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June  2022, 17(3): 333-357. doi: 10.3934/nhm.2022010

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

Received  May 2021 Revised  November 2021 Published  June 2022 Early access  March 2022

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.

Citation: Guillaume Cantin, Cristiana J. Silva, Arnaud Banos. Mathematical analysis of a hybrid model: Impacts of individual behaviors on the spreading of an epidemic. Networks and Heterogeneous Media, 2022, 17 (3) : 333-357. doi: 10.3934/nhm.2022010
References:
[1]

M. Ajelli, B. Gonçalves, D. Balcan, V. Colizza, H. Hu, J. J. Ramasco, S. Merler and A. Vespignani, Comparing wide-scale computational modeling Approaches to epidemic: Agent-based versus structured MetaPopulation models, BMC Infectious Diseases, 10 (2010), Article number: 190. doi: 10.1186/1471-2334-10-190.

[2]

L. J. S. Allen, An Introduction to Stochastic Processes with Applications to Biology, Second edition. CRC Press, Boca Raton, FL, 2011.

[3]

L. J. S. Allen and E. J. Allen, A comparison of three different stochastic population models with regard to persistence time, Theoretical Population Biology, 64 (2003), 439-449.  doi: 10.1016/S0040-5809(03)00104-7.

[4]

A. BanosN. CorsonB. GaudouV. Laperrière and S. R. Coyrehourcq, The importance of being hybrid for spatial epidemic models: a multi-scale approach, Systems, 3 (2015), 309-329.  doi: 10.3390/systems3040309.

[5]

A. Banos, C. Lang and M. Nicolas, Agent-based Spatial Simulation with NetLogo, Elsevier, 2017. doi: 10.1016/C2015-0-01299-0.

[6]

G. Cantin, Nonidentical coupled networks with a geographical model for human behaviors during catastrophic events, International Journal of Bifurcation and Chaos, 27 (2017), 1750213, 21pp. doi: 10.1142/S0218127417502133.

[7]

G. Cantin and C. J. Silva, Influence of the topology on the dynamics of a complex network of HIV/AIDS epidemic models, AIMS Mathematics, 4 (2019), 1145-1169.  doi: 10.3934/math.2019.4.1145.

[8]

Centers for Disease Control and Prevention, Coronavirus Disease 2019 (COVID-19), 2020. Available from: https://www.cdc.gov/coronavirus/2019-ncov/prevent-getting-sick/prevention.html.

[9]

V. Colizza and A. Vespignani, Epidemic modeling in metapopulation systems with heterogeneous coupling pattern: Theory and simulations, Journal of theoretical biology, 251 (2008), 450-467.  doi: 10.1016/j.jtbi.2007.11.028.

[10]

E. DelisleC. RousseauB. BrocheI. Leparc-Goffart and ot hers, Chikungunya outbreak in Montpellier, France, September to October 2014, Eurosurveillance, 20 (2015), 21108.  doi: 10.2807/1560-7917.ES2015.20.17.21108.

[11]

Direção Geral da Saúde – COVID-19, Ponto de Situação Atual em Portugal, 2021. Available from: https://covid19.min-saude.pt/ponto-de-situacao-atual-em-portugal/.

[12]

M. Dolfinand M. Lachowicz, Modeling opinion dynamics: how the network enhances consensus, Networks & Heterogeneous Media, 10 (2015), 877-896.  doi: 10.3934/nhm.2015.10.877.

[13]

A. Ducrot and P. Magal, Travelling wave solutions for an infection-age structured model with diffusion, Proc. Roy. Soc. Edinburgh Sect. A, 139 (2009), 459-482.  doi: 10.1017/S0308210507000455.

[14]

J. M. Epstein, J. Parker, D. Cummings and R. A. Hammond, Coupled contagion dynamics of fear and disease: Mathematical and computational explorations, PLoS One, 3 (2008), e3955. doi: 10.1371/journal.pone.0003955.

[15]

European Centre for Disease Prevention and Control, Guidelines for the Implementation of Non-Pharmaceutical Interventions Against COVID-19, 2020. Available from: https://www.ecdc.europa.eu/en/publications-data/covid-19-guidelines-non-pharmaceutical-interventions.

[16]

L. FahseC. Wissel and V. Grimm, Reconciling classical and individual-based approaches in theoretical population ecology: A protocol for extracting population parameters from individual-based models, The American Naturalist, 152 (1998), 832-856.  doi: 10.1086/286212.

