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June  2022, 17(3): 385-400. doi: 10.3934/nhm.2022012

## Vaccination strategies through intra—compartmental dynamics

 1 INdAM Unit and Department of Information Engineering, University of Brescia, Via Branze, 38, 25123 Brescia, Italy 2 Department of Sciences and Methods for Engineering, University of Modena and Reggio Emilia, Via Amendola, 2, 42122 Reggio Emilia, Italy

Received  April 2021 Revised  July 2021 Published  June 2022 Early access  March 2022

Fund Project: The authors were partly supported by the GNAMPA 2020 project "From Wellposedness to Game Theory in Conservation Laws". The IBM Power Systems Academic Initiative contributed to numerical integrations

We present a new epidemic model highlighting the roles of the immunization time and concurrent use of different vaccines in a vaccination campaign. To this aim, we introduce new intra-compartmental dynamics, a procedure that can be extended to various other situations, as detailed through specific case studies considered herein, where the dynamics within compartments are present and influence the whole evolution.

Citation: Rinaldo M. Colombo, Francesca Marcellini, Elena Rossi. Vaccination strategies through intra—compartmental dynamics. Networks and Heterogeneous Media, 2022, 17 (3) : 385-400. doi: 10.3934/nhm.2022012
##### References:
 [1] J. L. Aron, Mathematical modeling of immunity to malaria. Nonlinearity in biology and medicine, Math. Biosci., 90 (1988), 385-396.  doi: 10.1016/0025-5564(88)90076-4. [2] N. Bellomo, R. Bingham, M. A. J. Chaplain, G. Dosi, G. Forni and et al., A multiscale 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. [3] F. Brauer and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology, 2$^{nd}$ edition, Texts in Applied Mathematics, 40, Springer, New York, 2012. doi: 10.1007/978-1-4614-1686-9. [4] F. Brauer and P. van den Driessche, Models for transmission of disease with immigration of infectives, Math. Biosci., 171 (2001), 143-154.  doi: 10.1016/S0025-5564(01)00057-8. [5] R. M. Colombo and M. Garavello, Well posedness and control in a nonlocal SIR model, Appl. Math. Optim., 84 (2021), 737-771.  doi: 10.1007/s00245-020-09660-9. [6] R. M. Colombo, M. Garavello, F. Marcellini and E. Rossi, An age and space structured SIR model describing the Covid-19 pandemic, J. Math. Ind., 10 (2020), 20pp. doi: 10.1186/s13362-020-00090-4. [7] R. M. Colombo, M. Garavello, F. Marcellini and E. Rossi, IBVPs for inhomogeneous systems of balance laws in several space dimensions motivated by biology and epidemiology, preprint, 2021. [8] G. Dimarco, L. Pareschi, G. Toscani and M. Zanella, Wealth distribution under the spread of infectious diseases, Phys. Rev. E, 102 (2020), 14pp. doi: 10.1103/physreve.102.022303. [9] A. d'Onofrio, P. Manfredi and E. Salinelli, Bifurcation thresholds in an SIR model with information-dependent vaccination, Math. Model. Nat. Phenom., 2 (2007), 23-38.  doi: 10.1051/mmnp:2008009. [10] S. Ghosh and S. Bhattacharya, A data-driven understanding of COVID-19 dynamics using sequential genetic algorithm based probabilistic cellular automata, Appl. Soft Comput., 96 (2020). doi: 10.1016/j.asoc.2020.106692. [11] G. Giordano, F. Blanchini, R. Bruno, P. Colaneri, A. Di Filippo, A. Di Matteo and M. Colaneri, Modelling the COVID-19 epidemic and implementation of population-wide interventions in Italy, Nature Medicine, 26 (2020), 855-860.  