Different Approaches to the Modelling of COVID-19

In this paper some innovative aspects of the mathematical modelling of classic epidemiology problems for the study of models related to the COVID-19 pandemic dynamics are presented. In addition, they are compared to real-world data using numerical methods in order to approximate the solutions. One of these models includes a non-transmitting compartment and another one, a delay-differential equation in the SIR-type method. Finally, a comparative discussion of the results is also presented.

Keywords:
COVID-19; mathematical modelling; basic reproductive number; mathematical epidemiology; nonlinear systems of ODE

Authorship

J.F. C. A. MEYER ^**Corresponding author: J. F. C. A. Meyer - E-mail: jmeyer@unicamp.br

Department of Applied Mathematics, IMECC, University of Campinas, R. Sérgio Buarque de Holanda, 651, 13083-859, Campinas, SP, Brazil - E-mail: jmeyer@unicamp.brUniversity of CampinasBrazilCampinas, SP, BrazilDepartment of Applied Mathematics, IMECC, University of Campinas, R. Sérgio Buarque de Holanda, 651, 13083-859, Campinas, SP, Brazil - E-mail: jmeyer@unicamp.br

http://orcid.org/0000-0001-8002-7075

M. LIMA

Department of Applied Mathematics, IMECC, University of Campinas, R. Sérgio Buarque de Holanda, 651, 13083-859, Campinas, SP, Brazil - E-mail: marina@ime.unicamp.brUniversity of CampinasBrazilCampinas, SP, BrazilDepartment of Applied Mathematics, IMECC, University of Campinas, R. Sérgio Buarque de Holanda, 651, 13083-859, Campinas, SP, Brazil - E-mail: marina@ime.unicamp.br

http://orcid.org/0000-0002-8520-4306

C. C. ESPITIA

Department of Applied Mathematics, IMECC, University of Campinas, R. Sérgio Buarque de Holanda, 651, 13083-859, Campinas, SP, Brazil - E-mail: espitiacristian@gmail.comUniversity of CampinasBrazilCampinas, SP, BrazilDepartment of Applied Mathematics, IMECC, University of Campinas, R. Sérgio Buarque de Holanda, 651, 13083-859, Campinas, SP, Brazil - E-mail: espitiacristian@gmail.com

http://orcid.org/0000-0001-5500-0460

F. LONGO

Department of Applied Mathematics, IMECC, University of Campinas, R. Sérgio Buarque de Holanda, 651, 13083-859, Campinas, SP, Brazil - E-mail: longo@ime.unicamp.brUniversity of CampinasBrazilCampinas, SP, BrazilDepartment of Applied Mathematics, IMECC, University of Campinas, R. Sérgio Buarque de Holanda, 651, 13083-859, Campinas, SP, Brazil - E-mail: longo@ime.unicamp.br

http://orcid.org/0000-0002-2868-3937

B. LAIATE

Department of Applied Mathematics, IMECC, University of Campinas, R. Sérgio Buarque de Holanda, 651, Campinas, SP, 13083-859, Brazil - E-mail: beatrizlaiate@gmail.comUniversity of CampinasBrazilCampinas, SP, BrazilDepartment of Applied Mathematics, IMECC, University of Campinas, R. Sérgio Buarque de Holanda, 651, Campinas, SP, 13083-859, Brazil - E-mail: beatrizlaiate@gmail.com

http://orcid.org/0000-0002-7312-4843

A. N. GOIS

Federal Institute of Alagoas, Campus Piranhas, Av. Sergipe, S/N, Piranhas, AL, 57460-000, Brazil - E-mail: aedson.gois@gmail.comFederal Institute of AlagoasBrazilPiranhas, AL, BrazilFederal Institute of Alagoas, Campus Piranhas, Av. Sergipe, S/N, Piranhas, AL, 57460-000, Brazil - E-mail: aedson.gois@gmail.com

http://orcid.org/0000-0002-9118-6247

C. F. D. KUNZ

Frankfurt Institute for Advanced Studies, Goethe University Frankfurt, Ruth-Moufang-Straße 1, 60438, Frankfurt am Main, Germany - E-mail: camileknz@gmail.comGoethe University FrankfurtGermanyFrankfurt am Main, GermanyFrankfurt Institute for Advanced Studies, Goethe University Frankfurt, Ruth-Moufang-Straße 1, 60438, Frankfurt am Main, Germany - E-mail: camileknz@gmail.com

http://orcid.org/0000-0002-7193-9002

*Corresponding author: J. F. C. A. Meyer - E-mail: jmeyer@unicamp.br

SCIMAGO INSTITUTIONS RANKINGS

Figures | Formulas

Figures (11)
Formulas (21)

