American Institute of Mathematical Sciences

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June  2022, 17(3): 467-494. doi: 10.3934/nhm.2022017

An SIR–like kinetic model tracking individuals' viral load

Rossella Della Marca 1, , Nadia Loy 2, and  Andrea Tosin 3,,

1. 

Mathematics Area, SISSA – International School for Advanced Studies, Via Bonomea 265, I-34136 Trieste, Italy
2. 

Department of Mathematical, Physical and Computer Sciences, Università di Parma, Parco Area delle Scienze 53/A, 43124 Parma, Italy
3. 

Department of Mathematical Sciences "G. L. Lagrange", Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy

* Corresponding author: Andrea Tosin

Received  June 2021 Revised  January 2022 Published  June 2022 Early access  March 2022

In classical epidemic models, a neglected aspect is the heterogeneity of disease transmission and progression linked to the viral load of each infected individual. Here, we investigate the interplay between the evolution of individuals' viral load and the epidemic dynamics from a theoretical point of view. We propose a stochastic particle model describing the infection transmission and the individual physiological course of the disease. Agents have a double microscopic state: a discrete label, that denotes the epidemiological compartment to which they belong and switches in consequence of a Markovian process, and a microscopic trait, measuring their viral load, that changes in consequence of binary interactions or interactions with a background. Specifically, we consider Susceptible–Infected–Removed–like dynamics where infectious individuals may be isolated and the isolation rate may depend on the viral load–sensitivity and frequency of tests. We derive kinetic evolution equations for the distribution functions of the viral load of the individuals in each compartment, whence, via upscaling procedures, we obtain macroscopic equations for the densities and viral load momentum. We perform then a qualitative analysis of the ensuing macroscopic model. Finally, we present numerical tests in the case of both constant and viral load–dependent isolation control.

Keywords: Boltzmann–type equations, Markov–type jump processes, epidemic, SIR model, basic reproduction number, viral load, qualitative analysis.
Mathematics Subject Classification: Primary: 35Q20, 37N25; Secondary: 35Q70.
Citation: 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
References:
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F. Brauer, P. van den Driessche and J. Wu, Mathematical Epidemiology, Springer, Berlin, 2008. doi: 10.1007/978-3-540-78911-6.

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B. Buonomo and R. Della Marca, Effects of information–induced behavioural changes during the COVID–19 lockdowns: The case of Italy, Royal Society Open Science, 7 (2020), 201635.  doi: 10.1098/rsos.201635.

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C. Castillo-Chavez, Z. Feng and W. Huang, On the computation of $\mathcal{R}_0$ and its role on global stability, in Mathematical Approaches for Emerging and Reemerging Infectious Diseases: An Introduction, Springer, New York, 125 (2002), 229–250.

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G. Dimarco, L. Pareschi, G. Toscani and M. Zanella, Wealth distribution under the spread of infectious diseases, Physical Review E, 102 (2020), 022303, 14pp. doi: 10.1103/physreve.102.022303.

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G. Dimarco, B. Perthame, G. Toscani and M. Zanella, Kinetic models for epidemic dynamics with social heterogeneity, Journal of Mathematical Biology, 83 (2021), 1-32.  doi: 10.1007/s00285-021-01630-1.

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[14]

J. Fajnzylber, J. Regan, K. Coxen, H. Corry, C. Wong, A. Rosenthal, D. Worrall, F. Giguel, A. Piechocka-Trocha, C. Atyeo, S. Fischinger, A. Chan, K. T. Flaherty, K. Hall, M. Dougan, E. T. Ryan, E. Gillespie, R. Chishti, Y. Li, N. Jilg, D. Hanidziar, R. M. Baron, L. Baden, A. M. Tsibris, K. A. Armstrong, D. R. Kuritzkes, G. Alter, B. D. Walker, X. Yu and J. Z. Li, SARS–CoV–2 viral load is associated with increased disease severity and mortality, Nature Communications, 11 (2020), 5493.  doi: 10.1038/s41467-020-19057-5.

