Optimal Vaccination Policies for COVID-19 Considering Vaccine Doses Delays

GIANATTI, J.; LOTITO, P. A.; NEDER, J. G.; NUÑEZ, P. M.; PARENTE, L. A.

doi:10.5540/tcam.2022.024.01.00121

J. GIANATTI

CIFASIS-CONICET, Facultad de Ciencias Exactas, Ingeniería y Agrimensura, Universidad Nacional de Rosario, Ocampo y Esmeralda, Rosario, Argentina - E-mail: gianatti@cifasis-conicet.gov.ar

https://orcid.org/0000-0002-7279-1097

P. A. LOTITO

Universidad Nacional del Centro de la Provincia de Buenos Aires, Facultad de Ciencias Exactas, PLADEMA, Tandil, Argentina. - E-mail: plotito@exa.unicen.edu.ar

CONICET, Argentina.

https://orcid.org/0000-0002-7579-6942

J. G. NEDER

CIFASIS-CONICET, Facultad de Ciencias Exactas, Ingeniería y Agrimensura, Universidad Nacional de Rosario, Ocampo y Esmeralda, Rosario, Argentina - E-mail: jneder@fceia.unr.edu.ar

https://orcid.org/0000-0001-7434-0516

P. M. NUÑEZ

Universidad Nacional del Centro de la Provincia de Buenos Aires, Facultad de Ciencias Exactas, PLADEMA, Tandil, Argentina - Email: penunez@alumnos.exa.unicen.edu.ar

https://orcid.org/0000-0001-6466-1981

L. A. PARENTE ^**Corresponding author: Lisandro Armando Parente - E-mail: parente@cifasis-conicet.gov.ar

CIFASIS-CONICET, Facultad de Ciencias Exactas, Ingeniería y Agrimensura, Universidad Nacional de Rosario, Ocampo y Esmeralda, Rosario, Argentina - E-mail: parente@cifasis-conicet.gov.ar

https://orcid.org/0000-0003-3740-4546

*Corresponding author: Lisandro Armando Parente - E-mail: parente@cifasis-conicet.gov.ar

Variables		Coefficients
S	susceptible	β	transmission coefficient
E	exposed	T _EI	average time of incubation
I	infected	T _ID	average time to detection.
D	detected	T _DR	average recovery time for detected
H	hospitalized	T _DH	average time between detection and hospitalization
R	recovered	T _HR	average hospitalization time for recovering patients
F	dead	C _H	proportion of detected patients to be hospitalized
		C _F	proportion of hospitalized patients who die

Parameter	Value	Initial cond.	Value
β	0.310	S(0)	126347.102
T _EI	4.000	E(0)	187.717
T _ID	5.300	I(0)	196.708
T _DR	7.050	D(0)	177.337
T _DH	1.750	H(0)	40.257
T _HR	6.100	R(0)	10026.743
T _HF	11.700	F(0)	183.477
C _H	0.068
C _F	0.240

	Variables
S	susceptible population
v _a	susceptible vaccinated with the first dose since a days ago
E	exposed population without the effect of vaccine
E _V	exposed population under the effect of vaccine
I	infected population without the effect of vaccine
I _V	infected population under the effect of vaccine
D	detected population without the effect of vaccine
D _V	detected population under the effect of vaccine
H	hospitalized population
R	recovered population
F	dead population

Variables	Test 1	Test 2	Test 3	Test 4	Test 5
S	50409.371	28432.943	16163.757	25917.113	22569.913
$\sum_{a = 0}^{83} v_{a}$	0.000	3621.852	28161.028	3616.113	8455.930
E	5.486	0.290	0.276	0.251	0.258
E _v	0.000	0.017	0.063	0.009	0.001
I	9.727	0.691	0.687	0.619	0.617
I _v	0.000	0.057	0.128	0.032	0.002
D	14.874	1.452	1.463	1.348	1.298
D _v	0.000	0.169	0.266	0.098	0.012
H	5.927	0.834	0.843	0.800	0.750
R	84069.956	89893.511	91147.290	105851.179	104323.154
F	2644.001	1683.536	1683.539	1771.779	1807.405

Variables	Test 1	Test 2	Test 3	Test 4	Test
S	60357.753	51399.744	36953.365	49153.112	45272.549
$\sum_{a = 0}^{83} v_{a}$	0.000	2253.590	31161.275	2625.841	9377.586
E	255.172	18.329	18.047	19.426	22.372
E _v	0.000	0.437	2.378	0.329	0.052
I	406.576	31.397	31.606	33.914	37.985
I _v	0.000	1.122	3.521	0.928	0.055
D	547.647	45.745	46.372	50.393	54.774
D _v	0.000	2.370	4.954	2.099	0.171
H	184.436	16.898	17.178	19.035	20.045
R	73146.338	67629.179	67971.886	84235.667	81340.738
F	2261.420	948.700	948.7587	1018.596	1033.013

