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doi: 10.3934/era.2020108

On a model of COVID-19 dynamics

Jorge Rebaza

Missouri State University, 901 National Avenue, Springfield, MO 65897, USA

Received  June 2020 Revised  September 2020 Published  October 2020

A model of COVID-19 in an interconnected network of communities is studied. This model considers the dynamics of susceptible, asymptomatic and symptomatic individuals, deceased but not yet buried people, as well as the dynamics of the virus or pathogen at connected nodes or communities. People can move between communities carrying the virus to any node in the region of $ n $ communities (or patches). This model considers both virus direct (person to person) and indirect (contaminated environment to person) transmissions. Using either matrix and graph-theoretic methods and some combinatorial identities, appropriate Lyapunov functions are constructed to study global stability properties of both the disease-free and the endemic equilibrium of the corresponding system of $ 5n $ differential equations.

Keywords: Epidemics, COVID-19, global stability, Lyapunov functions.
Mathematics Subject Classification: Primary:37B25, 92D30;Secondary:92D25.
Citation: Jorge Rebaza. On a model of COVID-19 dynamics. Electronic Research Archive, doi: 10.3934/era.2020108
References:
[1]

A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Classics in Applied Mathematics, 9, SIAM, Philadelphia, PA, 1994. doi: 10.1137/1.9781611971262.  Google Scholar

[2]

K. Bessey, M. Mavis, J. Rebaza and J. Zhang, Global stability analysis of a general model of Zika virus, Nonauton. Dyn. Syst., 6 (2019), 18-34.  doi: 10.1515/msds-2019-0002.  Google Scholar

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M. Calmon, Considerations of coronavirus (COVID-19) impact and the management of the dead in Brazil, Forensic Science Internat: Reports, 2020. In press. doi: 10.1016/j.fsir.2020.100110.  Google Scholar

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C. Cross, A. Edwards, D. Mercadante and J. Rebaza, Dynamics of a networked connectivity model of epidemics, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 3379-3390.  doi: 10.3934/dcdsb.2016102.  Google Scholar

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R. Cui, Asymptotic profiles of the endemic equilibrium of a reaction-diffusion-advection SIS epidemic model with saturated incidence rate, Discrete Contin. Dyn. Syst. Ser. B, (2020). doi: 10.3934/dcdsb.2020217.  Google Scholar

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O. Diekmann, J. A. Heesterbeek and J. A. Metz, On the definition and the computation of the basic reproduction number ratio $R_0$ in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382.  doi: 10.1007/BF00178324.  Google Scholar

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M. Gatto, E. Bertuzzo, L. Mari, S. Miccoli, L. Carraro, R. Casagrandi and A. Rinaldo, Spread and dynamics of the COVID-19 epidemic in Italy: Effects of emergency containment measures, Proceed. Nat. Acad. of Scienc., 117 (2020), 10484-10491.  doi: 10.1073/pnas.2004978117.  Google Scholar

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M. Gatto, L. Mari, E. Bertuzzo, R. Casagrandi, L. Righetto, I. Rodriguez-Iturbe and A. Rinaldo, Generalized reproduction numbers and the prediction of patterns in waterborne disease, Proceed. Nat. Acad. of Scienc., 109 (2012), 19703-19708.  doi: 10.1073/pnas.1217567109.  Google Scholar

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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.  Google Scholar

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J. K. Hale, Ordinary Differential Equations, Second edition. Robert E. Krieger Publishing Co., Inc., Huntington, N.Y., 1980.  Google Scholar

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A. Iggidr, G. Sallet and M. O. Souza, On the dynamics of a class of multi-group models for vector-borne diseases, J. Math. Anal. Appl., 441 (2016), 723-743.  doi: 10.1016/j.jmaa.2016.04.003.  Google Scholar

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B. Ivorra, M. R. Ferrández, M. Vela-Pérez and A. M. Ramos, 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., 88 (2020), 105303, 21 pp. doi: 10.1016/j.cnsns.2020.105303.  Google Scholar

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M. Y. Li, J. R. Graef, L. Wang and J. Karsai, Global dynamics of a SEIR model with varying total population size, Math. Biosci., 160 (1999), 191-213.  doi: 10.1016/S0025-5564(99)00030-9.  Google Scholar

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M. Y. Li and Z. Shuai, Global stability of an epidemic model in a patchy environment, Can. Appl. Math. Q., 17 (2009), 175-187.   Google Scholar

[16]

