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:
[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. BonaschiJ. A. CarrilloM. 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. BrittonF. 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. GiordanoF. BlanchiniR. BrunoP. ColaneriA. Di FilippoA. 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. MetcalfD. 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. RuktanonchaiJ. R. FloydS. LaiC. W. RuktanonchaiA. SadilekP. Rente-LourencoX. BenA. CarioliJ. 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. VespignaniH. TianC. DyeJ. O. Lloyd-SmithR. M. EggoM. ShresthaS. V. ScarpinoB. GutierrezM. U. G. KraemerJ. 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. ZhangM. LitvinovaY. LiangY. WangW. WangS. ZhaoQ. WuS. MerlerC. 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. ZhangM. LitvinovaW. WangY. WangX. DengX. ChenM. LiW. ZhengL. 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. BonaschiJ. A. CarrilloM. 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. BrittonF. 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. GiordanoF. BlanchiniR. BrunoP. ColaneriA. Di FilippoA. 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. MetcalfD. 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. RuktanonchaiJ. R. FloydS. LaiC. W. RuktanonchaiA. SadilekP. Rente-LourencoX. BenA. CarioliJ. 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. VespignaniH. TianC. DyeJ. O. Lloyd-SmithR. M. EggoM. ShresthaS. V. ScarpinoB. GutierrezM. U. G. KraemerJ. 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. ZhangM. LitvinovaY. LiangY. WangW. WangS. ZhaoQ. WuS. MerlerC. 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. ZhangM. LitvinovaW. WangY. WangX. DengX. ChenM. LiW. ZhengL. 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.

[1]

Kaifa Wang, Yu Jin, Aijun Fan. The effect of immune responses in viral infections: A mathematical model view. Discrete and Continuous Dynamical Systems - B, 2014, 19 (10) : 3379-3396. doi: 10.3934/dcdsb.2014.19.3379

[2]

Rossella Della Marca, Nadia Loy, Andrea Tosin. An SIR–like kinetic model tracking individuals' viral load. Networks and Heterogeneous Media, 2022, 17 (3) : 467-494. doi: 10.3934/nhm.2022017

[3]

Qing Ma, Yanjun Wang. Distributionally robust chance constrained svm model with $\ell_2$-Wasserstein distance. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021212

[4]

Yang Yang, Yun-Rui Yang, Xin-Jun Jiao. Traveling waves for a nonlocal dispersal SIR model equipped delay and generalized incidence. Electronic Research Archive, 2020, 28 (1) : 1-13. doi: 10.3934/era.2020001

[5]

Xing Huang, Feng-Yu Wang. Mckean-Vlasov sdes with drifts discontinuous under wasserstein distance. Discrete and Continuous Dynamical Systems, 2021, 41 (4) : 1667-1679. doi: 10.3934/dcds.2020336

[6]

Monika Joanna Piotrowska, Urszula Foryś, Marek Bodnar, Jan Poleszczuk. A simple model of carcinogenic mutations with time delay and diffusion. Mathematical Biosciences & Engineering, 2013, 10 (3) : 861-872. doi: 10.3934/mbe.2013.10.861

[7]

Helen Moore, Weiqing Gu. A mathematical model for treatment-resistant mutations of HIV. Mathematical Biosciences & Engineering, 2005, 2 (2) : 363-380. doi: 10.3934/mbe.2005.2.363

[8]

Giuseppe Da Prato. An integral inequality for the invariant measure of some finite dimensional stochastic differential equation. Discrete and Continuous Dynamical Systems - B, 2016, 21 (9) : 3015-3027. doi: 10.3934/dcdsb.2016085

[9]

Chunhua Shan, Hongjun Gao, Huaiping Zhu. Dynamics of a delay Schistosomiasis model in snail infections. Mathematical Biosciences & Engineering, 2011, 8 (4) : 1099-1115. doi: 10.3934/mbe.2011.8.1099

[10]

Jinliang Wang, Jingmei Pang, Toshikazu Kuniya. A note on global stability for malaria infections model with latencies. Mathematical Biosciences & Engineering, 2014, 11 (4) : 995-1001. doi: 10.3934/mbe.2014.11.995

[11]

Qiyuan Wei, Liwei Zhang. An accelerated differential equation system for generalized equations. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021195

[12]

Ruiqiang He, Xiangchu Feng, Xiaolong Zhu, Hua Huang, Bingzhe Wei. RWRM: Residual Wasserstein regularization model for image restoration. Inverse Problems and Imaging, 2021, 15 (6) : 1307-1332. doi: 10.3934/ipi.2020069

[13]

E. Almaraz, A. Gómez-Corral. On SIR-models with Markov-modulated events: Length of an outbreak, total size of the epidemic and number of secondary infections. Discrete and Continuous Dynamical Systems - B, 2018, 23 (6) : 2153-2176. doi: 10.3934/dcdsb.2018229

[14]

Jonathan Zinsl. The gradient flow of a generalized Fisher information functional with respect to modified Wasserstein distances. Discrete and Continuous Dynamical Systems - S, 2017, 10 (4) : 919-933. doi: 10.3934/dcdss.2017047

[15]

Suqi Ma. Low viral persistence of an immunological model. Mathematical Biosciences & Engineering, 2012, 9 (4) : 809-817. doi: 10.3934/mbe.2012.9.809

[16]

Urszula Foryś, Beata Zduniak. Two-stage model of carcinogenic mutations with the influence of delays. Discrete and Continuous Dynamical Systems - B, 2014, 19 (8) : 2501-2519. doi: 10.3934/dcdsb.2014.19.2501

[17]

Vianney Perchet, Marc Quincampoix. A differential game on Wasserstein space. Application to weak approachability with partial monitoring. Journal of Dynamics and Games, 2019, 6 (1) : 65-85. doi: 10.3934/jdg.2019005

[18]

Chuchu Chen, Jialin Hong, Yulan Lu. Stochastic differential equation with piecewise continuous arguments: Markov property, invariant measure and numerical approximation. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022098

[19]

Helge Dietert, Josephine Evans, Thomas Holding. Contraction in the Wasserstein metric for the kinetic Fokker-Planck equation on the torus. Kinetic and Related Models, 2018, 11 (6) : 1427-1441. doi: 10.3934/krm.2018056

[20]

Narcisa Apreutesei, Nikolai Bessonov, Vitaly Volpert, Vitali Vougalter. Spatial structures and generalized travelling waves for an integro-differential equation. Discrete and Continuous Dynamical Systems - B, 2010, 13 (3) : 537-557. doi: 10.3934/dcdsb.2010.13.537

2021 Impact Factor: 1.41

Article outline

[Back to Top]