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June  2022, 17(3): 293-310. doi: 10.3934/nhm.2022008

Multiscale models of Covid-19 with mutations and variants

1. 

University of Granada, Departamento de Matemática Aplicada, 18071-Granada, Spain, Politecnico, Torino, Italy

2. 

University of Perugia, Italy

3. 

Faculty of Sciences Semlalia-UCA, LMDP, Morocco and UMMISCO (IRD-SU, France)

Received  June 2021 Revised  August 2021 Published  June 2022 Early access  March 2022

Fund Project: Nicola Bellomo acknowledges the support of the University of Granada, Project Modeling in Nature, https://www.modelingnature.org.

This paper focuses on the multiscale modeling of the COVID-19 pandemic and presents further developments of the model [7] with the aim of showing how relaxations of the confinement rules can generate sequential waves. Subsequently, the dynamics of mutations into new variants can be modeled. Simulations are developed also to support the decision making of crisis managers.

Citation: Nicola Bellomo, Diletta Burini, Nisrine Outada. Multiscale models of Covid-19 with mutations and variants. Networks and Heterogeneous Media, 2022, 17 (3) : 293-310. doi: 10.3934/nhm.2022008
References:
[1]

K. G. AndersenA. RambautW. Ian LipkinE. C. Holmes and R. F. Garry, The proximal origin of SARS-CoV-2, Nature Medicine, 26 (2020), 450-452.  doi: 10.1038/s41591-020-0820-9.

[2]

V. V. Aristov, Biological systems as nonequilibrium structures described by kinetic methods, Results in Physics, 13 (2019), 102232.  doi: 10.1016/j.rinp.2019.102232.

[3]

B. Avishai, The pandemic isn't a black swan but a portent of a more fragile global system, The New Yorker, (2020), https://www.newyorker.com/news/daily-comment/the-pandemic-isnt-a-black-swan-but-a-portent-of-a-more-fragile-global-system.

[4]

B. AylajN. BellomoL. Gibelli and A. Reali, On a unified multiscale vision of behavioral crowds, Math. Models Methods Appl. Sci., 30 (2020), 1-22.  doi: 10.1142/S0218202520500013.

[5]

P. Baldwin and B. W. Di Mauro, Economics in the Time of COVID-19, VoxEU.org Book, (2020).

[6]

N. Bellomo and A. Bellouquid, On multiscale models of pedestrian crowds from mesoscopic to macroscopic, Commun. Math. Sci., 13 (2015), 1649-1664.  doi: 10.4310/CMS.2015.v13.n7.a1.

[7]

N. BellomoR. BinghamM. ChaplainG. DosiG. ForniD. KnopoffJ. LowengrubR. Twarock and M. E. Virgillito, A multi-scale model of virus pandemic: Heterogeneous interactive entities in a globally connected world, Math. Models Methods Appl. Sci., 30 (2020), 1591-1651.  doi: 10.1142/S0218202520500323.

[8]

N. BellomoD. BuriniG. DosiL. GibelliD. A. KnopoffN. OutadaP. Terna and M. E. Virgillito, What is life? A perspective of the mathematical kinetic theory of active particles, Math. Models Methods Appl. Sci., 31 (2021), 1821-1866.  doi: 10.1142/S0218202521500408.

[9]

N. BellomoL. Gibelli and N. Outada, On the interplay between behavioral dynamics and social interactions in human crowds, Kinet. Relat. Models, 12 (2019), 397-409.  doi: 10.3934/krm.2019017.

[10]

N. BellomoK. J. PainterY. Tao and M. Winkler, Occurrence vs. absence of txis-driven instabilities in a May–Nowak Model for virus infection, SIAM J. Appl. Math., 79 (2019), 1990-2010.  doi: 10.1137/19M1250261.

[11]

A. Bellouquid and M. Delitala, Modelling Complex Biological Systems - A Kinetic Theory Approach, Modeling and Simulation in Science, Engineering and Technology. Birkhäuser Boston, Inc., Boston, MA, 2006.