doi: 10.1038/s41591-020-0883-7. [12] A. Godio, F. Pace and A. Vergnano, SEIR modeling of the Italian epidemic of SARS-CoV-2 using computational swarm intelligence, Internat. J. Environ. Res. Public Health, 17 (2020). doi: 10.3390/ijerph17103535. [13] D. Greenhalgh, Some results for an SEIR epidemic model with density dependence in the death rate, IMA J. Math. Appl. Med. Biol., 9 (1992), 67-106.  doi: 10.1093/imammb/9.2.67. [14] H. Inaba, Age-structured SIR epidemic model, in Age-Structured Population Dynamics in Demography and Epidemiology, Springer, 2017,287–331. [15] W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics. II. The problem of endemicity, Proc. Roy. Soc. Lond. A, 138 (1932), 55-83.  doi: 10.1098/rspa.1932.0171. [16] W. O. Kermack, A. G. McKendrick and G. T. Walker, A contribution to the mathematical theory of epidemics, Proc. Roy. Soc. Lond. A, 115 (1927), 700-721.  doi: 10.1098/rspa.1927.0118. [17] C. M. Kribs-Zaleta and J. X. Velasco-Hernández, A simple vaccination model with multiple endemic states, Math. Biosci., 164 (2000), 183-201.  doi: 10.1016/S0025-5564(00)00003-1. [18] G. Li and Z. Jin, Global stability of a SEIR epidemic model with infectious force in latent, infected and immune period, Chaos Solitons Fractals, 25 (2005), 1177-1184.  doi: 10.1016/j.chaos.2004.11.062. [19] M. Y. Li, H. L. Smith and L. Wang, Global dynamics an SEIR epidemic model with vertical transmission, SIAM J. Appl. Math., 62 (2001), 58-69.  doi: 10.1137/S0036139999359860. [20] X. Liu, Y. Takeuchi and S. Iwami, SVIR epidemic models with vaccination strategies, J. Theoret. Biol., 253 (2008), 1-11.  doi: 10.1016/j.jtbi.2007.10.014. [21] J. Mena-Lorca and H. W. Hethcote, Dynamic models of infectious diseases as regulators of population sizes, J. Math. Biol., 30 (1992), 693-716.  doi: 10.1007/BF00173264. [22] J. D. Murray, Mathematical Biology. I. An Introduction, 3$^{rd}$ edition, Interdisciplinary Applied Mathematics, 17, Springer-Verlag, New York, 2002. doi: 10.1007/b98868. [23] B. Perthame, Transport Equations in Biology, Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2007. doi: 10.1007/978-3-7643-7842-4. [24] C. Piazzola, L. Tamellini and R. Tempone, A note on tools for prediction under uncertainty and identifiability of SIR-like dynamical systems for epidemiology, Math. Biosci., 332 (2021), 21pp. doi: 10.1016/j.mbs.2020.108514. [25] H. R. Thieme, Epidemic and demographic interaction in the spread of potentially fatal diseases in growing populations, Math. Biosci., 111 (1992), 99-130.  doi: 10.1016/0025-5564(92)90081-7. [26] H. Wackerhage, R. Everett, K. Krüger, M. Murgia and P. Simon, et al., Sport, exercise and COVID-19, the disease caused by the SARS-CoV-2 coronavirus, Dtsch. Z. Sportmed., 71 (2020), E1–E12. doi: 10.5960/dzsm.2020.441. [27] P. Yarsky, Using a genetic algorithm to fit parameters of a COVID-19 SEIR model for US states, Math. Comput. Simulation, 185 (2021), 687-695.  doi: 10.1016/j.matcom.2021.01.022.