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Figure 1
Diagram for SIRD model

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Figure 2
Diagram for SEIRD model

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Figure 3
Diagram for SCAIRD model

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Figure 4
Diagram for SCEIRD model

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Figure 5
Diagram for SCEAIRD model

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Figure 6
Active cases of COVID-19 in Brazil. This scenario was generated using data from²⁰20 W.H.O. WHO. Coronavirus disease (COVID-19) Situation Report - 204. URL URL https://covid19.who.int . [accessed 04-November-2020].
https://covid19.who.int... and the presented classic SIRD model with parameters: α _S = 4.96 × 10⁻⁹ (person ⁻¹ day ⁻¹ ) and η = 0.02(day ⁻¹ ), both estimated, and γ _I = 0.97(day ⁻¹ ) and µ _I = 0.03(day ⁻¹ ) from²¹21 Worldometers. Coronavirus incubation period. URL URL https://www.worldometers.info/coronavirus/coronavirus-incubation-period . [accessed 04-November-2020].
https://www.worldometers.info/coronaviru... . In order to obtain coherent values for unknown parameters, the solutions were adjusted to the existing data provided in²¹21 Worldometers. Coronavirus incubation period. URL URL https://www.worldometers.info/coronavirus/coronavirus-incubation-period . [accessed 04-November-2020].
https://www.worldometers.info/coronaviru... . The estimate of R ₀ obtained is 1.08.

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Figure 7
Active cases of COVID-19 in Brazil. This scenario was generated using data from²⁰20 W.H.O. WHO. Coronavirus disease (COVID-19) Situation Report - 204. URL URL https://covid19.who.int . [accessed 04-November-2020].
https://covid19.who.int... and the proposed SEIRD model with parameters: α _S = 5.16 × 10⁻⁹ (person ⁻¹ day ⁻¹ ), α _E = 0.65(day ⁻¹ ) and η = 0.02(day ⁻¹ ), which were estimated, and γ _I = 0.97(day ⁻¹ ) and µ _I = 0.03(day ⁻¹ ) from²¹21 Worldometers. Coronavirus incubation period. URL URL https://www.worldometers.info/coronavirus/coronavirus-incubation-period . [accessed 04-November-2020].
https://www.worldometers.info/coronaviru... . In order to obtain coherent values for unknown parameters, the solutions were adjusted to the existing data provided in²¹21 Worldometers. Coronavirus incubation period. URL URL https://www.worldometers.info/coronavirus/coronavirus-incubation-period . [accessed 04-November-2020].
https://www.worldometers.info/coronaviru... . The estimate of R ₀ is 1.13.

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Figure 8
Active cases of COVID-19 in Brazil. This scenario was generated using data from²⁰20 W.H.O. WHO. Coronavirus disease (COVID-19) Situation Report - 204. URL URL https://covid19.who.int . [accessed 04-November-2020].
https://covid19.who.int... and the proposed SCAIRD model with parameters: α _A = 4.71 × 10⁻⁹ (person ⁻¹ day ⁻¹ ), α _I = 7.16 × 10⁻⁹ (person ⁻¹ day ⁻¹ ), α _C = 0.4(day ⁻¹ ), β _A = 4.71 × 10⁻¹⁰ (person ⁻¹ day ⁻¹ ), β _I = 2.83 × 10⁻⁹ (person ⁻¹ day ⁻¹ ), ε _C = 0.7(day ⁻¹ ), γ _A = 0.4(day ⁻¹ ), δ _A = 0.5(day ⁻¹ ), µ _A = 0.003(day ⁻¹ ) and η = 0.02(day ⁻¹ ), which were estimated, and τ = 5.2(day), γ _I = 0.97(day ⁻¹ ) and µ _I = 0.03(day ⁻¹ ) from²¹21 Worldometers. Coronavirus incubation period. URL URL https://www.worldometers.info/coronavirus/coronavirus-incubation-period . [accessed 04-November-2020].
https://www.worldometers.info/coronaviru... . In order to obtain coherent values for unknown parameters, the solutions were adjusted to the existing data provided in²¹21 Worldometers. Coronavirus incubation period. URL URL https://www.worldometers.info/coronavirus/coronavirus-incubation-period . [accessed 04-November-2020].
https://www.worldometers.info/coronaviru... . The estimate of R ₀ is 1.80.