[12]

A. L. BertozziJ. RosadoM. B. Short and L. Wang, Contagion shocks in one dimension, J. Stat. Phys., 158 (2015), 647-664.  doi: 10.1007/s10955-014-1019-6.

[13]

R. BinghamE. Dykeman and R. Twarock, RNA virus evolution via a quasispecies-based model reveals a drug target with a high barrier to resistance, Viruses, 9 (2017), 347.  doi: 10.3390/v9110347.

[14]

W. BoscheriG. Dimarco and L. Pareschi, Modeling and simulating the spatial spread of an epidemic through multiscale kinetic transport equations, Math. Models Methods Appl. Sci., 31 (2021), 1059-1097.  doi: 10.1142/S0218202521400017.

[15]

D. Burini and N. Chouhad, A Multiscale view of nonlinear diffusion in biology: From cells to tissues, Math. Models Methods Appl. Sci., 29 (2019), 791-823.  doi: 10.1142/S0218202519400062.

[16]

E. Callaway, Coronavirus vaccines: Five key questions as trials begin, Nature, 592 (2021), 670-671.  doi: 10.1038/d41586-020-00798-8.

[17]

M. Cecconi, G. Forni and A. Mantovani, COVID-19: An executive report April 2020 update, Accademia Nazionale dei Lincei, Commissione Salute, (2020), https://www.lincei.it/sites/default/files/documenti/Commissioni.

[18]

M. Cecconi, G. Forni and A. Mantovani, COVID-19: An executive report Summer 2020 update, Accademia Nazionale dei Lincei, Commissione Salute, (2002), https://www.lincei.it/sites/default/files/documenti/Commissioni.

[19]

M. CecconiG. Forni and A. Mantovani, Ten things we learned about COVID-19, Intensive Care Medicine, 46 (2020), 1590-1593.  doi: 10.1007/s00134-020-06140-0.

[20]

E. L. Cooper, Evolution of immune system from self/not self to danger to artificial immune system, Physics of Life Review, 7 (2010), 55-78.  doi: 10.1016/j.plrev.2009.12.001.

[21]

D. Cyranoski, Profile of a killer: The complex biology powering the coronavirus pandemic, Nature, (2020), 22-26.  doi: 10.1038/d41586-020-01315-7.

[22]

S. De LilloM. Delitala and M. Salvatori, Modelling epidemics and virus mutations by methods of the mathematical kinetic theory for active particles, Math. Models Methods Appl. Sci., 19 (2009), 1405-1425.  doi: 10.1142/S0218202509003838.

[23]

G. Dimarco, L. Pareschi and G. Toscani, et al., Wealth distribution under the spread of infectious diseases, Phys. Rev. E, 102 (2020), Article 022303, 14 pp. doi: 10.1103/physreve.102.022303.

[24]

G. DosiL. Fanti and M. E. Virgillito, Unequal societies in usual times, unjust societies in pandemic ones, Journal of Industrial and Business Economics, 47 (2020), 371-389.  doi: 10.1007/s40812-020-00173-8.

[25]

I. Echeverria-HuarteA. GarcimartinR. C. HidalgoC. Martin-Gomez and I. Zurigue, Estimating density limits for walking pedestrians keeping a safe interpersonal distancing, Scientific Reports, 11 (2021), 1534.  doi: 10.1038/s41598-020-79454-0.

[26]

E. Estrada, COVID-19 and SARS-CoV-2. Modeling the present, looking at the future, Phys. Rep., 869 (2020), 1-51.  doi: 10.1016/j.physrep.2020.07.005.

[27]

F. FlandoliE. La Fauci and M. Riva, Individual-based Markov model of virus diffusion: Comparison with COVID-19 incubation period, serial interval and regional time series, Math. Models Methods Appl. Sci., 31 (2021), 907-939.  doi: 10.1142/S0218202521500226.

[28]

J. F. Fontanari, A stochastic model for the influence of social distancing on loneliness, Phys. A, 584 (2021), Paper No. 126367, 10 pp. doi: 10.1016/j.physa.2021.126367.

