An SIR–like kinetic model tracking individuals' viral load

Rossella Della Marca; Nadia Loy; Andrea Tosin

doi:10.3934/nhm.2022017

Article Contents

2022, Volume 17, Issue 3: 467-494. Doi: 10.3934/nhm.2022017

This issue Previous Article Optimization of vaccination for COVID-19 in the midst of a pandemic Next Article

An SIR–like kinetic model tracking individuals' viral load

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
^* Corresponding author: Andrea Tosin

Received: June 2021

Revised: January 2022

Early access: March 2022

Published: June 2022

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Abstract

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.

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Mathematics Subject Classification: Primary: 35Q20, 37N25; Secondary: 35Q70.

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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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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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