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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. Banos, N. Corson, B. Gaudou, V. 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. Delisle, C. Rousseau, B. Broche, I. 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. Fahse, C. 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. Grauwin, E. Bertin, R. 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. Jones, H. 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. Marilleau, C. 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. Banos, N. Corson, B. Gaudou, V. 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. Delisle, C. Rousseau, B. Broche, I. 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. Fahse, C. 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. Grauwin, E. Bertin, R. 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. Jones, H. 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. Marilleau, C. 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.
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
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
Basic reproduction number $R_0(p, u)$ of the $SAIRP$ model (3), with $0.25 \leq p \leq 0.675$ and $0 \leq u \leq 0.4$
Local stability condition of the endemic equilibrium $\Sigma_+$ is satisfied for $R_0(p, u) > 1$. Considering the parameter values (Pfixed) and varying $0.25 \leq p \leq 0.675$, $0 \leq u \leq 0.4$
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
A geographical network with 5 regions and the main connections. Individual displacements from one region to another occur along these connections
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
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 $S$) $[0, 73]$ $\beta_1 = 1.502$ $p_1 = 0.675$ $f_1 = 0.066$ $[73, 90]$ $\beta_2 = 0.600$ $p_2 = 0.650$ $f_2 = 0.090$ $[90, 130]$ $\beta_3 = 1.240$ $p_3 = 0.580$ $f_3 = 0.180$ $[130, 163]$ $\beta_4 = 0.936$ $p_4 = 0.610$ $f_4 = 0.160$ $[163, 200]$ $\beta_5 = 1.531$ $p_5 = 0.580$ $f_5 = 0.170$ $[200, 253]$ $\beta_6 = 0.886$ $p_6 = 0.290$ $f_6 = 0.140$ $[253, 304]$ $\beta_7 = 0.250$ $p_7 = 0.370$ $f_7 = 0.379$ $[304, 329]$ $\beta_8 = 0.793$ $p_8 = 0.370$ $f_8 = 0.090$ $[329, 410]$ $\beta_9 = 0.100$ $p_9 = 0.550$ $f_9 = 0.090$
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 $P(0) = 0$
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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