[29]

G. Forni and and A. Mantovani, COVID-19 vaccines: Where we stand and challenges ahead, Cell Death & Differentiation, 28 (2021), 626-639.  doi: 10.1038/s41418-020-00720-9.

[30]

Q. GaoJ. ZhuangT. Wu and H. Shen, Transmission dynamics and quarantine control of COVID-19 in cluster community: A new transmission-quarantine model with case study for diamond princess, Math. Models Methods Appl. Sci., 31 (2021), 619-648.  doi: 10.1142/S0218202521500147.

[31]

M. GattoE. BertuzzoL. MariS. MiccoliL. CarraroR. Casagrandi and A. Rinaldo, Spread and dynamics of the COVID-19 epidemic in Italy: Effects of emergency containment measures, Proceedings of the National Academy of Sciences, 117 (2020), 10484-10491.  doi: 10.1073/pnas.2004978117.

[32]

P. HardyL. S. Marcolino and J. F. Fontanari, The paradox of productivity during quarantine: An agent-based simulation, European Physical Journal B, 94 (2021), 40.  doi: 10.1140/epjb/s10051-020-00016-4.

[33]

H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653.  doi: 10.1137/S0036144500371907.

[34]

D. Kim and A. Quaini, A kinetic theory approach to model pedestrian dynamics in bounded domains with obstacles, Kinet. Relat. Models, 12 (2019), 1273-1296.  doi: 10.3934/krm.2019049.

[35]

D. Kim and A. Quaini, Coupling kinetic theory approaches for pedestrian dynamics and disease contagion in a confined environment, Math. Models Methods Appl. Sci., 30 (2020), 1893-1915.  doi: 10.1142/S0218202520400126.

[36]

S. M. KisslerC. TedijantoE. GoldsteinY. H. Grad and M. Lipsitch, Projecting the transmission dynamics of SARS-CoV-2 through the postpandemic period, Science, 368 (2020), 860-868.  doi: 10.1126/science.abb5793.

[37]

W. Lee, S. Liu, W. Li and S. Osher, Mean field control problems for vaccine distribution, arXiv preprint, arXiv: 2104.11887, (2021).

[38]

O. A. MacLeanR. J. OrtonJ. B. Singer and D. L. Robertson, No evidence for distinct types in the evolution of SARS-CoV-2, Virus Evolution, 6 (2020).  doi: 10.1093/ve/veaa034.

[39]

S. T. McQuadeR. WeightmanN. J. MerrilA. YadavE. TrélatS. R. Allred and B. Piccoli, Control of COVID-19 outbreack using an extended SEIR model, Math. Models Methods Appl. Sci., 31 (2021), 2399-2424.  doi: 10.1142/S0218202521500512.

[40]

P. M. MatricardiR. W. Dal Negro and R. Nisini, The first, holistic immunological model of COVID-19: Implications for prevention, diagnosis, and public health measures, Pediatric Allergy Imunology, 31 (2020), 454-470.  doi: 10.1111/pai.13271.

[41]

J. B. Moore and C. H. June, Cytokine release syndrome in severe COVID-19, Science, 368 (2020), 473-474.  doi: 10.1126/science.abb8925.

[42]

P. Musiani, and G. Forni, Basic Immunology, Piccin, Padua, 2018.

[43]

P. S. A. SalamW. BockA. Klar and S. Tiwari, Disease contagion models coupled to crowd motion and mesh-free simulation, Math. Models Methods Appl. Sci., 31 (2021), 1277-1295.  doi: 10.1142/S0218202521400066.

[44]

R. Sanjuán and P. Domingo-Calap, Mechanisms of viral mutation, Cellular and Molecular Life Sciences, 73 (2016), 4433-4448. 

[45]

A. Syed FarazA. A. Quadeer and M. R. McKay, Preliminary identification of potential vaccine targets for the COVID-19 coronavirus (SARS-CoV-2) based on SARS-CoV immunological studies, Viruses, 12 (2020), 254. 

