# On an SEIR Epidemic Model with Vaccination of Newborns and Periodic Impulsive Vaccination with Eventual On-Line Adapted Vaccination Strategies to the Varying Levels of the Susceptible Subpopulation

^{*}

## Abstract

**:**

## 1. Introduction

## 2. SEIR Epidemic Model with Eventual Vaccination of Newborns

#### 2.1. The Epidemic Model

- -
- Susceptible: individuals who have not contracted the disease but are at risk for it, since they do not have the necessary antibodies to cope with it.
- -
- Exposed: this is the group of individuals who are already infected, but who are not yet infectious, so they cannot transmit the infection. This state lasts for a limited period of time that varies according to the disease, and is known as the latent period [1] (Chap. Two).
- -
- Infectious: those individuals that are infected and can transmit the disease.
- -
- Recovered: individuals who have managed to defeat the disease. The body generates immunity as an adaptive response to the infectious agent [15], so that a recovered individual is no longer susceptible.

- S: Proportion of susceptible individuals
- E: Proportion of exposed individuals
- I: Proportion of infectious individuals
- R: Proportion of recovered individuals
- A: Proportion of new individuals per unit of time
- β: Transmission rate
- μ: Natural mortality rate
- σ: Inverse of the latent period
- γ: Recovery rate
- α: Mortality rate caused by infection
- q: Fraction of vaccinated newborns

- New individuals enter the susceptible compartment per unit of time, either by birth or by immigration. βSI indicates the proportion of infections [4], that is, the proportion of individuals who are no longer susceptible and who become exposed; and finally, μS is the proportion of individuals who die from causes not related to the infection, also per unit of time.
- The infected susceptible βSI pass into the exposed compartment and are removed from it once the latent period has passed, that is, σE individuals are removed per unit of time. Naturally, the fraction μE of those exposed individuals who die from causes unrelated to infection must also be considered.
- At the end of the latent period, the exposed σE become infectious. Moreover, γI is the proportion of individuals who overcome the disease and who, therefore, pass to the recovered compartment; instead, αI represents the individuals who die from the infection, and who are removed from the population.
- γI individuals enter the recovered compartment per unit of time, and μR are withdrawn due to mortality from causes not related to the infection.
- The fraction of vaccinated newborns is directly added to the recovered compartment and withdrawn from the susceptible one.

#### 2.2. Non-Negativity of the Solution

**Theorem**

**1.**

- (i)
- The solution trajectory of the epidemic model (1) is non-negative for all time for any given non-negative initial conditions.
- (ii)
- The solution trajectory of Equation (1) is bounded for all time for any given finite non-negative initial conditions and$\underset{t\to \infty}{\mathrm{lim}\text{}\mathrm{sup}}I\left(t\right)\le A\mathrm{min}\left(1/\alpha \text{},\text{}1/\mu \right)$. As a result, the epidemic model is globally stable for any given arbitrary finite non-negative initial conditions.

**Proof.**

#### 2.3. Equilibrium Points

**Proposition**

**1.**

#### 2.4. Basic Reproduction Number and Stability of the Equilibrium Points

**F**and

**–V**defined from the second and third equations of (1) by:

**Theorem**

**2.**

**Proof.**

**Alternative**

**proof.**

- (a)
- it is the unique attainable equilibrium point since the endemic one is not reachable from Proposition 1.
- (b)
- the first right-hand-side term of Equation (2) tends asymptotically to zero as time tends to infinity so that, irrespective of the initial conditions, either the following limit exists:$$\underset{t\to \infty}{\mathrm{lim}}\left(S\left(t\right)-\left(1-q\right)A{\displaystyle {\int}_{0}^{t}{e}^{-{\displaystyle {\int}_{\tau}^{t}\left(\beta I\left(s\right)+\mu \right)ds}}}d\tau \right)=0$$
- (c)
- it converges to a periodic solution of some period $T>0$, that is, $S\left(nkT+\tau \right)\to {S}^{*}\left(\rho ,q\right)$ for some continuous $\rho :\left[0\text{},\text{}T\right)\to {R}_{0+}$ as $n\left(\in {Z}_{0+}\right)\to \infty $; $\forall k\in {Z}_{+}$. □

**Remark**

**1.**

**Remark**

**2.**

**Remark**

**3.**

**Proposition**

**2.**

**Theorem**

**3.**

**Proof.**

**Remark**

**4.**

- (a)
- If${R}_{0}\left(q\right)<1$, then the disease-free equilibrium point is locally asymptotically stable and the endemic one is not reachable. A limit cycle surrounding it in any phase plane of two subpopulations, if any, would be unstable from the combined topology of equilibrium points and limit cycles. Therefore, it would be irrelevant for stability considerations. As a result, the disease-free equilibrium point is globally asymptotically stable (Theorem 2).
- (b)
- If${R}_{0}\left(q\right)>1$, then the endemic equilibrium point is locally asymptotically stable and the disease-free equilibrium point is unstable. A limit cycle surrounding it leaving away the disease-free one, if any, would be unstable for stability considerations. No limit cycles can exist surrounding both equilibrium points since the combined Poincaré index of both of them would be 0 (if the disease-free equilibrium point were a saddle point) or +2 (if the disease-free equilibrium point were not a saddle point). So, in any case, such a combined Poincaré index is not +1 compatible with the existence of some limit cycle surrounding both equilibrium points. As a result, the endemic equilibrium point is globally asymptotically stable if${R}_{0}\left(q\right)>1$(Theorem 3).
- (c)
- If${R}_{0}\left(q\right)=1$, then the disease-free and the endemic equilibrium points are coincident. From the non-negativity property of the solution for any given non-negative initial conditions (Theorem 1 (i)), it follows that a limit cycle surrounding the equilibrium point cannot exist in the phase plane (E, I). Also, by inspection of (1), it follows that$S\left(t\right)$and$R\left(t\right)$can only have periodic asymptotic solutions if$I\left(t\right)$and$E\left(t\right)$have asymptotic periodic solutions. As a result, the stability is also asymptotic and global in this case.