[17]

S. Galam, Sociophysics: A Physicist's Modeling of Psycho-political Phenomena, Understanding Complex Systems. Springer, New York, 2012. doi: 10.1007/978-1-4614-2032-3.

[18]

S. GrauwinE. BertinR. Lemoy and P. Jensen, Competition between collective and individual dynamics, Proceedings of the National Academy of Sciences, 106 (2009), 20622-20626.  doi: 10.1073/pnas.0906263106.

[19]

J. K. Hale, Ordinary Differential Equations, Krieger Publishing Company (second edition), 1980.

[20]

D. Helbing, Quantitative Sociodynamics: Stochastic Methods and Models of Social Interaction Processes, Springer-Verlag Berlin Heidelberg, 2010. doi: 10.1007/978-3-642-11546-2.

[21]

A. J. Heppenstall, A. T. Crooks, L. M. See and M. Batty, Agent-based Models of Geographical Systems, Springer Science & Business Media, 2011. doi: 10.1007/978-90-481-8927-4.

[22]

H. W. Hethcote, Three basic epidemiological models, in Applied Mathematical Ecology, Springer, 18 (1989), 119–144. doi: 10.1007/978-3-642-61317-3_5.

[23]

H. W. Hethcote and P. Van den Driessche, Some epidemiological models with nonlinear incidence, Journal of Mathematical Biology, 29 (1991), 271-287.  doi: 10.1007/BF00160539.

[24]

D. A. JonesH. L. Smith and H. R. Thieme, Spread of viral infection of immobilized bacteria, Networks & Heterogeneous Media, 8 (2013), 327-342.  doi: 10.3934/nhm.2013.8.327.

[25]

W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics–I. 1927, Bulletin of mathematical biology, 53 (1991), 33-55. 

[26]

K. Klemm, M. Serrano, V. M. Eguíluz and M. San Miguel, A measure of individual role in collective dynamics, Scientific Reports, 2 (2012), Article number: 292, 8pp. doi: 10.1038/srep00292.

[27]

E. Logak and I. Passat, An epidemic model with nonlocal diffusion on networks, Networks & Heterogeneous Media, 11 (2016), 693-719.  doi: 10.3934/nhm.2016014.

[28]

N. MarilleauC. Lang and P. Giraudoux, Coupling agent-based with equation-based models to study spatially explicit megapopulation dynamics, Ecological Modelling, 384 (2018), 34-42.  doi: 10.1016/j.ecolmodel.2018.06.011.

[29]

C. McPhail and R. T. Wohlstein, Individual and collective behaviors within gatherings, demonstrations, and riots, Annual Review of Sociology, 9 (1983), 579-600.  doi: 10.1146/annurev.so.09.080183.003051.

[30]

J. D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications, Third edition. Interdisciplinary Applied Mathematics, 18. Springer-Verlag, New York, 2003.

[31]

M. E. J. Newman and D. J. Watts, Scaling and percolation in the small-world network model, Physical Review E, 60 (1999), 7332.  doi: 10.1515/9781400841356.310.

[32]

N. D. Nguyen, Coupling Equation-based and Individual-based Models in the Study of Complex Systems. A Case Study in Theoretical Population Ecology, Ph.D thesis, Pierre and Marie Curie University, 2010.

[33]

F. Schweitzer, Self-organization of Complex Structures: From Individual to Collective Dynamics, Gordon and Breach Science Publishers, Amsterdam, 1997.

[34]

C. J. Silva, G. Cantin, C. Cruz, R. Fonseca-Pinto, R. P. Fonseca, E. S. Santos and D. F. M. Torres, Complex network model for COVID-19: human behavior, pseudo-periodic solutions and multiple epidemic waves, Journal of Mathematical Analysis and Applications, (2021), 125171. doi: 10.1016/j.jmaa.2021.125171.

[35]

C. J. Silva, C. Cruz, D. F. M. Torres et al., Optimal control of the COVID-19 pandemic: Controlled sanitary deconfinement in Portugal, Scientific Reports, 11 (2021), Art. 3451, 15 pp. doi: 10.1038/s41598-021-83075-6.

[36]

H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence, American Mathematical Soc., Providence, RI, 2011. doi: 10.1090/gsm/118.