show all references

##### References:
 [1] J. L. Aron, Mathematical modeling of immunity to malaria. Nonlinearity in biology and medicine, Math. Biosci., 90 (1988), 385-396.  doi: 10.1016/0025-5564(88)90076-4. [2] N. Bellomo, R. Bingham, M. A. J. Chaplain, G. Dosi, G. Forni and et al., A multiscale 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. [3] F. Brauer and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology, 2$^{nd}$ edition, Texts in Applied Mathematics, 40, Springer, New York, 2012. doi: 10.1007/978-1-4614-1686-9. [4] F. Brauer and P. van den Driessche, Models for transmission of disease with immigration of infectives, Math. Biosci., 171 (2001), 143-154.  doi: 10.1016/S0025-5564(01)00057-8. [5] R. M. Colombo and M. Garavello, Well posedness and control in a nonlocal SIR model, Appl. Math. Optim., 84 (2021), 737-771.  doi: 10.1007/s00245-020-09660-9. [6] R. M. Colombo, M. Garavello, F. Marcellini and E. Rossi, An age and space structured SIR model describing the Covid-19 pandemic, J. Math. Ind., 10 (2020), 20pp. doi: 10.1186/s13362-020-00090-4. [7] R. M. Colombo, M. Garavello, F. Marcellini and E. Rossi, IBVPs for inhomogeneous systems of balance laws in several space dimensions motivated by biology and epidemiology, preprint, 2021. [8] G. Dimarco, L. Pareschi, G. Toscani and M. Zanella, Wealth distribution under the spread of infectious diseases, Phys. Rev. E, 102 (2020), 14pp. doi: 10.1103/physreve.102.022303. [9] A. d'Onofrio, P. Manfredi and E. Salinelli, Bifurcation thresholds in an SIR model with information-dependent vaccination, Math. Model. Nat. Phenom., 2 (2007), 23-38.  doi: 10.1051/mmnp:2008009. [10] S. Ghosh and S. Bhattacharya, A data-driven understanding of COVID-19 dynamics using sequential genetic algorithm based probabilistic cellular automata, Appl. Soft Comput., 96 (2020). doi: 10.1016/j.asoc.2020.106692. [11] G. Giordano, F. Blanchini, R. Bruno, P. Colaneri, A. Di Filippo, A. Di Matteo and M. Colaneri, Modelling the COVID-19 epidemic and implementation of population-wide interventions in Italy, Nature Medicine, 26 (2020), 855-860.  doi: 10.1038/s41591-020-0883-7. [12] A. Godio, F. Pace and A. Vergnano, SEIR modeling of the Italian epidemic of SARS-CoV-2 using computational swarm intelligence, Internat. J. Environ. Res. Public Health, 17 (2020). doi: 10.3390/ijerph17103535. [13] D. Greenhalgh, Some results for an SEIR epidemic model with density dependence in the death rate, IMA J. Math. Appl. Med. Biol., 9 (1992), 67-106.  doi: 10.1093/imammb/9.2.67. [14] H. Inaba, Age-structured SIR epidemic model, in Age-Structured Population Dynamics in Demography and Epidemiology, Springer, 2017,287–331. [15] W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics. II. The problem of endemicity, Proc. Roy. Soc. Lond. A, 138 (1932), 55-83.  doi: 10.1098/rspa.1932.0171. [16] W. O. Kermack, A. G. McKendrick and G. T. Walker, A contribution to the mathematical theory of epidemics, Proc. Roy. Soc. Lond. A, 115 (1927), 700-721.  doi: 10.1098/rspa.1927.0118. [17] C. M. Kribs-Zaleta and J. X. Velasco-Hernández, A simple vaccination model with multiple endemic states, Math. Biosci., 164 (2000), 183-201.  doi: 10.1016/S0025-5564(00)00003-1. [18] G. Li and Z. Jin, Global stability of a SEIR epidemic model with infectious force in latent, infected and immune period, Chaos Solitons Fractals, 25 (2005), 1177-1184.  doi: 10.1016/j.chaos.2004.11.062. [19] M. Y. Li, H. L. Smith and L. Wang, Global dynamics an SEIR epidemic model with vertical transmission, SIAM J. Appl. Math., 62 (2001), 58-69.  doi: 10.1137/S0036139999359860. [20] X. Liu, Y. Takeuchi and S. Iwami, SVIR epidemic models with vaccination strategies, J. Theoret. Biol., 253 (2008), 1-11.  