Mathematical analysis of a hybrid model: Impacts of individual behaviors on the spreading of an epidemic

Guillaume Cantin; Cristiana J. Silva; Arnaud Banos

doi:10.3934/nhm.2022010

Article Contents

2022, Volume 17, Issue 3: 333-357. Doi: 10.3934/nhm.2022010

This issue Previous Article A martingale formulation for stochastic compartmental susceptible-infected-recovered (SIR) models to analyze finite size effects in COVID-19 case studies Next Article A study of computational and conceptual complexities of compartment and agent based models

Mathematical analysis of a hybrid model: Impacts of individual behaviors on the spreading of an epidemic

1.
Laboratoire des Sciences du Numérique, LS2N UMR CNRS 6004, Université de Nantes, France
2.
Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal
3.
UMR IDEES, CNRS, 25 rue Philippe Lebon BP 1123, 76063 Le Havre Cedex, UK

^* Corresponding author: Cristiana J. Silva
^* Corresponding author: Cristiana J. Silva

Received: May 2021

Revised: November 2021

Early access: March 2022

Published: June 2022

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Abstract

In this paper, we investigate the well-posedness and dynamics of a class of hybrid models, obtained by coupling a system of ordinary differential equations and an agent-based model. These hybrid models intend to integrate the microscopic dynamics of individual behaviors into the macroscopic evolution of various population dynamics models, and can be applied to a great number of complex problems arising in economics, sociology, geography and epidemiology. Here, in particular, we apply our general framework to the current COVID-19 pandemic. We establish, at a theoretical level, sufficient conditions which lead to particular solutions exhibiting irregular oscillations and interpret those particular solutions as pandemic waves. We perform numerical simulations of a set of relevant scenarios which show how the microscopic processes impact the macroscopic dynamics.

Keywords:

Mathematics Subject Classification: Primary: 34F05, 34A38; Secondary: 91C99.

Citation:

FullText(HTML)

Figure 1. Timeline of the hybrid model $ \text{(AHP)} $. At $ t = t_0 $, the initial condition $ ( \mathfrak{IC}) $ gives $ (X_0, \lambda_0) \in E \times J $. On each interval $ [t_s, t_{s+1}] $, the macroscopic part $ ( \mathfrak{M}_s) $ is determined by an ordinary differential equation. At each time step $ t = t_s $, the microscopic part $ ( \mathfrak{m}_s) $ follows from a discrete mapping which is derived from an agent-based model

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Figure 2. Social network generated over a finite set of agents, by running a Newman–Watts-Strogatz graph generation algorithm: each vertex represents an agent, and each edge models a social connection between two agents. Different colors correspond to the different epidemic sub-classes of the population. In such a social network, each agent can observe the types and the behaviors of its neighbors and can make decisions with respect to its observations

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Figure 3. Basic reproduction number $ R_0(p, u) $ of the $ SAIRP $ model (3), with $ 0.25 \leq p \leq 0.675 $ and $ 0 \leq u \leq 0.4 $

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Figure 4. Local stability condition of the endemic equilibrium $ \Sigma_+ $ is satisfied for $ R_0(p, u) > 1 $. Considering the parameter values (Pfixed) and varying $ 0.25 \leq p \leq 0.675 $, $ 0 \leq u \leq 0.4 $

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Figure 5. Model $ SAIRP $ with parameter values from Tables 1-2 (colored continuous line) fitting the real data (discontinuous line) of active infected individuals with COVID-19 in Portugal, from March 2, 2020 until April 15, 2021

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Figure 6. A geographical network with 5 regions and the main connections. Individual displacements from one region to another occur along these connections

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Figure 7. Numerical simulations of the hybrid model (8)-(13), for four relevant scenarios. Each sub-figure shows the number $ I_i(t) $ of infected individuals in each region $ D_i $ ($ 1 \leq i \leq 5 $) of the geographical network depicted in Figure 6

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Table 1. Piecewise parameter values $ \beta_i $, $ p_i $, $ m_i $, for $ i = 1, \ldots, 9 $, of the $ SAIRP $ model