[37]

R. H. Turner, L. M. Killian and others, Collective Behavior, Prentice-Hall Englewood Cliffs, NJ, 1957.

[38]

S. Wright, Crowds and Riots: A Study in Social Organization, Sage Publications Beverly Hills, CA, 1978.

[39]

P. Yan and G. Chowell, Beyond the initial phase: Compartment models for disease transmission, in Quantitative Methods for Investigating Infectious Disease Outbreaks (Texts in Applied Mathematics), Springer, Cham, 70 (2019), 135–182.

show all references

References:
[1]

M. Ajelli, B. Gonçalves, D. Balcan, V. Colizza, H. Hu, J. J. Ramasco, S. Merler and A. Vespignani, Comparing wide-scale computational modeling Approaches to epidemic: Agent-based versus structured MetaPopulation models, BMC Infectious Diseases, 10 (2010), Article number: 190. doi: 10.1186/1471-2334-10-190.

[2]

L. J. S. Allen, An Introduction to Stochastic Processes with Applications to Biology, Second edition. CRC Press, Boca Raton, FL, 2011.

[3]

L. J. S. Allen and E. J. Allen, A comparison of three different stochastic population models with regard to persistence time, Theoretical Population Biology, 64 (2003), 439-449.  doi: 10.1016/S0040-5809(03)00104-7.

[4]

A. BanosN. CorsonB. GaudouV. Laperrière and S. R. Coyrehourcq, The importance of being hybrid for spatial epidemic models: a multi-scale approach, Systems, 3 (2015), 309-329.  doi: 10.3390/systems3040309.

[5]

A. Banos, C. Lang and M. Nicolas, Agent-based Spatial Simulation with NetLogo, Elsevier, 2017. doi: 10.1016/C2015-0-01299-0.

[6]

G. Cantin, Nonidentical coupled networks with a geographical model for human behaviors during catastrophic events, International Journal of Bifurcation and Chaos, 27 (2017), 1750213, 21pp. doi: 10.1142/S0218127417502133.

[7]

G. Cantin and C. J. Silva, Influence of the topology on the dynamics of a complex network of HIV/AIDS epidemic models, AIMS Mathematics, 4 (2019), 1145-1169.  doi: 10.3934/math.2019.4.1145.

[8]

Centers for Disease Control and Prevention, Coronavirus Disease 2019 (COVID-19), 2020. Available from: https://www.cdc.gov/coronavirus/2019-ncov/prevent-getting-sick/prevention.html.

[9]

V. Colizza and A. Vespignani, Epidemic modeling in metapopulation systems with heterogeneous coupling pattern: Theory and simulations, Journal of theoretical biology, 251 (2008), 450-467.  doi: 10.1016/j.jtbi.2007.11.028.

[10]

E. DelisleC. RousseauB. BrocheI. Leparc-Goffart and ot hers, Chikungunya outbreak in Montpellier, France, September to October 2014, Eurosurveillance, 20 (2015), 21108.  doi: 10.2807/1560-7917.ES2015.20.17.21108.

[11]

Direção Geral da Saúde – COVID-19, Ponto de Situação Atual em Portugal, 2021. Available from: https://covid19.min-saude.pt/ponto-de-situacao-atual-em-portugal/.

[12]

M. Dolfinand M. Lachowicz, Modeling opinion dynamics: how the network enhances consensus, Networks & Heterogeneous Media, 10 (2015), 877-896.  doi: 10.3934/nhm.2015.10.877.

[13]

A. Ducrot and P. Magal, Travelling wave solutions for an infection-age structured model with diffusion, Proc. Roy. Soc. Edinburgh Sect. A, 139 (2009), 459-482.  doi: 10.1017/S0308210507000455.

[14]

J. M. Epstein, J. Parker, D. Cummings and R. A. Hammond, Coupled contagion dynamics of fear and disease: Mathematical and computational explorations, PLoS One, 3 (2008), e3955. doi: 10.1371/journal.pone.0003955.

[15]

European Centre for Disease Prevention and Control, Guidelines for the Implementation of Non-Pharmaceutical Interventions Against COVID-19, 2020. Available from: https://www.ecdc.europa.eu/en/publications-data/covid-19-guidelines-non-pharmaceutical-interventions.

[16]

L. FahseC. Wissel and V. Grimm, Reconciling classical and individual-based approaches in theoretical population ecology: A protocol for extracting population parameters from individual-based models, The American Naturalist, 152 (1998), 832-856.  doi: 10.1086/286212.

[17]

S. Galam, Sociophysics: A Physicist's Modeling of Psycho-political Phenomena, Understanding Complex Systems. Springer, New York, 2012. doi: 10.1007/978-1-4614-2032-3.

