schematic flow diagram representing a Susceptible-Infected-Recovered-Dead (SIRD) COVID-19 transmission.

**Figure 2.**A schematic representation of the Physics informed neural network, which takes an input of time (t) and outputs Susceptible (S), Infected (I), Recovered (R) and Deceased D. The output is subjected to PINN.

**Figure 3.**A Mathematica generated graph simulation of an example SIRD model. The green represents Susceptible population, blue represents the recoveries, red is the active infected population and orange is the deceased population.

**Figure 4.**The resulting graph of the predicted values of the Infected population and the actual values of the infected population from the Mathematica generated data.

**Figure 5.**The graphs shows the results of the predicted values of the Recovered and the actual values of the recovered from the Mathematica generated data.

**Figure 6.**The resulting graph of the predicted values of the Deceased population and the actual values of the deceased population from the Mathematica generated data.

**Figure 7.**This graph shows a comparison of the predicted values of susceptible population and the actual data of susceptible population for the State of Alabama.

**Figure 8.**The graph shows a comparison of the predicted infected population values and the actual data infected population for the State of Alabama.

**Figure 9.**This graph shows a comparison of the predicted values of recovered population and the actual data of recovered population for the State of Alabama.

**Figure 10.**The graph shows a comparison of the predicted deceased population values and the actual data deceased population for the State of Alabama.

**Figure 11.**This graph shows a comparison of the predicted values of susceptible population and the actual data of susceptible population for a 130 data points.

**Figure 12.**The graph shows a comparison of the predicted infected population values and the actual data of the infected population for a 130 data points.

**Figure 13.**This graph shows a comparison of the predicted values of recovered population and the actual data of the recovered population for a 130 data points.

**Figure 14.**The graph shows a comparison of the predicted deceased population values and the actual data of the deceased population for a 130 data points.

**Figure 15.**This graph shows a comparison of the predicted values of susceptible population and the actual data of the susceptible population for a 530 data points.

**Figure 16.**The graph shows a comparison of the predicted infected population values and the actual data of the infected population for a 530 data points.

**Figure 17.**This graph shows a comparison of the predicted values of recovered population and the actual data of the recovered population for a 530 data points.

**Figure 18.**The graph shows a comparison of the predicted deceased population values and the actual data of the deceased population for a 530 data points.

**Figure 19.**This graph shows a comparison of the predicted values of susceptible population and the actual data of susceptible population for a SIRD model with future predictions.

**Figure 20.**The graph shows a comparison of the predicted infected population values and the actual data infected population for a SIRD model with future predictions.

**Figure 21.**This graph shows a comparison of the predicted values of recovered population and the actual data of recovered population for a SIRD model with future predictions.

**Figure 22.**The graph shows a comparison of the predicted deceased population values and the actual data deceased population for a SIRD model with future predictions.

**Table 1.**Results of the mean square error analyzing of varying number of iterations and number of layers in the simulations.

Number of Layers | |||
---|---|---|---|

Iterations | 2 | 4 | 8 |

100,000 | 3.397 × 10${}^{-6}$ | 1.996 × 10${}^{-7}$ | 5.461 × 10${}^{-8}$ |

200,000 | 2.434 × 10${}^{-6}$ | 1.871 × 10${}^{-7}$ | 3.866 × 10${}^{-8}$ |

400,000 | 2.098 × 10${}^{-7}$ | 2.340 × 10${}^{-8}$ | 3.584 × 10${}^{-9}$ |

800,000 | 1.454 × 10${}^{-7}$ | 1.621 × 10${}^{-8}$ | 3.055 × 10${}^{-9}$ |

**Table 2.**Results of the mean square error analysing of varying number of nodes in a layer and number of layers in the simulations.

Number of Layers | |||
---|---|---|---|

Nodes | 2 | 4 | 8 |

10 | 1.870 × 10${}^{-7}$ | 2.098 × 10${}^{-8}$ | 4.783 × 10${}^{-9}$ |

20 | 2.494 × 10${}^{-7}$ | 2.243 × 10${}^{-8}$ | 4.131 × 10${}^{-9}$ |

40 | 2.144 × 10${}^{-7}$ | 2.830 × 10${}^{-8}$ | 4.723 × 10${}^{-9}$ |

80 | 2.789 × 10${}^{-7}$ | 2.941 × 10${}^{-8}$ | 8.794 × 10${}^{-9}$ |

**Table 3.**Results of the mean square error analysing of varying sizes of data points and number of layers in the simulations.

