# American Institute of Mathematical Sciences

June  2022, 17(3): 427-442. doi: 10.3934/nhm.2022015

## A measure model for the spread of viral infections with mutations

 1 School of Mathematical and Statistical Science, Arizona State University, Tempe, AZ, 85281, USA 2 Department of Mathematical Sciences and Center for Computational and Integrative Biology, Rutgers University, Camden, NJ, 08102, USA

* Corresponding author: Benedetto Piccoli

Received  May 2021 Revised  September 2021 Published  June 2022 Early access  March 2022

Genetic variations in the COVID-19 virus are one of the main causes of the COVID-19 pandemic outbreak in 2020 and 2021. In this article, we aim to introduce a new type of model, a system coupled with ordinary differential equations (ODEs) and measure differential equation (MDE), stemming from the classical SIR model for the variants distribution. Specifically, we model the evolution of susceptible $S$ and removed $R$ populations by ODEs and the infected $I$ population by a MDE comprised of a probability vector field (PVF) and a source term. In addition, the ODEs for $S$ and $R$ contains terms that are related to the measure $I$. We establish analytically the well-posedness of the coupled ODE-MDE system by using generalized Wasserstein distance. We give two examples to show that the proposed ODE-MDE model coincides with the classical SIR model in case of constant or time-dependent parameters as special cases.

Citation: Xiaoqian Gong, Benedetto Piccoli. A measure model for the spread of viral infections with mutations. Networks and Heterogeneous Media, 2022, 17 (3) : 427-442. doi: 10.3934/nhm.2022015
##### References:
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Del Valle, Epidemic forecasting is messier than weather forecasting: The role of human behavior and internet data streams in epidemic forecast, The Journal of Infectious Diseases, 214 (2016), S404–S408. doi: 10.1093/infdis/jiw375. [15] B. Piccoli, Measure differential equations, Archive for Rational Mechanics and Analysis, 233 (2019), 1289-1317.  doi: 10.1007/s00205-019-01379-4. [16] B. Piccoli and F. Rossi, Generalized Wasserstein distance and its application to transport equations with source, Archive for Rational Mechanics and Analysis, 211 (2014), 335-358.  doi: 10.1007/s00205-013-0669-x. [17] B. Piccoli and F. Rossi, On properties of the generalized Wasserstein distance, Archive for Rational Mechanics and Analysis, 222 (2016), 1339-1365.  doi: 10.1007/s00205-016-1026-7. [18] B. Piccoli and F. Rossi, Measure dynamics with probability vector fields and sources, Discrete Contin. Dyn. Syst., 39 (2019), 6207-6230.  doi: 10.3934/dcds.2019270. [19] N. W. Ruktanonchai, J. R. Floyd, S. Lai, C. W. Ruktanonchai, A. Sadilek, P. Rente-Lourenco, X. Ben, A. Carioli, J. Gwinn and J. E. Steele, Assessing the impact of coordinated COVID-19 exit strategies across europe, Science, 369 (2020), 1465-1470.  doi: 10.1126/science.abc5096. [20] A. Vespignani, H. Tian, C. Dye, J. O. Lloyd-Smith, R. M. Eggo, M. Shrestha, S. V. Scarpino, B. Gutierrez, M. U. G. Kraemer, J. Wu and et al., Modelling covid-19, Nature Reviews Physics, 2 (2020), 279-281. [21] C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics, 58. American Mathematical Society, Providence, RI, 2003. doi: 10.1090/gsm/058. [22] J. Zhang, M. Litvinova, Y. Liang, Y. Wang, W. Wang, S. Zhao, Q. Wu, S. Merler, C. Viboud and A. Vespignani, Changes in contact patterns shape the dynamics of the COVID-19 outbreak in China, Science, 368 (2020), 1481-1486.  doi: 10.1126/science.abb8001. [23] J. Zhang, M. Litvinova, W. Wang, Y. Wang, X. Deng, X. Chen, M. Li, W. Zheng, L. Yi and X. Chen, Evolving epidemiology and transmission dynamics of coronavirus disease 2019 outside Hubei province, China: A descriptive and modelling study, The Lancet Infectious Diseases, 20 (2020), 793-802.  doi: 10.1016/S1473-3099(20)30230-9.