Time sub-interval	$ \beta_i $	$ p_i $	$ f_i $
	(transmission rate)	(transfer from $ S $ to $ P $)	(transfer from $ P $ to $ S $)
$ [0, 73] $	$ \beta_1 = 1.502 $	$ p_1 = 0.675 $	$ f_1 = 0.066 $
$ [73, 90] $	$ \beta_2 = 0.600 $	$ p_2 = 0.650 $	$ f_2 = 0.090 $
$ [90, 130] $	$ \beta_3 = 1.240 $	$ p_3 = 0.580 $	$ f_3 = 0.180 $
$ [130, 163] $	$ \beta_4 = 0.936 $	$ p_4 = 0.610 $	$ f_4 = 0.160 $
$ [163, 200] $	$ \beta_5 = 1.531 $	$ p_5 = 0.580 $	$ f_5 = 0.170 $
$ [200, 253] $	$ \beta_6 = 0.886 $	$ p_6 = 0.290 $	$ f_6 = 0.140 $
$ [253, 304] $	$ \beta_7 = 0.250 $	$ p_7 = 0.370 $	$ f_7 = 0.379 $
$ [304, 329] $	$ \beta_8 = 0.793 $	$ p_8 = 0.370 $	$ f_8 = 0.090 $
$ [329, 410] $	$ \beta_9 = 0.100 $	$ p_9 = 0.550 $	$ f_9 = 0.090 $

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Table 2. Constant parameter values and initial conditions for $ SAIRP $ model, see [34]

Parameter	Description	Value
$ \Lambda $	Recruitment rate	$ \frac{0.19\%\times N_0}{365} $
$ \mu $	Natural death rate	$ \frac{1}{81 \times 365} $
$ \theta $	Modification parameter	$ 1 $
$ v $	Transfer rate from $ A $ to $ I $	$ 1 $
$ q $	Fraction of $ A $ individuals confirmed as infected	$ 0.15 $
$ \phi $	Transfer rate from $ S $ to $ P $	$ \frac{1}{12} $
$ \delta $	Transfer rate from $ I $ to $ R $	$ \frac{1}{27} $
$ w $	Transfer rate from $ P $ to $ S $	$ \frac{1}{45} $
Class of individuals	Initial condition value
Susceptible	$ S(0) = 10295894 $
Asymptomatic	$ A(0) = \tfrac{2}{0.15} $
Active infected	$ I(0) = 2 $
Removed	$ R(0) = 0 $
Protected	$ P(0) = 0 $

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Table 3. Values of the parameters for the numerical simulations of the hybrid model (8)

Parameter	Region 1	Region 2	Region 3	Region 4	Region 5
$ \beta $	$ 2 $	$ 2 $	$ 2 $	$ 0.1 $	$ 0.1 $
$ p $	$ 0.0 $	$ 0.0 $	$ 0.0 $	$ 0.5 $	$ 0.5 $
$ \theta $	$ 1 $	$ 1 $	$ 1 $	$ 1 $	$ 1 $
$ \Lambda $	$ 1000 $	$ 1000 $	$ 1000 $	$ 1 $	$ 1 $
$ \phi $	$ 1 $	$ 1 $	$ 1 $	$ 1 $	$ 1 $
$ \omega $	$ 1 $	$ 1 $	$ 1 $	$ 1 $	$ 1 $
$ \mu $	$ 1 $	$ 1 $	$ 1 $	$ 1 $	$ 1 $
$ \nu $	$ 1 $	$ 1 $	$ 1 $	$ 1 $	$ 1 $
$ \delta $	$ 1 $	$ 1 $	$ 1 $	$ 1 $	$ 1 $
$ u $	$ 0.2 $	$ 0.2 $	$ 0.2 $	$ 0.2 $	$ 0.2 $
$ R_0 $	$ 1.3269 $	$ 1.3269 $	$ 1.3269 $	$ 0.0375 $	$ 0.0375 $
Initial condition	Region 1	Region 2	Region 3	Region 4	Region 5
$ S_0 $	$ 297.24 $	$ 140.39 $	$ 358.32 $	$ 151.39 $	$ 443.33 $
$ A_0 $	$ 13.33 $	$ 6.66 $	$ 6.66 $	$ 13.33 $	$ 6.66 $
$ I_0 $	$ 2 $	$ 1 $	$ 1 $	$ 2 $	$ 1 $
$ R_0 $	$ 0 $	$ 0 $	$ 0 $	$ 0 $	$ 0 $
$ P_0 $	$ 0 $	$ 0 $	$ 0 $	$ 0 $	$ 0 $

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