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Figure 9
Active cases of COVID-19 in Brazil. This scenario was generated using data from²⁰20 W.H.O. WHO. Coronavirus disease (COVID-19) Situation Report - 204. URL URL https://covid19.who.int . [accessed 04-November-2020].
https://covid19.who.int... and the proposed SCEIRD model with parameters: α _S = 7.35 × 10⁻⁹ (person ⁻¹ day ⁻¹ ), α _E = 4.71 × 10⁻⁹ (person ⁻¹ day ⁻¹ ), α _C = 0.4(day ⁻¹ ), ε _C = 0.6(day ⁻¹ ), β _I = 0.5(day ⁻¹ ), δ _E = 0.1(day ⁻¹ ) and η = 0.02(day ⁻¹ ), which were estimated, and γ _I = 0.97(day ⁻¹ ) and µ _I = 0.03(day ⁻¹ ) from²¹21 Worldometers. Coronavirus incubation period. URL URL https://www.worldometers.info/coronavirus/coronavirus-incubation-period . [accessed 04-November-2020].
https://www.worldometers.info/coronaviru... . In order to obtain coherent values for unknown parameters, the solutions were adjusted to the existing data provided in²¹21 Worldometers. Coronavirus incubation period. URL URL https://www.worldometers.info/coronavirus/coronavirus-incubation-period . [accessed 04-November-2020].
https://www.worldometers.info/coronaviru... . The estimate of R ₀ is 1.33.

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Figure 10
Active cases of COVID-19 in Brazil. This scenario was generated using data from²⁰20 W.H.O. WHO. Coronavirus disease (COVID-19) Situation Report - 204. URL URL https://covid19.who.int . [accessed 04-November-2020].
https://covid19.who.int... and the proposed SCEAIRD model with parameters: α _A = 4.00 × 10⁻⁹(person ⁻¹ day ⁻¹), α _I = 6.36 × 10⁻⁹(person ⁻¹ day ⁻¹), α _C = 0.4(day ⁻¹), ε _C = 0.8(day ⁻¹), β _A = 1.88 × 10⁻⁹(person ⁻¹ day ⁻¹), β _I = 3.77 × 10⁻⁹(person ⁻¹ day ⁻¹), δ _A = 0.4(day ⁻¹), γ _A = 0.55(day ⁻¹), λ _A = 0.7(day ⁻¹) and η = 0.02(day ⁻¹), which were estimated, and γ _I = 0.97(day ⁻¹) and µ _I = 0.03(day ⁻¹) from²¹21 Worldometers. Coronavirus incubation period. URL URL https://www.worldometers.info/coronavirus/coronavirus-incubation-period . [accessed 04-November-2020].
https://www.worldometers.info/coronaviru... . In order to obtain coherent values for unknown parameters, the solutions were adjusted to the existing data provided in²¹21 Worldometers. Coronavirus incubation period. URL URL https://www.worldometers.info/coronavirus/coronavirus-incubation-period . [accessed 04-November-2020].
https://www.worldometers.info/coronaviru... . The estimate of R ₀ is 1.46.

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Figure 11
Active cases of COVID-19 in Brazil and the comparison between the models presented in this work.