Number of Layers | |||
---|---|---|---|

Data Size | 2 | 4 | 8 |

100 | 2.269 × 10${}^{-7}$ | 2.070 × 10${}^{-8}$ | 3.438 × 10${}^{-9}$ |

150 | 2.121 × 10${}^{-7}$ | 3.473 × 10${}^{-8}$ | 4.123 × 10${}^{-9}$ |

200 | 3.364 × 10${}^{-7}$ | 2.904 × 10${}^{-8}$ | 1.015 × 10${}^{-9}$ |

350 | 3.214 × 10${}^{-7}$ | 4.470 × 10${}^{-8}$ | 3.440 × 10${}^{-9}$ |

**Table 4.**Results of the mean square error analysing of varying number of iterations and number of nodes per layer in the simulations.

Number of Iterations | |||
---|---|---|---|

Nodes | 100,000 | 400,000 | 800,000 |

10 | 8.991 × 10${}^{-7}$ | 3.655 × 10${}^{-8}$ | 5.824 × 10${}^{-8}$ |

20 | 2.361 × 10${}^{-6}$ | 2.323 × 10${}^{-8}$ | 2.390 × 10${}^{-8}$ |

40 | 5.144 × 10${}^{-7}$ | 7.846 × 10${}^{-8}$ | 2.632 × 10${}^{-8}$ |

80 | 7.642 × 10${}^{-7}$ | 5.921 × 10${}^{-8}$ | 3.327 × 10${}^{-8}$ |

**Table 5.**Results of the mean square error analysis of varying number of iterations and data size per layer in the simulations.

Number of Iterations | |||
---|---|---|---|

Data Size | 100,000 | 400,000 | 800,000 |

100 | 1.292 × 10${}^{-6}$ | 1.188 × 10${}^{-7}$ | 2.752 × 10${}^{-8}$ |

150 | 2.536 × 10${}^{-6}$ | 3.273 × 10${}^{-8}$ | 2.843 × 10${}^{-8}$ |

200 | 1.110 × 10${}^{-6}$ | 1.491 × 10${}^{-8}$ | 2.064 × 10${}^{-8}$ |

350 | 1.063 × 10${}^{-6}$ | 3.399 × 10${}^{-8}$ | 1.661 × 10${}^{-8}$ |

**Table 6.**Results of the mean square error analyzing of varying sizes of data points iterations and number of nodes in layers in the simulations.

Number of Nodes | |||
---|---|---|---|

Data Size | 10 | 40 | 80 |

100 | 6.102 × 10${}^{-8}$ | 4.882 × 10${}^{-8}$ | 1.533 × 10${}^{-7}$ |

150 | 4.574 × 10${}^{-8}$ | 3.127 × 10${}^{-8}$ | 1.823 × 10${}^{-7}$ |

200 | 4.053 × 10${}^{-8}$ | 3.386 × 10${}^{-8}$ | 5.401 × 10${}^{-8}$ |

350 | 1.277 × 10${}^{-7}$ | 3.318 × 10${}^{-8}$ | 1.231 × 10${}^{-8}$ |

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**MDPI and ACS Style**

Malinzi, J.; Gwebu, S.; Motsa, S.
Determining COVID-19 Dynamics Using Physics Informed Neural Networks. *Axioms* **2022**, *11*, 121.
https://doi.org/10.3390/axioms11030121

**AMA Style**

Malinzi J, Gwebu S, Motsa S.
Determining COVID-19 Dynamics Using Physics Informed Neural Networks. *Axioms*. 2022; 11(3):121.
https://doi.org/10.3390/axioms11030121

**Chicago/Turabian Style**

Malinzi, Joseph, Simanga Gwebu, and Sandile Motsa.
2022. "Determining COVID-19 Dynamics Using Physics Informed Neural Networks" *Axioms* 11, no. 3: 121.
https://doi.org/10.3390/axioms11030121