[46]

R. TwarockR. J. BinghamE. C. Dykeman and P. G. Stockley, A modelling paradigm for RNA virus assembly, Current Opinion in Virology, 31 (2018), 74-81.  doi: 10.1016/j.coviro.2018.07.003.

[47]

R. TwarockG. Leonov and P. G. Stockley, Hamiltonian path analysis of viral genomes, Nature Communications, 9 (2018).  doi: 10.1038/s41467-018-03713-y.

[48]

N. Vabret and G. J. Britton, Immunology of COVID-19: Current state of the science, Immunity 2020, 52 (2020), 910-941.  doi: 10.1016/j.immuni.2020.05.002.

[49]

N. van Doremalen and T. Bushmaker, Aerosol and surface stability of SARS-CoV-2 as compared with SARS-CoV, N. Engl. J. Med., 382 (2020), 1564-1567.  doi: 10.1056/NEJMc2004973.

show all references

References:
[1]

K. G. AndersenA. RambautW. Ian LipkinE. C. Holmes and R. F. Garry, The proximal origin of SARS-CoV-2, Nature Medicine, 26 (2020), 450-452.  doi: 10.1038/s41591-020-0820-9.

[2]

V. V. Aristov, Biological systems as nonequilibrium structures described by kinetic methods, Results in Physics, 13 (2019), 102232.  doi: 10.1016/j.rinp.2019.102232.

[3]

B. Avishai, The pandemic isn't a black swan but a portent of a more fragile global system, The New Yorker, (2020), https://www.newyorker.com/news/daily-comment/the-pandemic-isnt-a-black-swan-but-a-portent-of-a-more-fragile-global-system.

[4]

B. AylajN. BellomoL. Gibelli and A. Reali, On a unified multiscale vision of behavioral crowds, Math. Models Methods Appl. Sci., 30 (2020), 1-22.  doi: 10.1142/S0218202520500013.

[5]

P. Baldwin and B. W. Di Mauro, Economics in the Time of COVID-19, VoxEU.org Book, (2020).

[6]

N. Bellomo and A. Bellouquid, On multiscale models of pedestrian crowds from mesoscopic to macroscopic, Commun. Math. Sci., 13 (2015), 1649-1664.  doi: 10.4310/CMS.2015.v13.n7.a1.

[7]

N. BellomoR. BinghamM. ChaplainG. DosiG. ForniD. KnopoffJ. LowengrubR. Twarock and M. E. Virgillito, A multi-scale model of virus pandemic: Heterogeneous interactive entities in a globally connected world, Math. Models Methods Appl. Sci., 30 (2020), 1591-1651.  doi: 10.1142/S0218202520500323.

[8]

N. BellomoD. BuriniG. DosiL. GibelliD. A. KnopoffN. OutadaP. Terna and M. E. Virgillito, What is life? A perspective of the mathematical kinetic theory of active particles, Math. Models Methods Appl. Sci., 31 (2021), 1821-1866.  doi: 10.1142/S0218202521500408.

[9]

N. BellomoL. Gibelli and N. Outada, On the interplay between behavioral dynamics and social interactions in human crowds, Kinet. Relat. Models, 12 (2019), 397-409.  doi: 10.3934/krm.2019017.

[10]

N. BellomoK. J. PainterY. Tao and M. Winkler, Occurrence vs. absence of txis-driven instabilities in a May–Nowak Model for virus infection, SIAM J. Appl. Math., 79 (2019), 1990-2010.  doi: 10.1137/19M1250261.

[11]

A. Bellouquid and M. Delitala, Modelling Complex Biological Systems - A Kinetic Theory Approach, Modeling and Simulation in Science, Engineering and Technology. Birkhäuser Boston, Inc., Boston, MA, 2006.

[12]

A. L. BertozziJ. RosadoM. B. Short and L. Wang, Contagion shocks in one dimension, J. Stat. Phys., 158 (2015), 647-664.  doi: 10.1007/s10955-014-1019-6.