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(2.1)

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Figure 1 Diagram for SIRD model

Figure 2 Diagram for SEIRD model

Figure 3 Diagram for SCAIRD model

Figure 4 Diagram for SCEIRD model

Figure 5 Diagram for SCEAIRD model

Figure 6 Active cases of COVID-19 in Brazil. This scenario was generated using data from²⁰20 W.H.O. WHO. Coronavirus disease (COVID-19) Situation Report - 204. URL URL https://covid19.who.int . [accessed 04-November-2020].
https://covid19.who.int... and the presented classic SIRD model with parameters: α _S = 4.96 × 10⁻⁹ (person ⁻¹ day ⁻¹ ) and η = 0.02(day ⁻¹ ), both estimated, and γ _I = 0.97(day ⁻¹ ) and µ _I = 0.03(day ⁻¹ ) from²¹21 Worldometers. Coronavirus incubation period. URL URL https://www.worldometers.info/coronavirus/coronavirus-incubation-period . [accessed 04-November-2020].
https://www.worldometers.info/coronaviru... . In order to obtain coherent values for unknown parameters, the solutions were adjusted to the existing data provided in²¹21 Worldometers. Coronavirus incubation period. URL URL https://www.worldometers.info/coronavirus/coronavirus-incubation-period . [accessed 04-November-2020].
https://www.worldometers.info/coronaviru... . The estimate of R ₀ obtained is 1.08.

Figure 7 Active cases of COVID-19 in Brazil. This scenario was generated using data from²⁰20 W.H.O. WHO. Coronavirus disease (COVID-19) Situation Report - 204. URL URL https://covid19.who.int . [accessed 04-November-2020].
https://covid19.who.int... and the proposed SEIRD model with parameters: α _S = 5.16 × 10⁻⁹ (person ⁻¹ day ⁻¹ ), α _E = 0.65(day ⁻¹ ) and η = 0.02(day ⁻¹ ), which were estimated, and γ _I = 0.97(day ⁻¹ ) and µ _I = 0.03(day ⁻¹ ) from²¹21 Worldometers. Coronavirus incubation period. URL URL https://www.worldometers.info/coronavirus/coronavirus-incubation-period . [accessed 04-November-2020].
https://www.worldometers.info/coronaviru... . In order to obtain coherent values for unknown parameters, the solutions were adjusted to the existing data provided in²¹21 Worldometers. Coronavirus incubation period. URL URL https://www.worldometers.info/coronavirus/coronavirus-incubation-period . [accessed 04-November-2020].
https://www.worldometers.info/coronaviru... . The estimate of R ₀ is 1.13.

Figure 8 Active cases of COVID-19 in Brazil. This scenario was generated using data from²⁰20 W.H.O. WHO. Coronavirus disease (COVID-19) Situation Report - 204. URL URL https://covid19.who.int . [accessed 04-November-2020].
https://covid19.who.int... and the proposed SCAIRD model with parameters: α _A = 4.71 × 10⁻⁹ (person ⁻¹ day ⁻¹ ), α _I = 7.16 × 10⁻⁹ (person ⁻¹ day ⁻¹ ), α _C = 0.4(day ⁻¹ ), β _A = 4.71 × 10⁻¹⁰ (person ⁻¹ day ⁻¹ ), β _I = 2.83 × 10⁻⁹ (person ⁻¹ day ⁻¹ ), ε _C = 0.7(day ⁻¹ ), γ _A = 0.4(day ⁻¹ ), δ _A = 0.5(day ⁻¹ ), µ _A = 0.003(day ⁻¹ ) and η = 0.02(day ⁻¹ ), which were estimated, and τ = 5.2(day), γ _I = 0.97(day ⁻¹ ) and µ _I = 0.03(day ⁻¹ ) from²¹21 Worldometers. Coronavirus incubation period. URL URL https://www.worldometers.info/coronavirus/coronavirus-incubation-period . [accessed 04-November-2020].
https://www.worldometers.info/coronaviru... . In order to obtain coherent values for unknown parameters, the solutions were adjusted to the existing data provided in²¹21 Worldometers. Coronavirus incubation period. URL URL https://www.worldometers.info/coronavirus/coronavirus-incubation-period . [accessed 04-November-2020].
https://www.worldometers.info/coronaviru... . The estimate of R ₀ is 1.80.