[15]

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J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, Berlin, 1983. doi: 10.1007/978-1-4612-1140-2.

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X. He, E. H. Y. Lau, P. Wu, X. Deng, W. Jian, X. Hao, Y. C. Lau, J. Y. Wong, Y. Guan, X. Tan, X. Mo, Y. Chen, B. Liao, W. Chen, F. Hu, Q. Zhang, M. Zhong, Y. Wu, L. Zhao, F. Zhang, B. J. Cowling, F. Li and G. M. Leung, Temporal dynamics in viral shedding and transmissibility of COVID–19, Nature Medicine, 26 (2020), 672-675.  doi: 10.1038/s41591-020-0869-5.

[18]

W. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proceedings of the Royal Society of London A, 115 (1927), 700-721. 

[19] J. La Salle, Stability by Liapunov's Direct Method with Applications, Academic Press, New York–London, 1961. 
[20]

D. B. Larremore, B. Wilder, E. Lester, S. Shehata, J. M. Burke, J. A. Hay, M. Tambe, M. J. Mina and R. Parker, Test sensitivity is secondary to frequency and turnaround time for COVID–19 screening, Science Advances, 7 (2021), eabd5393. doi: 10.1126/sciadv.abd5393.

[21]

N. Lee, P. K. S. Chan, D. S. C. Hui, T. H. Rainer, E. Wong, K.-W. Choi, G. C. Y. Lui, B. C. K. Wong, R. Y. K. Wong, W.-Y. Lam, I. M. T. Chu, R. W. M. Lai, C. S. Cockram and J. J. Y. Sung, Viral loads and duration of viral shedding in adult patients hospitalized with influenza, The Journal of Infectious Diseases, 200 (2009), 492-500.  doi: 10.1086/600383.

[22]

N. Loy and L. Preziosi, Stability of a non–local kinetic model for cell migration with density dependent orientation bias, Kinetic and Related Models, 13 (2020), 1007-1027.  doi: 10.3934/krm.2020035.

[23]

N. Loy and A. Tosin, Markov jump processes and collision–like models in the kinetic description of multi–agent systems, Communications in Mathematical Sciences, 18 (2020), 1539-1568.  doi: 10.4310/CMS.2020.v18.n6.a3.

[24]

N. Loy and A. Tosin, Boltzmann–type equations for multi–agent systems with label switching, Kinetic and Related Models, 14 (2021), 867-894.  doi: 10.3934/krm.2021027.

[25]

N. Loy and A. Tosin, A viral load–based model for epidemic spread on spatial networks, Mathematical Biosciences and Engineering, 18 (2021), 5635-5663.  doi: 10.3934/mbe.2021285.

[26]

MATLAB, Matlab release 2020a. The MathWorks, Inc., Natick, MA, 2020.

[27] L. Pareschi and G. Toscani, Interacting Multiagent Systems: Kinetic Equations and Monte Carlo Methods, Oxford University Press, Oxford, 2013. 
[28]

M. Simmonds, D. Brown and L. Jin, Measles viral load may reflect SSPE disease progression, Virology Journal, 3 (2006), 49.  doi: 10.1186/1743-422X-3-49.

[29]

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

[30]

Z. Wang, C. T. Bauch, S. Bhattacharyya, A. d'Onofrio, P. Manfredi, M. Perc, N. Perra, M. Salathé and D. Zhao, Statistical physics of vaccination, Physics Reports, 664 (2016), 1-113.  doi: 10.1016/j.physrep.2016.10.006.

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[33]

WHO, World Health Organization, Coronavirus disease (COVID–19) pandemic, https://www.who.int/emergencies/diseases/novel-coronavirus-2019, 2021, (Accessed on April 2021).

show all references

References:
[1]

F. Brauer, P. van den Driessche and J. Wu, Mathematical Epidemiology, Springer, Berlin, 2008. doi: 10.1007/978-3-540-78911-6.

[2]

B. Buonomo and R. Della Marca, Effects of information–induced behavioural changes during the COVID–19 lockdowns: The case of Italy, Royal Society Open Science, 7 (2020), 201635.  doi: 10.1098/rsos.201635.