[18]

S. GrauwinE. BertinR. Lemoy and P. Jensen, Competition between collective and individual dynamics, Proceedings of the National Academy of Sciences, 106 (2009), 20622-20626.  doi: 10.1073/pnas.0906263106.

[19]

J. K. Hale, Ordinary Differential Equations, Krieger Publishing Company (second edition), 1980.

[20]

D. Helbing, Quantitative Sociodynamics: Stochastic Methods and Models of Social Interaction Processes, Springer-Verlag Berlin Heidelberg, 2010. doi: 10.1007/978-3-642-11546-2.

[21]

A. J. Heppenstall, A. T. Crooks, L. M. See and M. Batty, Agent-based Models of Geographical Systems, Springer Science & Business Media, 2011. doi: 10.1007/978-90-481-8927-4.

[22]

H. W. Hethcote, Three basic epidemiological models, in Applied Mathematical Ecology, Springer, 18 (1989), 119–144. doi: 10.1007/978-3-642-61317-3_5.

[23]

H. W. Hethcote and P. Van den Driessche, Some epidemiological models with nonlinear incidence, Journal of Mathematical Biology, 29 (1991), 271-287.  doi: 10.1007/BF00160539.

[24]

D. A. JonesH. L. Smith and H. R. Thieme, Spread of viral infection of immobilized bacteria, Networks & Heterogeneous Media, 8 (2013), 327-342.  doi: 10.3934/nhm.2013.8.327.

[25]

W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics–I. 1927, Bulletin of mathematical biology, 53 (1991), 33-55. 

[26]

K. Klemm, M. Serrano, V. M. Eguíluz and M. San Miguel, A measure of individual role in collective dynamics, Scientific Reports, 2 (2012), Article number: 292, 8pp. doi: 10.1038/srep00292.

[27]

E. Logak and I. Passat, An epidemic model with nonlocal diffusion on networks, Networks & Heterogeneous Media, 11 (2016), 693-719.  doi: 10.3934/nhm.2016014.

[28]

N. MarilleauC. Lang and P. Giraudoux, Coupling agent-based with equation-based models to study spatially explicit megapopulation dynamics, Ecological Modelling, 384 (2018), 34-42.  doi: 10.1016/j.ecolmodel.2018.06.011.

[29]

C. McPhail and R. T. Wohlstein, Individual and collective behaviors within gatherings, demonstrations, and riots, Annual Review of Sociology, 9 (1983), 579-600.  doi: 10.1146/annurev.so.09.080183.003051.

[30]

J. D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications, Third edition. Interdisciplinary Applied Mathematics, 18. Springer-Verlag, New York, 2003.

[31]

M. E. J. Newman and D. J. Watts, Scaling and percolation in the small-world network model, Physical Review E, 60 (1999), 7332.  doi: 10.1515/9781400841356.310.

[32]

N. D. Nguyen, Coupling Equation-based and Individual-based Models in the Study of Complex Systems. A Case Study in Theoretical Population Ecology, Ph.D thesis, Pierre and Marie Curie University, 2010.

[33]

F. Schweitzer, Self-organization of Complex Structures: From Individual to Collective Dynamics, Gordon and Breach Science Publishers, Amsterdam, 1997.

[34]

C. J. Silva, G. Cantin, C. Cruz, R. Fonseca-Pinto, R. P. Fonseca, E. S. Santos and D. F. M. Torres, Complex network model for COVID-19: human behavior, pseudo-periodic solutions and multiple epidemic waves, Journal of Mathematical Analysis and Applications, (2021), 125171. doi: 10.1016/j.jmaa.2021.125171.

[35]

C. J. Silva, C. Cruz, D. F. M. Torres et al., Optimal control of the COVID-19 pandemic: Controlled sanitary deconfinement in Portugal, Scientific Reports, 11 (2021), Art. 3451, 15 pp. doi: 10.1038/s41598-021-83075-6.

[36]

H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence, American Mathematical Soc., Providence, RI, 2011. doi: 10.1090/gsm/118.

[37]

R. H. Turner, L. M. Killian and others, Collective Behavior, Prentice-Hall Englewood Cliffs, NJ, 1957.

[38]

S. Wright, Crowds and Riots: A Study in Social Organization, Sage Publications Beverly Hills, CA, 1978.

[39]

P. Yan and G. Chowell, Beyond the initial phase: Compartment models for disease transmission, in Quantitative Methods for Investigating Infectious Disease Outbreaks (Texts in Applied Mathematics), Springer, Cham, 70 (2019), 135–182.

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 $
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 $
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 $
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 $
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 $
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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