[13]

R. BinghamE. Dykeman and R. Twarock, RNA virus evolution via a quasispecies-based model reveals a drug target with a high barrier to resistance, Viruses, 9 (2017), 347.  doi: 10.3390/v9110347.

[14]

W. BoscheriG. Dimarco and L. Pareschi, Modeling and simulating the spatial spread of an epidemic through multiscale kinetic transport equations, Math. Models Methods Appl. Sci., 31 (2021), 1059-1097.  doi: 10.1142/S0218202521400017.

[15]

D. Burini and N. Chouhad, A Multiscale view of nonlinear diffusion in biology: From cells to tissues, Math. Models Methods Appl. Sci., 29 (2019), 791-823.  doi: 10.1142/S0218202519400062.

[16]

E. Callaway, Coronavirus vaccines: Five key questions as trials begin, Nature, 592 (2021), 670-671.  doi: 10.1038/d41586-020-00798-8.

[17]

M. Cecconi, G. Forni and A. Mantovani, COVID-19: An executive report April 2020 update, Accademia Nazionale dei Lincei, Commissione Salute, (2020), https://www.lincei.it/sites/default/files/documenti/Commissioni.

[18]

M. Cecconi, G. Forni and A. Mantovani, COVID-19: An executive report Summer 2020 update, Accademia Nazionale dei Lincei, Commissione Salute, (2002), https://www.lincei.it/sites/default/files/documenti/Commissioni.

[19]

M. CecconiG. Forni and A. Mantovani, Ten things we learned about COVID-19, Intensive Care Medicine, 46 (2020), 1590-1593.  doi: 10.1007/s00134-020-06140-0.

[20]

E. L. Cooper, Evolution of immune system from self/not self to danger to artificial immune system, Physics of Life Review, 7 (2010), 55-78.  doi: 10.1016/j.plrev.2009.12.001.

[21]

D. Cyranoski, Profile of a killer: The complex biology powering the coronavirus pandemic, Nature, (2020), 22-26.  doi: 10.1038/d41586-020-01315-7.

[22]

S. De LilloM. Delitala and M. Salvatori, Modelling epidemics and virus mutations by methods of the mathematical kinetic theory for active particles, Math. Models Methods Appl. Sci., 19 (2009), 1405-1425.  doi: 10.1142/S0218202509003838.

[23]

G. Dimarco, L. Pareschi and G. Toscani, et al., Wealth distribution under the spread of infectious diseases, Phys. Rev. E, 102 (2020), Article 022303, 14 pp. doi: 10.1103/physreve.102.022303.

[24]

G. DosiL. Fanti and M. E. Virgillito, Unequal societies in usual times, unjust societies in pandemic ones, Journal of Industrial and Business Economics, 47 (2020), 371-389.  doi: 10.1007/s40812-020-00173-8.

[25]

I. Echeverria-HuarteA. GarcimartinR. C. HidalgoC. Martin-Gomez and I. Zurigue, Estimating density limits for walking pedestrians keeping a safe interpersonal distancing, Scientific Reports, 11 (2021), 1534.  doi: 10.1038/s41598-020-79454-0.

[26]

E. Estrada, COVID-19 and SARS-CoV-2. Modeling the present, looking at the future, Phys. Rep., 869 (2020), 1-51.  doi: 10.1016/j.physrep.2020.07.005.

[27]

F. FlandoliE. La Fauci and M. Riva, Individual-based Markov model of virus diffusion: Comparison with COVID-19 incubation period, serial interval and regional time series, Math. Models Methods Appl. Sci., 31 (2021), 907-939.  doi: 10.1142/S0218202521500226.

[28]

J. F. Fontanari, A stochastic model for the influence of social distancing on loneliness, Phys. A, 584 (2021), Paper No. 126367, 10 pp. doi: 10.1016/j.physa.2021.126367.

[29]

G. Forni and and A. Mantovani, COVID-19 vaccines: Where we stand and challenges ahead, Cell Death & Differentiation, 28 (2021), 626-639.  doi: 10.1038/s41418-020-00720-9.