show all references

##### References:
 [1] S. Anita and V. Capasso, Reaction-diffusion systems in epidemiology, 2017. [2] N. Bellomo, R. Bingham, M. A. J. Chaplain, G. Dosi, G. Forni, D. A. Knopoff, J. Lowengrub, R. Twarock and M. E. Virgillito, A multi-scale model of virus pandemic: Heterogeneous interactive entities in a globally connected world, Math. Models Methods Appl. Sci., 30 (2020), 1591–1651, arXiv: 2006.03915. doi: 10.1142/S0218202520500323. [3] G. A. Bonaschi, J. A. Carrillo, M. Di Francesco and M. A. Peletier, Equivalence of gradient flows and entropy solutions for singular nonlocal interaction equations in 1D, ESAIM Control Optim. Calc. Var., 21 (2015), 414-441.  doi: 10.1051/cocv/2014032. [4] T. Britton, F. Ball and P. Trapman, A mathematical model reveals the influence of population heterogeneity on herd immunity to SARS-CoV-2, Science, 369 (2020), 846-849.  doi: 10.1126/science.abc6810. [5] Y.-C. Chen, P.-E. Lu and C.-S. Chang, A time-dependent SIR model for COVID-19, IEEE Trans. Network Sci. Eng., 7 (2020), 3279–3294, arXiv: 2003.00122. doi: 10.1109/TNSE.2020.3024723. [6] R. M. Colombo and M. Garavello, Well posedness and control in a nonlocal SIR model, Appl. Math. Optim., 84 (2021), 737-771.  doi: 10.1007/s00245-020-09660-9. [7] R. M. Colombo, M. Garavello, F. Marcellini and E. Rossi, An age and space structured SIR model describing the COVID-19 pandemic, Journal of Mathematics in Industry, 10 (2020), Paper No. 22, 20 pp. doi: 10.1186/s13362-020-00090-4. [8] G. Giordano, F. Blanchini, R. Bruno, P. Colaneri, A. Di Filippo, A. Di Matteo and M. Colaneri, Modelling the COVID-19 epidemic and implementation of population-wide interventions in Italy, Nature Medicine, 26 (2020), 855-860.  doi: 10.1038/s41591-020-0883-7. [9] V. Kala, K. Guo, E. Swantek, A. Tong, M.. Chyba, Y. Mileyko, C. Gray, T. Lee and A. E. Koniges, Pandemics in Hawai'i: 1918 Influenza and COVID-19, The Ninth International Conference on Global Health Challenges GLOBAL HEALTH 2020, IARIA, 2020. [10] A. Keimer and L. Pflug, Modeling infectious diseases using integro-differential equations: Optimal control strategies for policy decisions and applications in covid-19, 2020. [11] W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics. II.-the problem of endemicity, Proceedings of the Royal Society of London. Series A, containing papers of a mathematical and physical character, 138 (1932), 55-83.  doi: 10.1098/rspa.1932.0171. [12] K. Kupferschmidt, Evolving threat, Science, 373 (2021), 844-849.  doi: 10.1126/science.373.6557.844. [13] C. J. E. Metcalf, D. H. Morris and S. W. Park, Mathematical models to guide pandemic response, Science, 369 (2020), 368-369.  doi: 10.1126/science.abd1668. [14] K. R. Moran, G. Fairchild, N. Generous, K. Hickmann, D. Osthus, R. Priedhorsky, J. Hyman and S. Y. Del Valle, Epidemic forecasting is messier than weather forecasting: The role of human behavior and internet data streams in epidemic forecast, The Journal of Infectious Diseases, 214 (2016), S404–S408. doi: 10.1093/infdis/jiw375. [15] B. Piccoli, Measure differential equations, Archive for Rational Mechanics and Analysis, 233 (2019), 1289-1317.  doi: 10.1007/s00205-019-01379-4. [16] B. Piccoli and F. Rossi, Generalized Wasserstein distance and its application to transport equations with source, Archive for Rational Mechanics and Analysis, 211 (2014), 335-358.  doi: 10.1007/s00205-013-0669-x. [17] B. Piccoli and F. Rossi, On properties of the generalized Wasserstein distance, Archive for Rational Mechanics and Analysis, 222 (2016), 1339-1365.  doi: 10.1007/s00205-016-1026-7. [18] B. Piccoli and F. Rossi, Measure dynamics with probability vector fields and sources, Discrete Contin. Dyn. Syst., 39 (2019), 6207-6230.  doi: 10.3934/dcds.2019270. [19] N. W. Ruktanonchai, J. R. Floyd, S. Lai, C. W. Ruktanonchai, A. Sadilek, P. Rente-Lourenco, X. Ben, A. Carioli, J. Gwinn and J. E. Steele, Assessing the impact of coordinated COVID-19 exit strategies across europe, Science, 369 (2020), 1465-1470.  doi: 10.1126/science.abc5096. [20] A. Vespignani, H. Tian, C. Dye, J. O. Lloyd-Smith, R. M. Eggo, M. Shrestha, S. V. Scarpino, B. Gutierrez, M. U. G. Kraemer, J. Wu and et al., Modelling covid-19, Nature Reviews Physics, 2 (2020), 279-281. [21] C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics, 58. American Mathematical Society, Providence, RI, 2003. doi: 10.1090/gsm/058. [22] J. Zhang, M. Litvinova, Y. Liang, Y. Wang, W. Wang, S. Zhao, Q. Wu, S. Merler, C. Viboud and A. Vespignani, Changes in contact patterns shape the dynamics of the COVID-19 outbreak in China, Science, 368 (2020), 1481-1486.  doi: 10.1126/science.abb8001. [23] J. Zhang, M. Litvinova, W. Wang, Y. Wang, X. Deng, X. Chen, M. Li, W. Zheng, L. Yi and X. Chen, Evolving epidemiology and transmission dynamics of coronavirus disease 2019 outside Hubei province, China: A descriptive and modelling study, The Lancet Infectious Diseases, 20 (2020), 793-802.  doi: 10.1016/S1473-3099(20)30230-9.
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