Figure 9 Active cases of COVID-19 in Brazil. This scenario was generated using data from²⁰20 W.H.O. WHO. Coronavirus disease (COVID-19) Situation Report - 204. URL URL https://covid19.who.int . [accessed 04-November-2020].
https://covid19.who.int... and the proposed SCEIRD model with parameters: α _S = 7.35 × 10⁻⁹ (person ⁻¹ day ⁻¹ ), α _E = 4.71 × 10⁻⁹ (person ⁻¹ day ⁻¹ ), α _C = 0.4(day ⁻¹ ), ε _C = 0.6(day ⁻¹ ), β _I = 0.5(day ⁻¹ ), δ _E = 0.1(day ⁻¹ ) and η = 0.02(day ⁻¹ ), which were estimated, and γ _I = 0.97(day ⁻¹ ) and µ _I = 0.03(day ⁻¹ ) from²¹21 Worldometers. Coronavirus incubation period. URL URL https://www.worldometers.info/coronavirus/coronavirus-incubation-period . [accessed 04-November-2020].
https://www.worldometers.info/coronaviru... . In order to obtain coherent values for unknown parameters, the solutions were adjusted to the existing data provided in²¹21 Worldometers. Coronavirus incubation period. URL URL https://www.worldometers.info/coronavirus/coronavirus-incubation-period . [accessed 04-November-2020].
https://www.worldometers.info/coronaviru... . The estimate of R ₀ is 1.33.

Figure 10 Active cases of COVID-19 in Brazil. This scenario was generated using data from²⁰20 W.H.O. WHO. Coronavirus disease (COVID-19) Situation Report - 204. URL URL https://covid19.who.int . [accessed 04-November-2020].
https://covid19.who.int... and the proposed SCEAIRD model with parameters: α _A = 4.00 × 10⁻⁹(person ⁻¹ day ⁻¹), α _I = 6.36 × 10⁻⁹(person ⁻¹ day ⁻¹), α _C = 0.4(day ⁻¹), ε _C = 0.8(day ⁻¹), β _A = 1.88 × 10⁻⁹(person ⁻¹ day ⁻¹), β _I = 3.77 × 10⁻⁹(person ⁻¹ day ⁻¹), δ _A = 0.4(day ⁻¹), γ _A = 0.55(day ⁻¹), λ _A = 0.7(day ⁻¹) and η = 0.02(day ⁻¹), which were estimated, and γ _I = 0.97(day ⁻¹) and µ _I = 0.03(day ⁻¹) from²¹21 Worldometers. Coronavirus incubation period. URL URL https://www.worldometers.info/coronavirus/coronavirus-incubation-period . [accessed 04-November-2020].
https://www.worldometers.info/coronaviru... . In order to obtain coherent values for unknown parameters, the solutions were adjusted to the existing data provided in²¹21 Worldometers. Coronavirus incubation period. URL URL https://www.worldometers.info/coronavirus/coronavirus-incubation-period . [accessed 04-November-2020].
https://www.worldometers.info/coronaviru... . The estimate of R ₀ is 1.46.

Figure 11 Active cases of COVID-19 in Brazil and the comparison between the models presented in this work.

S (t) + I (t) + R (t) + D (t) = N = constant .

\frac{d S (t)}{d t} = - α_{S} S (t) I (t) + η R (t); \frac{d I (t)}{d t} = α_{S} S (t) I (t) - γ_{I} I (t) - μ_{I} I (t); \frac{d R (t)}{d t} = γ_{I} I (t) - η R (t); \frac{d D (t)}{d t} = μ_{I} I (t) .

(2.1)

R_{0} = \frac{α_{S} S_{0}}{γ_{I} + μ_{I}} .

S (t) + E (t) + I (t) + R (t) + D (t) = N = constant .

\frac{d S (t)}{d t} = - α_{S} S (t) I (t) + η R (t); \frac{d E (t)}{d t} = α_{S} S (t) I (t) - α_{E} E (t); \frac{d I (t)}{d t} = α_{E} E (t) - γ_{I} I (t) - μ_{I} I (t); \frac{d R (t)}{d t} = γ_{I} I (t) - η R (t); \frac{d D (t)}{d t} = μ_{I} I (t) .