[30]

Q. GaoJ. ZhuangT. Wu and H. Shen, Transmission dynamics and quarantine control of COVID-19 in cluster community: A new transmission-quarantine model with case study for diamond princess, Math. Models Methods Appl. Sci., 31 (2021), 619-648.  doi: 10.1142/S0218202521500147.

[31]

M. GattoE. BertuzzoL. MariS. MiccoliL. CarraroR. Casagrandi and A. Rinaldo, Spread and dynamics of the COVID-19 epidemic in Italy: Effects of emergency containment measures, Proceedings of the National Academy of Sciences, 117 (2020), 10484-10491.  doi: 10.1073/pnas.2004978117.

[32]

P. HardyL. S. Marcolino and J. F. Fontanari, The paradox of productivity during quarantine: An agent-based simulation, European Physical Journal B, 94 (2021), 40.  doi: 10.1140/epjb/s10051-020-00016-4.

[33]

H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653.  doi: 10.1137/S0036144500371907.

[34]

D. Kim and A. Quaini, A kinetic theory approach to model pedestrian dynamics in bounded domains with obstacles, Kinet. Relat. Models, 12 (2019), 1273-1296.  doi: 10.3934/krm.2019049.

[35]

D. Kim and A. Quaini, Coupling kinetic theory approaches for pedestrian dynamics and disease contagion in a confined environment, Math. Models Methods Appl. Sci., 30 (2020), 1893-1915.  doi: 10.1142/S0218202520400126.

[36]

S. M. KisslerC. TedijantoE. GoldsteinY. H. Grad and M. Lipsitch, Projecting the transmission dynamics of SARS-CoV-2 through the postpandemic period, Science, 368 (2020), 860-868.  doi: 10.1126/science.abb5793.

[37]

W. Lee, S. Liu, W. Li and S. Osher, Mean field control problems for vaccine distribution, arXiv preprint, arXiv: 2104.11887, (2021).

[38]

O. A. MacLeanR. J. OrtonJ. B. Singer and D. L. Robertson, No evidence for distinct types in the evolution of SARS-CoV-2, Virus Evolution, 6 (2020).  doi: 10.1093/ve/veaa034.

[39]

S. T. McQuadeR. WeightmanN. J. MerrilA. YadavE. TrélatS. R. Allred and B. Piccoli, Control of COVID-19 outbreack using an extended SEIR model, Math. Models Methods Appl. Sci., 31 (2021), 2399-2424.  doi: 10.1142/S0218202521500512.

[40]

P. M. MatricardiR. W. Dal Negro and R. Nisini, The first, holistic immunological model of COVID-19: Implications for prevention, diagnosis, and public health measures, Pediatric Allergy Imunology, 31 (2020), 454-470.  doi: 10.1111/pai.13271.

[41]

J. B. Moore and C. H. June, Cytokine release syndrome in severe COVID-19, Science, 368 (2020), 473-474.  doi: 10.1126/science.abb8925.

[42]

P. Musiani, and G. Forni, Basic Immunology, Piccin, Padua, 2018.

[43]

P. S. A. SalamW. BockA. Klar and S. Tiwari, Disease contagion models coupled to crowd motion and mesh-free simulation, Math. Models Methods Appl. Sci., 31 (2021), 1277-1295.  doi: 10.1142/S0218202521400066.

[44]

R. Sanjuán and P. Domingo-Calap, Mechanisms of viral mutation, Cellular and Molecular Life Sciences, 73 (2016), 4433-4448. 

[45]

A. Syed FarazA. A. Quadeer and M. R. McKay, Preliminary identification of potential vaccine targets for the COVID-19 coronavirus (SARS-CoV-2) based on SARS-CoV immunological studies, Viruses, 12 (2020), 254. 

[46]

R. TwarockR. J. BinghamE. C. Dykeman and P. G. Stockley, A modelling paradigm for RNA virus assembly, Current Opinion in Virology, 31 (2018), 74-81.  doi: 10.1016/j.coviro.2018.07.003.