M. Y. Li and Z. Shuai, Global stability problems for coupled systems of differential equation on networks, J. Differential Equations, 248 (2010), 1-20.  doi: 10.1016/j.jde.2009.09.003.  Google Scholar

[17]

S. Li and S. Guo, Permanence and extinction of a stochastic SIS epidemic model with three independent Brownian motions, Discrete Contin. Dyn. Syst. Ser. B, (2020). doi: 10.3934/dcdsb.2020201.  Google Scholar

[18]

M. Mandal, S. Jana, S. K. Nandi, A. Khatua, S. Adak and T. K. Kar, A model based study on the dynamics of COVID-19: Prediction and control, Chaos Solitons Fractals, 136 (2020), 109889, 12 pp. doi: 10.1016/j.chaos.2020.109889.  Google Scholar

[19]

T. Sasaki and T. Suzuki, Asymptotic behaviour of the solutions to a virus dynamics model with diffusion, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 525-541.  doi: 10.3934/dcdsb.2017206.  Google Scholar

[20]

Z. Shuai and P. van den Driessche, Global stability of infectious disease models using Lyapunov functions, SIAM J. Appl. Math., 73 (2013), 1513-1532.  doi: 10.1137/120876642.  Google Scholar

[21]

Z. Shuai and P. van den Driessche, Modelling and control of cholera on networks with a common water source, J. Biol. Dyn., 9 (2015), 90-103.  doi: 10.1080/17513758.2014.944226.  Google Scholar

[22] H. L. Smith and P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition, Cambridge University Press, Cambridge, 1995.  doi: 10.1017/CBO9780511530043.  Google Scholar
[23]

J. P. Tian, S. Liao and J. Wang, Analyzing the infection dynamics and control strategies of cholera, Discrete Contin. Dyn. Syst., (2013), 747–757. doi: 10.3934/proc.2013.2013.747.  Google Scholar

[24]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar

[25]

P. van den Driessche and J. Watmough, Further notes on the basic reproduction number, Lecture Notes in Mathematics, 1945 (2008), 159-178.  doi: 10.1007/978-3-540-78911-6_6.  Google Scholar

[26]

C. Yang and J. Wang, A mathematical model for the novel coronavirus epidemic in Wuhan, China, Math. Biosci. Eng., 17 (2020), 2708-2724.  doi: 10.3934/mbe.2020148.  Google Scholar

[27]

R. Zhang, Y. Li, A. L. Zhang, Y. Wang and M. J. Molina, Identifying airborne transmission as the dominant route for the spread of COVID-19, Proc. of National Acad. of Sciences, 117 (2020), 14857-14863.  doi: 10.1073/pnas.2009637117.  Google Scholar

show all references

References:
[1]

A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Classics in Applied Mathematics, 9, SIAM, Philadelphia, PA, 1994. doi: 10.1137/1.9781611971262.  Google Scholar

[2]

K. Bessey, M. Mavis, J. Rebaza and J. Zhang, Global stability analysis of a general model of Zika virus, Nonauton. Dyn. Syst., 6 (2019), 18-34.  doi: 10.1515/msds-2019-0002.  Google Scholar

[3]

M. Calmon, Considerations of coronavirus (COVID-19) impact and the management of the dead in Brazil, Forensic Science Internat: Reports, 2020. In press. doi: 10.1016/j.fsir.2020.100110.  Google Scholar

[4]

Center for Disease Control and Prevention (CDC), Coronavirus Disease (COVID-19), 2020. Available from: https://www.cdc.gov/coronavirus/2019-ncov/index.html. Google Scholar

[5]

C. Cross, A. Edwards, D. Mercadante and J. Rebaza, Dynamics of a networked connectivity model of epidemics, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 3379-3390.  doi: 10.3934/dcdsb.2016102.  Google Scholar

[6]

R. Cui, Asymptotic profiles of the endemic equilibrium of a reaction-diffusion-advection SIS epidemic model with saturated incidence rate, Discrete Contin. Dyn. Syst. Ser. B, (2020). doi: 10.3934/dcdsb.2020217.  Google Scholar

[7]

O. Diekmann, J. A. Heesterbeek and J. A. Metz, On the definition and the computation of the basic reproduction number ratio $R_0$ in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382.  doi: 10.1007/BF00178324.  Google Scholar

[8]

M. Gatto, E. Bertuzzo, L. Mari, S. Miccoli, L. Carraro, R. Casagrandi and A. Rinaldo, Spread and dynamics of the COVID-19 epidemic in Italy: Effects of emergency containment measures, Proceed. Nat. Acad. of Scienc., 117 (2020), 10484-10491.  doi: 10.1073/pnas.2004978117.  Google Scholar