[3]

C. Castillo-Chavez, Z. Feng and W. Huang, On the computation of $\mathcal{R}_0$ and its role on global stability, in Mathematical Approaches for Emerging and Reemerging Infectious Diseases: An Introduction, Springer, New York, 125 (2002), 229–250.

[4]

CDC, Centers for Disease Control and Prevention, 2014–2016 Ebola outbreak in West Africa, https://www.cdc.gov/vhf/ebola/history/2014-2016-outbreak/index.html, 2016, (Accessed on April 2021).

[5]

CDC, Centers for Disease Control and Prevention, Rubella–Laboratory Testing, https://www.cdc.gov/rubella/lab/rna-detection.html, 2020, (Accessed on June 2021).

[6]

M. Cevik, K. Kuppalli, J. Kindrachuk and M. Peiris, Virology, transmission, and pathogenesis of SARS–CoV–2, British Medical Journal, 371 (2020), m3862. doi: 10.1136/bmj.m3862.

[7]

T. Day, On the evolution of virulence and the relationship between various measures of mortality, Proceedings of the Royal Society of London B, 269 (2002), 1317-1323.  doi: 10.1098/rspb.2002.2021.

[8]

G. Dimarco, L. Pareschi, G. Toscani and M. Zanella, Wealth distribution under the spread of infectious diseases, Physical Review E, 102 (2020), 022303, 14pp. doi: 10.1103/physreve.102.022303.

[9]

G. Dimarco, B. Perthame, G. Toscani and M. Zanella, Kinetic models for epidemic dynamics with social heterogeneity, Journal of Mathematical Biology, 83 (2021), 1-32.  doi: 10.1007/s00285-021-01630-1.

[10]

J. Dushoff, W. Huang and C. Castillo-Chavez, Backwards bifurcations and catastrophe in simple models of fatal diseases, Journal of Mathematical Biology, 36 (1998), 227-248.  doi: 10.1007/s002850050099.

[11]

ECDC, European Centre for Disease Prevention and Control, Latest evidence on COVID–19 – Infection, https://www.ecdc.europa.eu/en/covid-19/latest-evidence/infection, 2020, (Accessed on June 2021).

[12]

European Commission - eurostat, Deaths and crude death rate, https://ec.europa.eu/eurostat/databrowser/view/tps00029/default/table?lang=en, 2021, (Accessed on April 2021).

[13]

European Commission - eurostat, Live births and crude birth rate, https://ec.europa.eu/eurostat/databrowser/view/TPS00204/bookmark/table?lang=en&bookmarkId=5b6e67ac-186d-4081-aa98-1453b77ec260, 2021, (Accessed on April 2021).

[14]

J. Fajnzylber, J. Regan, K. Coxen, H. Corry, C. Wong, A. Rosenthal, D. Worrall, F. Giguel, A. Piechocka-Trocha, C. Atyeo, S. Fischinger, A. Chan, K. T. Flaherty, K. Hall, M. Dougan, E. T. Ryan, E. Gillespie, R. Chishti, Y. Li, N. Jilg, D. Hanidziar, R. M. Baron, L. Baden, A. M. Tsibris, K. A. Armstrong, D. R. Kuritzkes, G. Alter, B. D. Walker, X. Yu and J. Z. Li, SARS–CoV–2 viral load is associated with increased disease severity and mortality, Nature Communications, 11 (2020), 5493.  doi: 10.1038/s41467-020-19057-5.

[15]

A. Goyal, D. B. Reeves, E. F. Cardozo-Ojeda, J. T. Schiffer and B. T. Mayer, Viral load and contact heterogeneity predict SARS–CoV–2 transmission and super–spreading events, eLife, 10 (2021), e63537. doi: 10.7554/eLife.63537.sa2.

[16]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, Berlin, 1983. doi: 10.1007/978-1-4612-1140-2.

