Vaccination strategies through intra—compartmental dynamics

Rinaldo M. Colombo; Francesca Marcellini; Elena Rossi

doi:10.3934/nhm.2022012

Article Contents

2022, Volume 17, Issue 3: 385-400. Doi: 10.3934/nhm.2022012

This issue Previous Article A study of computational and conceptual complexities of compartment and agent based models Next Article Bi-fidelity stochastic collocation methods for epidemic transport models with uncertainties

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

Early access: March 2022

Published: June 2022

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

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Abstract

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.

Keywords:

Compartmental models,
vaccination strategies,
differential equations in epidemic modeling,
Covid-19 modeling.

Mathematics Subject Classification: Primary: 92D30; Secondary: 35L65.

Citation:

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Figure 1. Solutions to (3)–(7)–(8) in the $ 4 $ cases $ T_* = 7, \, 21, \, 35, \, 49 $

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Figure 2. 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 $

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Figure 3. Diagrams of the solutions to (4)–(7)–(8)–(10). On the left with $ \omega = 0.1 $ and, on the right, with $ \omega = 0.4 $

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Figure 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)

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Figure 5. 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

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Table 1. 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	0.70

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Table 2. 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

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Table 3. 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

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