F (x) = (\begin{matrix} α_{S} S I \\ 0 \end{matrix}), V (x) = (\begin{matrix} α_{E} E \\ - α_{E} E + (γ_{I} + μ_{I}) I \end{matrix}), F = (\begin{matrix} 0 & α_{S} S_{0} \\ 0 & 0 \end{matrix}) V = (\begin{matrix} α_{E} & 0 \\ - α_{E} & γ_{I} + μ_{I} \end{matrix}), V^{- 1} = (\begin{matrix} \frac{1}{α_{E}} & 0 \\ \frac{1}{γ_{I} + μ_{I}} & \frac{1}{γ_{I} + μ_{I}} \end{matrix}), F V^{- 1} = (\begin{matrix} \frac{α_{S} S_{0}}{γ_{I} + μ_{I}} & \frac{α_{S} S_{0}}{γ_{I} + μ_{I}} \\ 0 & 0 \end{matrix}) .

R_{0} = \frac{α_{S} S_{0}}{γ_{I} + μ_{I}},

S (t) + C (t) + A (t) + I (t) + R (t) + D (t) = N = constant .

\frac{d S (t)}{d t} = η R (t) - α_{A} S (t) A (t - τ) - α_{I} S (t) I (t) - α_{C} S (t) + ϵ_{C} C (t); \frac{d C (t)}{d t} = α_{C} S (t) - β_{A} C (t) A (t - τ) - β_{I} C (t) I (t) - ϵ_{C} C (t); \frac{d A (t)}{d t} = α_{A} S (t) A (t - τ) + α_{I} S (t) I (t) + β_{A} C (t) A (t - τ) + β_{I} C (t) I (t) - γ_{A} A (t) - δ_{A} A (t) - μ_{A} A (t); \frac{d I (t)}{d t} = γ_{A} A (t) - δ_{R} I (t) - μ_{I} I (t); \frac{d R (t)}{d t} = δ_{A} A (t) + δ_{R} I (t) - η R (t); \frac{d D (t)}{d t} = μ_{I} I (t) + μ_{A} A (t) .

J (E_{0}) = (\begin{matrix} - α_{C} & ϵ_{C} & - α_{A} S_{0} & - α_{I} S_{0} & η & 0 \\ α_{C} & - ϵ_{C} & 0 & 0 & 0 & 0 \\ 0 & 0 & α_{A} S_{0} - (γ_{A} + δ_{A} + μ_{A}) & α_{I} S_{0} & 0 & 0 \\ 0 & 0 & γ_{A} & - (δ_{R} + μ_{I}) & 0 & 0 \\ 0 & 0 & δ_{A} & δ_{R} & - η & 0 \\ 0 & 0 & μ_{A} & μ_{I} & 0 & 0 \end{matrix}) .

p (λ) = λ (λ + η) [λ^{2} + λ (ϵ_{C} + α_{C}) + ϵ_{C} (α_{A} + α_{C})] [λ^{2} + (- S_{0} α_{A} + γ_{A} + δ_{A} + δ_{R} + μ_{I} + μ_{A}) λ + ((δ_{R} + μ_{I}) (γ_{A} + δ_{A} + μ_{A}) - S_{0} (α_{I} γ_{A} + α_{A} (δ_{R} + μ_{I})))

\Rightarrow p (λ) = λ (λ + η) [λ^{2} + λ (ϵ_{C} + α_{C}) + ϵ_{C} (α_{A} + α_{C})] [λ^{2} + (- S_{0} α_{A} + γ_{A} + δ_{A} + δ_{R} + μ_{I} + μ_{A}) λ + ((δ_{R} + μ_{I}) (γ_{A} + δ_{A} + μ_{A})) (1 - \frac{S_{0} (α_{I} γ_{A} + α_{A} (δ_{R} + μ_{I}))}{(δ_{R} + μ_{I}) (γ_{A} + δ_{A} + μ_{A})})]

R_{0} = \frac{[α_{I} γ_{A} + α_{A} (δ_{R} + μ_{I})] S_{0}}{(δ_{R} + μ_{I}) (γ_{A} + δ_{A} + μ_{A})} .

S (t) + C (t) + E (t) + I (t) + R (t) + D (t) = N = constant .