\{\begin{array}{rcl} \dot{S} (t) & = & - \frac{β}{N} S (t) I (t), \\ \dot{E} (t) & = & \frac{β}{N} S (t) I (t) - \frac{1}{T_{E I}} E (t), \\ \dot{I} (t) & = & \frac{1}{T_{E I}} E (t) - \frac{1}{T_{I D}} I (t), \\ \dot{D} (t) & = & \frac{1}{T_{I D}} I (t) - \frac{C_{H}}{T_{D H}} D (t) - \frac{(1 - C_{H})}{T_{D R}} D (t), \\ \dot{H} (t) & = & \frac{C_{H}}{T_{D H}} D (t) - \frac{C_{F}}{T_{H F}} H (t) - \frac{(1 - C_{F})}{T_{H R}} H (t), \\ \dot{R} (t) & = & \frac{(1 - C_{H})}{T_{D R}} D (t) + \frac{1 - C_{F}}{T_{H R}} H (t), \\ \dot{F} (t) & = & \frac{C_{F}}{T_{H F}} H (t) . \end{array}

\{\begin{array}{rcll} S^{k + 1} & = & (1 - β (I^{k} + I_{V}^{k}) - u_{1}^{k}) S^{k} \\ v_{0}^{k + 1} & = & u_{1}^{k} S^{k} \\ v_{a + 1}^{k + 1} & = & (1 - β (I^{k} + I_{V}^{k})) v_{a}^{k}, a = 0, \dots, 21 \\ v_{a + 1}^{k + 1} & = & (1 - β_{V} (I^{k} + I_{V}^{k}) - u_{2}^{k}) v_{a}^{k}, a = 22, \dots, 82 \\ E^{k + 1} & = & (1 - \frac{1}{T_{E}}) E^{k} + β (I^{k} + I_{V}^{k}) (S^{k} + \sum_{a = 0}^{21} v_{a}^{k}) \\ E_{V}^{k + 1} & = & (1 - \frac{1}{T_{E}}) E_{V}^{k} + β_{V} (I^{k} + I_{V}^{k}) \sum_{a = 22}^{82} v_{a}^{k} \\ I^{k + 1} & = & (1 - \frac{1}{T_{I D}}) I^{k} + \frac{1}{T_{E}} E^{k} \\ I_{V}^{k + 1} & = & (1 - \frac{1}{T_{I D}}) I_{V}^{k} + \frac{1}{T_{E}} E_{V}^{k} \\ D^{k + 1} & = & (1 - \frac{C_{H}}{T_{D H}} - \frac{(1 - C_{H})}{T_{D R}}) D^{k} + \frac{1}{T_{I D}} I^{k} \\ D_{V}^{k + 1} & = & (1 - \frac{C_{H}^{V}}{T_{D H}} - \frac{(1 - C_{H}^{V})}{T_{D R}}) D_{V}^{k} + \frac{1}{T_{I D}} I_{V}^{k} \\ H^{k + 1} & = & (1 - \frac{C_{F}}{T_{H F}} - \frac{(1 - C_{F})}{T_{H R}}) H^{k} + \frac{C_{H}}{T_{D H}} D^{k} + \frac{C_{H}^{V}}{T_{D H}} D_{V}^{k} \\ F^{k + 1} & = & F^{k} + \frac{C_{F}}{T_{H F}} H^{k} \\ R^{k + 1} & = & R^{k} + \frac{(1 - C_{H})}{T_{D R}} D^{k} + \frac{(1 - C_{H}^{V})}{T_{D R}} D_{V}^{k} + \frac{(1 - C_{F})}{T_{H R}} H^{k} + \sum_{a = 22}^{82} u_{2}^{k} v_{a}^{k} + v_{83}^{k} . \end{array}

\{\begin{array}{rcl} S^{k + 1} & = & (1 - w^{k} β (I^{k} + I_{V}^{k}) - u_{1}^{k}) S^{k} \\ v_{a + 1}^{k + 1} & = & (1 - w^{k} β (I^{k} + I_{V}^{k})) v_{a}^{k}, a = 0, \dots, 21 \\ v_{a + 1}^{k + 1} & = & (1 - w^{k} β_{V} (I^{k} + I_{V}^{k}) - u_{2}^{k}) v_{a}^{k}, a = 22, \dots, 82 \\ E^{k + 1} & = & (1 - \frac{1}{T_{E}}) E^{k} + w^{k} β (I^{k} + I_{V}^{k}) (S^{k} + \sum_{a = 0}^{21} v_{a}^{k}) \\ E_{V}^{k + 1} & = & (1 - \frac{1}{T_{E}}) E_{V}^{k} + w^{k} β_{V} (I^{k} + I_{V}^{k}) \sum_{a = 22}^{82} v_{a}^{k} . \end{array}

[1] *Corresponding author: Lisandro Armando Parente - E-mail: parente@cifasis-conicet.gov.ar

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Optimal Vaccination Policies for COVID-19 Considering Vaccine Doses Delays

ABSTRACT