doi: 10.1016/j.jtbi.2007.10.014. [21] J. Mena-Lorca and H. W. Hethcote, Dynamic models of infectious diseases as regulators of population sizes, J. Math. Biol., 30 (1992), 693-716.  doi: 10.1007/BF00173264. [22] J. D. Murray, Mathematical Biology. I. An Introduction, 3$^{rd}$ edition, Interdisciplinary Applied Mathematics, 17, Springer-Verlag, New York, 2002. doi: 10.1007/b98868. [23] B. Perthame, Transport Equations in Biology, Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2007. doi: 10.1007/978-3-7643-7842-4. [24] C. Piazzola, L. Tamellini and R. Tempone, A note on tools for prediction under uncertainty and identifiability of SIR-like dynamical systems for epidemiology, Math. Biosci., 332 (2021), 21pp. doi: 10.1016/j.mbs.2020.108514. [25] H. R. Thieme, Epidemic and demographic interaction in the spread of potentially fatal diseases in growing populations, Math. Biosci., 111 (1992), 99-130.  doi: 10.1016/0025-5564(92)90081-7. [26] H. Wackerhage, R. Everett, K. Krüger, M. Murgia and P. Simon, et al., Sport, exercise and COVID-19, the disease caused by the SARS-CoV-2 coronavirus, Dtsch. Z. Sportmed., 71 (2020), E1–E12. doi: 10.5960/dzsm.2020.441. [27] P. Yarsky, Using a genetic algorithm to fit parameters of a COVID-19 SEIR model for US states, Math. Comput. Simulation, 185 (2021), 687-695.  doi: 10.1016/j.matcom.2021.01.022.
Solutions to (3)–(7)–(8) in the $4$ cases $T_* = 7, \, 21, \, 35, \, 49$
Diagrams of the solutions to (3)–(7)–(8) with a suspension in the vaccination campaign as detailed in (9) in the $4$ cases $T_* = 7, \, 21, \, 35, \, 49$
Diagrams of the solutions to (4)–(7)–(8)–(10). On the left with $\omega = 0.1$ and, on the right, with $\omega = 0.4$
Above, the integrations of (1) and (15), below on the left that of (16) (19). The rightmost diagram on the second line displays the total number of living individuals in the three cases, showing that, with respect to mortality, the ODE–PDE model (16) can be seen in some senses in the middle between the ODE models (1) and (15)
Above, from left to right, the integrations of Case $(i)$, Case $(ii)$ and Case $(iii)$ in (20) with parameters and data as prescribed in (19). Below, the corresponding choices of the $\rho$ function as detailed in (20). The differences in the displayed evolutions are due to the intra–compartmental dynamics in the $I$ population
Times necessary for the vaccination to provide immunity and corresponding casualties according to model (3)–(7)–(8). The initial total population is $100$
 $T_*$ (days) 1 7 14 21 28 35 42 49 Deaths: 0.28 0.32 0.37 0.43 0.49 0.56 0.63 0.7
 $T_*$ (days) 1 7 14 21 28 35 42 49 Deaths: 0.28 0.32 0.37 0.43 0.49 0.56 0.63 0.7
Times necessary for the vaccination to provide immunity and corresponding casualties, according to model (3)–(7)–(8), in the case vaccinations are suspended as detailed in (9). The initial total population is $100$
 $T_*$ 1 7 14 21 28 35 42 49 Deaths: 1.11 1.18 1.25 1.32 1.38 1.43 1.48 1.53
 $T_*$ 1 7 14 21 28 35 42 49 Deaths: 1.11 1.18 1.25 1.32 1.38 1.43 1.48 1.53
Populations in model (21) from [11]
 $S$ Susceptible healthy can be infected $I$ Infected asymptomatic infective undetected $D$ Diagnosed asymptomatic infective detected $A$ Ailing symptomatic infective undetected $R$ Recognized symptomatic infective detected $T$ Threatened acutely symptomatic infected detected $H$ Healed healthy immune $E$ Extinct
 $S$ Susceptible healthy can be infected $I$ Infected asymptomatic infective undetected $D$ Diagnosed asymptomatic infective detected $A$ Ailing symptomatic infective undetected $R$ Recognized symptomatic infective detected $T$ Threatened acutely symptomatic infected detected $H$ Healed healthy immune $E$ Extinct
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