\frac{d S (t)}{d t} = ϵ_{C} C (t) - α_{C} S (t) - α_{S} S (t) I (t) + η R (t); \frac{d C (t)}{d t} = α_{C} S (t) - ϵ_{C} C (t) - α_{E} C (t) I (t); \frac{d E (t)}{d t} = α_{S} S (t) I (t) + α_{E} C (t) I (t) - β_{I} E (t) - δ_{E} E (t); \frac{d I (t)}{d t} = β_{I} E (t) - γ_{I} I (t) - μ_{I} I (t); \frac{d R (t)}{d t} = γ_{I} I (t) + δ_{E} E (t) - η R (t); \frac{d D (t)}{d t} = μ_{I} I (t) .

F (x) = (\begin{matrix} α_{S} S I + α_{E} C I \\ 0 \end{matrix}), V (x) = (\begin{matrix} (β_{I} + δ_{E}) E \\ - β_{I} E + (γ_{I} + μ_{I}) I \end{matrix}), F = (\begin{matrix} 0 & α_{S} S_{0} \\ 0 & 0 \end{matrix}) V = (\begin{matrix} β_{I} + δ_{E} & 0 \\ - β_{I} & γ_{I} + μ_{I} \end{matrix}), F V^{- 1} = (\begin{matrix} \frac{β_{I} α_{S} S_{0}}{(β_{I} + δ_{E}) (γ_{I} + μ_{I})} & \frac{α_{S} S_{0}}{γ_{I} + μ_{I}} \\ 0 & 0 \end{matrix}),

R_{0} = \frac{β_{I} α_{S} S_{0}}{(β_{I} + δ_{E}) (γ_{I} + μ_{I})} .

S (t) + C (t) + E (t) + A (t) + I (t) + R (t) + D (t) = N = constant .

\frac{d S (t)}{d t} = ϵ_{C} C (t) - α_{C} S (t) - α_{A} S (t) A (t) - α_{I} S (t) I (t) + η R (t) \frac{d C (t)}{d t} = α_{C} S (t) - ϵ_{C} C (t) - β_{A} C (t) A (t) - β_{I} C (t) I (t) \frac{d E (t)}{d t} = α_{A} S (t) A (t) + α_{I} S (t) I (t) + β_{A} C (t) A (t) + β_{I} C (t) I (t) - δ_{A} E (t) \frac{d A (t)}{d t} = δ_{A} E (t) - γ_{A} A (t) - λ_{A} A (t) - μ_{A} A (t) \frac{d I (t)}{d t} = λ_{A} A (t) - γ_{I} I (t) - μ_{I} I (t) \frac{d R (t)}{d t} = γ_{A} A (t) + γ_{I} I (t) - η R (t) \frac{d D (t)}{d t} = μ_{I} I (t) + μ_{A} A (t) .

F (x) = (\begin{matrix} α_{A} S A + α_{I} S I + β_{A} C A + β_{I} C I \\ 0 \\ 0 \end{matrix}), V (x) = (\begin{matrix} δ_{A} E \\ - δ_{A} E + (γ_{A} + λ_{A} + μ_{A}) A \\ - λ_{A} A + (γ_{I} + μ_{I}) I \end{matrix}) F = (\begin{matrix} 0 & α_{A} S_{0} & α_{I} S_{0} \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}), V = (\begin{matrix} δ_{A} & 0 & 0 \\ - δ_{A} γ_{A} & + λ_{A} + μ_{A} & 0 \\ 0 & - λ_{A} & γ_{I} + μ_{I} \end{matrix}), F V^{- 1} = (\begin{matrix} \frac{[α_{I} λ_{A} + α_{A} (γ_{I} + μ_{I})] S_{0}}{(γ_{I} + μ_{I}) (γ_{A} + λ_{A} + μ_{A})} & \frac{[α_{I} λ_{A} + α_{A} (γ_{I} + μ_{I})] S_{0}}{(γ_{I} + μ_{I}) (γ_{A} + λ_{A} + μ_{A})} & \frac{α_{I} S_{0}}{γ_{I} + μ_{I}} \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}) .

R_{0} = \frac{[α_{I} λ_{A} + α_{A} (γ_{I} + μ_{I})] S_{0}}{(γ_{I} + μ_{I}) (γ_{A} + λ_{A} + μ_{A})} .

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[1] *Corresponding author: J. F. C. A. Meyer - E-mail: jmeyer@unicamp.br