[17]

X. He, E. H. Y. Lau, P. Wu, X. Deng, W. Jian, X. Hao, Y. C. Lau, J. Y. Wong, Y. Guan, X. Tan, X. Mo, Y. Chen, B. Liao, W. Chen, F. Hu, Q. Zhang, M. Zhong, Y. Wu, L. Zhao, F. Zhang, B. J. Cowling, F. Li and G. M. Leung, Temporal dynamics in viral shedding and transmissibility of COVID–19, Nature Medicine, 26 (2020), 672-675.  doi: 10.1038/s41591-020-0869-5.

[18]

W. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proceedings of the Royal Society of London A, 115 (1927), 700-721. 

[19] J. La Salle, Stability by Liapunov's Direct Method with Applications, Academic Press, New York–London, 1961. 
[20]

D. B. Larremore, B. Wilder, E. Lester, S. Shehata, J. M. Burke, J. A. Hay, M. Tambe, M. J. Mina and R. Parker, Test sensitivity is secondary to frequency and turnaround time for COVID–19 screening, Science Advances, 7 (2021), eabd5393. doi: 10.1126/sciadv.abd5393.

[21]

N. Lee, P. K. S. Chan, D. S. C. Hui, T. H. Rainer, E. Wong, K.-W. Choi, G. C. Y. Lui, B. C. K. Wong, R. Y. K. Wong, W.-Y. Lam, I. M. T. Chu, R. W. M. Lai, C. S. Cockram and J. J. Y. Sung, Viral loads and duration of viral shedding in adult patients hospitalized with influenza, The Journal of Infectious Diseases, 200 (2009), 492-500.  doi: 10.1086/600383.

[22]

N. Loy and L. Preziosi, Stability of a non–local kinetic model for cell migration with density dependent orientation bias, Kinetic and Related Models, 13 (2020), 1007-1027.  doi: 10.3934/krm.2020035.

[23]

N. Loy and A. Tosin, Markov jump processes and collision–like models in the kinetic description of multi–agent systems, Communications in Mathematical Sciences, 18 (2020), 1539-1568.  doi: 10.4310/CMS.2020.v18.n6.a3.

[24]

N. Loy and A. Tosin, Boltzmann–type equations for multi–agent systems with label switching, Kinetic and Related Models, 14 (2021), 867-894.  doi: 10.3934/krm.2021027.

[25]

N. Loy and A. Tosin, A viral load–based model for epidemic spread on spatial networks, Mathematical Biosciences and Engineering, 18 (2021), 5635-5663.  doi: 10.3934/mbe.2021285.

[26]

MATLAB, Matlab release 2020a. The MathWorks, Inc., Natick, MA, 2020.

[27] L. Pareschi and G. Toscani, Interacting Multiagent Systems: Kinetic Equations and Monte Carlo Methods, Oxford University Press, Oxford, 2013. 
[28]

M. Simmonds, D. Brown and L. Jin, Measles viral load may reflect SSPE disease progression, Virology Journal, 3 (2006), 49.  doi: 10.1186/1743-422X-3-49.

[29]

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

[30]

Z. Wang, C. T. Bauch, S. Bhattacharyya, A. d'Onofrio, P. Manfredi, M. Perc, N. Perra, M. Salathé and D. Zhao, Statistical physics of vaccination, Physics Reports, 664 (2016), 1-113.  doi: 10.1016/j.physrep.2016.10.006.

[31]

WHO, World Health Organization, Severe acute respiratory syndrome (SARS), https://www.who.int/csr/don/archive/disease/severe_acute_respiratory_syndrome/en/, 2004, (Accessed on April 2021).

[32]

WHO, World Health Organization, Diagnostic testing for SARS–CoV–2. Interim guidance., file:///C:/Users/rosde/AppData/Local/Temp/WHO-2019-nCoV-laboratory-2020.6-eng-1.pdf, 2020, (Accessed on May 2021).

[33]

WHO, World Health Organization, Coronavirus disease (COVID–19) pandemic, https://www.who.int/emergencies/diseases/novel-coronavirus-2019, 2021, (Accessed on April 2021).