[9]

M. Gatto, L. Mari, E. Bertuzzo, R. Casagrandi, L. Righetto, I. Rodriguez-Iturbe and A. Rinaldo, Generalized reproduction numbers and the prediction of patterns in waterborne disease, Proceed. Nat. Acad. of Scienc., 109 (2012), 19703-19708.  doi: 10.1073/pnas.1217567109.  Google Scholar

[10]

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.  Google Scholar

[11]

J. K. Hale, Ordinary Differential Equations, Second edition. Robert E. Krieger Publishing Co., Inc., Huntington, N.Y., 1980.  Google Scholar

[12]

A. Iggidr, G. Sallet and M. O. Souza, On the dynamics of a class of multi-group models for vector-borne diseases, J. Math. Anal. Appl., 441 (2016), 723-743.  doi: 10.1016/j.jmaa.2016.04.003.  Google Scholar

[13]

B. Ivorra, M. R. Ferrández, M. Vela-Pérez and A. M. Ramos, 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., 88 (2020), 105303, 21 pp. doi: 10.1016/j.cnsns.2020.105303.  Google Scholar

[14]

M. Y. Li, J. R. Graef, L. Wang and J. Karsai, Global dynamics of a SEIR model with varying total population size, Math. Biosci., 160 (1999), 191-213.  doi: 10.1016/S0025-5564(99)00030-9.  Google Scholar

[15]

M. Y. Li and Z. Shuai, Global stability of an epidemic model in a patchy environment, Can. Appl. Math. Q., 17 (2009), 175-187.   Google Scholar

[16]

M. Y. Li and Z. Shuai, Global stability problems for coupled systems of differential equation on networks, J. Differential Equations, 248 (2010), 1-20.  doi: 10.1016/j.jde.2009.09.003.  Google Scholar

[17]

S. Li and S. Guo, Permanence and extinction of a stochastic SIS epidemic model with three independent Brownian motions, Discrete Contin. Dyn. Syst. Ser. B, (2020). doi: 10.3934/dcdsb.2020201.  Google Scholar

[18]

M. Mandal, S. Jana, S. K. Nandi, A. Khatua, S. Adak and T. K. Kar, A model based study on the dynamics of COVID-19: Prediction and control, Chaos Solitons Fractals, 136 (2020), 109889, 12 pp. doi: 10.1016/j.chaos.2020.109889.  Google Scholar

[19]

T. Sasaki and T. Suzuki, Asymptotic behaviour of the solutions to a virus dynamics model with diffusion, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 525-541.  doi: 10.3934/dcdsb.2017206.  Google Scholar

[20]

Z. Shuai and P. van den Driessche, Global stability of infectious disease models using Lyapunov functions, SIAM J. Appl. Math., 73 (2013), 1513-1532.  doi: 10.1137/120876642.  Google Scholar

[21]

Z. Shuai and P. van den Driessche, Modelling and control of cholera on networks with a common water source, J. Biol. Dyn., 9 (2015), 90-103.  doi: 10.1080/17513758.2014.944226.  Google Scholar

[22] H. L. Smith and P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition, Cambridge University Press, Cambridge, 1995.  doi: 10.1017/CBO9780511530043.  Google Scholar
[23]

J. P. Tian, S. Liao and J. Wang, Analyzing the infection dynamics and control strategies of cholera, Discrete Contin. Dyn. Syst., (2013), 747–757. doi: 10.3934/proc.2013.2013.747.  Google Scholar

[24]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar

[25]

P. van den Driessche and J. Watmough, Further notes on the basic reproduction number, Lecture Notes in Mathematics, 1945 (2008), 159-178.  doi: 10.1007/978-3-540-78911-6_6.  Google Scholar

[26]

C. Yang and J. Wang, A mathematical model for the novel coronavirus epidemic in Wuhan, China, Math. Biosci. Eng., 17 (2020), 2708-2724.  doi: 10.3934/mbe.2020148.  Google Scholar

[27]

R. Zhang, Y. Li, A. L. Zhang, Y. Wang and M. J. Molina, Identifying airborne transmission as the dominant route for the spread of COVID-19, Proc. of National Acad. of Sciences, 117 (2020), 14857-14863.  doi: 10.1073/pnas.2009637117.  Google Scholar

Figure 1.  The digraph $ \Gamma(A) $
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Figure 2.  Simulation for $ n = 2 $. Populations in one community of 200K people.
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