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

  • * Corresponding author: Cristiana J. Silva

    * Corresponding author: Cristiana J. Silva 
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  • 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.

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

    Citation:

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  • 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

    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

    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 $

    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 $

    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

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

    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

    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 $
     | Show Table
    DownLoad: CSV

    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 $
     | Show Table
    DownLoad: CSV

    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 $
     | Show Table
    DownLoad: CSV
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