Figure 2.  Epidemic dynamics in absence of isolation control ($ \alpha_H\equiv 0 $). Compartment sizes (grey scale colour) and mean viral loads (blue scale colour) as predicted by the model (16) (solid lines) and by the particle model (5) (markers). Panel (a): susceptible, $ S $. Panel (b): recovered, $ R $. Panel (c): infectious with increasing viral load, $ I_1 $. Panel (d): infectious with decreasing viral load, $ I_2 $. Initial conditions and other parameters values are given in (26) and Table 1, respectively
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Figure 1.  Contour plot of the basic reproduction number $ \mathcal{R}_0 $, as given in (21), versus the decay rate of viral load, $ \lambda_\gamma\nu_1 $, and the increase rate of viral load, $ \lambda_\gamma\nu_2 $. Intersection between dotted black lines indicates the value corresponding to the baseline scenario. Other parameters values are given in Table 1
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Figure 3.  Viral load–dependent vs. constant isolation control. Numerical solutions as predicted by the model (16) (solid lines) and by the particle model (5) (markers) in scenarios S1 (grey scale colour) and S2 (blue scale colour). Panel (a): compartment size of infectious individuals with increasing viral load, $ I_1 $. Panel (b): compartment size of infectious individuals with decreasing viral load, $ I_2 $. Panel (c): compartment size of isolated individuals with increasing viral load, $ H_1 $. Panel (d): compartment size of isolated individuals with decreasing viral load, $ H_2 $. Initial conditions and other parameters values are given in (26) and Table 1, respectively
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Figure 4.  Viral load–dependent vs. constant isolation control. Numerical solutions as predicted by the model (16) (solid lines) and by the particle model (5) (markers) in scenarios S1 (grey scale colour) and S2 (blue scale colour). Panel (a): mean viral load of infectious individuals with increasing viral load, $ I_1 $. Panel (b): mean viral load of infectious individuals with decreasing viral load, $ I_2 $. Panel (c): mean viral load of isolated individuals with increasing viral load, $ H_1 $. Panel (d): mean viral load of isolated individuals with decreasing viral load, $ H_2 $. Initial conditions and other parameters values are given in (26) and Table 1, respectively
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Figure 5.  Viral load evolution from the time of infection exposure to the final time $ t_f = 1 $ year for five system agents, as predicted by the stochastic particle model (5) with viral load–dependent isolation control (scenario S1). Different line markers and/or colours refer to the different epidemiological compartments the agent passes trough; the meaning is specified in the legend. Initial conditions and other parameters values are given in (26) and Table 1, respectively
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Table 1.  List of model parameters with corresponding description and baseline value
Parameter Description Baseline value
$ \lambda_b $ Frequency of new births or immigration 1 days$ ^{-1} $
$ b $ Newborns probability parameter $ 2.58\cdot 10^{-5} $
$ \lambda_\mu $ Frequency of natural deaths 0.01 days$ ^{-1} $
$ \mu $ Probability of dying of natural causes $ 2.79\cdot 10^{-3} $
$ \lambda_\beta $ Frequency of binary interactions 1 days$ ^{-1} $
$ \nu_\beta $ Transmission probability parameter 0.29