[47]

R. TwarockG. Leonov and P. G. Stockley, Hamiltonian path analysis of viral genomes, Nature Communications, 9 (2018).  doi: 10.1038/s41467-018-03713-y.

[48]

N. Vabret and G. J. Britton, Immunology of COVID-19: Current state of the science, Immunity 2020, 52 (2020), 910-941.  doi: 10.1016/j.immuni.2020.05.002.

[49]

N. van Doremalen and T. Bushmaker, Aerosol and surface stability of SARS-CoV-2 as compared with SARS-CoV, N. Engl. J. Med., 382 (2020), 1564-1567.  doi: 10.1056/NEJMc2004973.

Figure 1.  Transfer diagram of the model. Boxes represent functional subsystems and arrows indicate transition of individuals
Figure 2.  Infected population $ n_2 = n_2(t) $ : $ \varepsilon = 0.001 $, $ \kappa = 0.16 $, $ \alpha_\ell = 0.2 $ (red) and $ \alpha_\ell = 0.3 $ (black)
Figure 3.  Infected population $ n_2 = n_2(t) $ : $ \varepsilon = 0.001 $, $ \alpha_\ell = 0.3 $, $ \kappa = 0.1 $ (blue), $ \kappa = 0.2 $ (black)
Figure 4.  Infected population $ n_2 = n_2(t) $ for $ \varepsilon = 0.001 $, $ \kappa = 0.1 $, $ T_d = 1 $, $ \alpha_\ell = 0.1 $, $ \alpha_d = 0.40 $ (black), $ \alpha_d = 0.45 $ (red), and $ \alpha_d = 0.50 $ (blue)
Figure 5.  Infected population $ n_2 = n_2(t) $ for $ \varepsilon = 0.001 $, $ \kappa = 0.1 $, $ T_d = 1 $, $ \alpha_\ell = 0.1 $, $ \alpha_d = 0.20 $ (black), $ \alpha_d = 0.25 $ (red), and $ \alpha_d = 0.30 $ (blue)
Figure 6.  Infected population $ n_2 = n_2(t) $ and $ n_5 = n_5(t) $ for $ \varepsilon = 0.01 $, $ \varepsilon_v = 0.005 $, $ \kappa = 0.1 $, $ \lambda = 1.5 $, $ T_d = 0.5 $, $ \alpha_\ell = 0.1 $, $ \alpha_d = 0.50 $
Figure 7.  Infected population $ n_2 = n_2(t) $ and $ n_5 = n_5(t) $ for $ \varepsilon = 0.01 $, $ \varepsilon_v = 0.005 $, $ \kappa = 0.1 $, $ \lambda = 1.5 $, $ T_d = 1 $, $ \alpha_\ell = 0.1 $, $ \alpha_d = 0.50 $
Figure 8.  Infected population $ n_2 = n_2(t) $ and $ n_5 = n_5(t) $ for $ \varepsilon = 0.01 $, $ \varepsilon_v = 0.005 $, $ \kappa = 0.1 $, $ \lambda = 1.5 $, $ T_d = 0.75 $, $ \alpha_\ell = 0.1 $, $ \alpha_d = 0.50 $
Figure 9.  Death in the case of absence of mutations: $ n_4 = n_4(t) $ for $ \varepsilon = 0.001 $, $ \kappa = 0.1 $, $ T_d = 1 $, $ \alpha_\ell = 0.1 $, $ \alpha_d = 0.40 $ (black), $ \alpha_d = 0.50 $ (red) and $ \alpha_d = 0.60 $ (blue)
Figure 10.  Death in the case of mutations: $ n_4 = n_4(t) $ for $ \varepsilon = 0.01 $, $ \varepsilon_v = 0.005 $, $ \kappa = 0.1 $, $ T_d = 1 $, $ \alpha_\ell = 0.1 $, $ \alpha_d = 0.50 $ $ \lambda = 1.0 $ (red), $ \lambda = 1.5 $ (blue)
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