$ v_0 $ Initial viral load of infected individuals 0.01
$ \lambda_{H_1, I_1}(t) $ Frequency of isolation for $ I_1 $ members See Section 5.3
$ \lambda_{H_2, I_2}(t) $ Frequency of isolation for $ I_2 $ members See Section 5.3
$ \alpha_H(v) $ Probability for an infectious individual to be isolated See Section 5.3
$ \lambda_\gamma $ Frequency of viral load evolution 0.50 days$ ^{-1} $
$ \nu_1 $ Factor of increase of the viral load 0.40
$ \nu_2 $ Factor of decay of the viral load 0.20
$ \eta(v) $ Probability of having passed the viral load peak $ \nu_1 $
$ \gamma(v) $ Probability of recovering $ \nu_2 $
$ \lambda_d $ Frequency of disease–induced deaths 0.01 days$ ^{-1} $
$ d $ Probability of dying from the disease 0.10
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Parameter Description Baseline value
$ \lambda_b $ Frequency of new births or immigration 1 days$ ^{-1} $
$ b $ Newborns probability parameter $ 2.58\cdot 10^{-5} $
$ \lambda_\mu $ Frequency of natural deaths 0.01 days$ ^{-1} $
$ \mu $ Probability of dying of natural causes $ 2.79\cdot 10^{-3} $
$ \lambda_\beta $ Frequency of binary interactions 1 days$ ^{-1} $
$ \nu_\beta $ Transmission probability parameter 0.29
$ v_0 $ Initial viral load of infected individuals 0.01
$ \lambda_{H_1, I_1}(t) $ Frequency of isolation for $ I_1 $ members See Section 5.3
$ \lambda_{H_2, I_2}(t) $ Frequency of isolation for $ I_2 $ members See Section 5.3
$ \alpha_H(v) $ Probability for an infectious individual to be isolated See Section 5.3
$ \lambda_\gamma $ Frequency of viral load evolution 0.50 days$ ^{-1} $
$ \nu_1 $ Factor of increase of the viral load 0.40
$ \nu_2 $ Factor of decay of the viral load 0.20
$ \eta(v) $ Probability of having passed the viral load peak $ \nu_1 $
$ \gamma(v) $ Probability of recovering $ \nu_2 $
$ \lambda_d $ Frequency of disease–induced deaths 0.01 days$ ^{-1} $
$ d $ Probability of dying from the disease 0.10
Table 2.  Relevant quantities as predicted by the model (16) in the case of viral load–dependent isolation S1 (first column) and in the case of constant isolation S2 (second column). First line: infectious prevalence peak, $ \max(\rho_{I_1}+\rho_{I_2})N_{tot} $. Second line: time of infectious prevalence peak, arg$ \max(\rho_{I_1}+\rho_{I_2}) $. Third line: cumulative incidence at $ t_f = 1 $ year, CI$ (t_f) $. Fourth line: cumulative isolated individuals at $ t_f = 1 $ year, CH$ (t_f) $. Fifth line: cumulative deaths at $ t_f = 1 $ year, CD$ (t_f) $. Sixth line: endemic infectious prevalence, $ (\rho_{I_1}^E+\rho_{I_2}^E)N_{tot} $. Initial conditions and other parameters values are given in (26) and Table 1, respectively
Scenario S1 Scenario S2
$ \max(\rho_{I_1}+\rho_{I_2})N_{tot} $ $ 8.20 \cdot 10^4 $ $ 15.60 \cdot 10^4 $
arg$ \max(\rho_{I_1}+\rho_{I_2}) $ 55.08 days 90.62 days
CI$ (t_f) $ $ 7.87 \cdot 10^5 $ $ 7.70 \cdot 10^5 $
CH$ (t_f) $ $ 5.14\cdot 10^5 $ $ 5.13\cdot 10^5 $
CD$ (t_f) $ $ 6.33\cdot 10^3 $ $ 6.34\cdot 10^3 $
$ (\rho_{I_1}^E+\rho_{I_2}^E)N_{tot} $ 289.35 75.92
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Scenario S1 Scenario S2
$ \max(\rho_{I_1}+\rho_{I_2})N_{tot} $ $ 8.20 \cdot 10^4 $ $ 15.60 \cdot 10^4 $
arg$ \max(\rho_{I_1}+\rho_{I_2}) $ 55.08 days 90.62 days
CI$ (t_f) $ $ 7.87 \cdot 10^5 $ $ 7.70 \cdot 10^5 $
CH$ (t_f) $ $ 5.14\cdot 10^5 $ $ 5.13\cdot 10^5 $
CD$ (t_f) $ $ 6.33\cdot 10^3 $ $ 6.34\cdot 10^3 $
$ (\rho_{I_1}^E+\rho_{I_2}^E)N_{tot} $ 289.35 75.92
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