## 3. SEIR Model with Impulsive Vaccination

#### 3.1. The Epidemic Model Under Periodic Impulsive Vaccination

**Remark**

**5.**

#### 3.2. Periodic Solutions and Stability Results

**Theorem**

**4.**

- (i)
- Any solution of Equation (21) is non-negative for all time for any given set of non-negative initial conditions.
- (ii)
- There exists a non-negative periodic disease-free solution of Equation (21) for each given set of non-negative finite initial conditions of the form$$\widehat{E}\left(t\right)=\widehat{I}\left(t\right)=0$$$$\widehat{S}\left(t\right)=\frac{A}{\mu}\left[1-\frac{p}{1-\left(1-p\right){e}^{-\mu T}}\left({e}^{-\mu \left(t-{t}_{0}\right)}+\left(1-{e}^{-\mu T}\right){\displaystyle {\int}_{{t}_{0}}^{t}\delta \left(\tau -nT\right)d\tau}\right)\right]\phantom{\rule{0ex}{0ex}}=\frac{A}{\mu}\left[1-\left(1-\frac{\left(1-p\right)\left(1-{e}^{-\mu T}\right)}{1-\left(1-p\right){e}^{-\mu T}}\right){e}^{-\mu \left(t-\left(n-1\right)T\right)}\right]$$$$\widehat{R}\left(t\right)=\frac{A}{\mu}\frac{p}{1-\left(1-p\right){e}^{-\mu T}}\left({e}^{-\mu \left(t-{t}_{0}\right)}+\left(1-{e}^{-\mu T}\right){\displaystyle {\int}_{{t}_{0}}^{t}\delta \left(\tau -nT\right)}d\tau \right)\phantom{\rule{0ex}{0ex}}=\frac{A}{\mu}\left(1-\frac{\left(1-p\right)\left(1-{e}^{-\mu T}\right)}{1-\left(1-p\right){e}^{-\mu T}}\right){e}^{-\mu \left(t-\left(n-1\right)T\right)}$$$$;\text{}\forall t\in \left[{t}_{0}=\left(n-1\right)T\text{},\text{}nT\right),\text{}\forall n\in {Z}_{+}$$
- (iii)
- The above periodic disease-free solution converges asymptotically to finite non-negative left and right limits in Equations (22) and (23) with ${E}_{*}={E}_{*}^{-}={I}_{*}={I}_{*}^{-}=0$ at the impulsive time instants ${t}_{n}=nT$ as $n\to \infty $.
- (iv)
- The above periodic disease-free solution is locally asymptotically stable if the following constraint holds:$$T\left(\alpha +\sigma +\gamma +2\mu \right)>{\displaystyle {\int}_{0}^{T}\sqrt{{\left(\alpha +\gamma -\sigma \right)}^{2}+4\beta \sigma \widehat{S}\left(t\right)}}\text{}dt;\text{}\forall t\in \left[{t}_{0}=\left(n-1\right)T\text{},\text{}nT\right),\text{}\forall n\in {Z}_{+}$$
- (v)
- A sufficient condition for Property (iv) to hold is that the transmission rate be sufficiently small according to:$$\beta \le {\beta}_{c}\left(p,T\right)=\frac{\mu \text{}\left[1-\left(1-p\right){e}^{-\mu T}\right]}{2A\sigma \left(1-{e}^{-\mu T}\right)}\left[{\left(\alpha +\sigma +\gamma +2\mu \right)}^{2}-{\left(\alpha +\gamma -\sigma \right)}^{2}\right]\phantom{\rule{0ex}{0ex}}=\frac{\mu \text{}\left[1-\left(1-p\right){e}^{-\mu T}\right]}{2A\sigma \left(1-{e}^{-\mu T}\right)}\left[2\mu \left(\mu +\sigma \right)+\left(\alpha +\gamma \right)\left(2\mu +3\sigma \right)\right]$$

**Proof.**

**Remark**

**6.**

## 4. Numerical Simulations

#### 4.1. Newborn Vaccination

#### 4.2. Impulsive Vaccination

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Keeling, M.J.; Rohani, P. Modelling Infectious Diseases; Princeton University Press: Princeton, NJ, USA, 2007. [Google Scholar]
- Kermack, W.O.; McKendrick, A.G. A contribution to the mathematical theory of epidemics. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci.
**1927**, 115, 700–721. [Google Scholar] [CrossRef] [Green Version] - Biswas, H.A.; Paiva, L.T.; De Pinho, M.D.R. A SEIR model for control of infectious diseases with constraints. Math. Biosci. Eng.
**2014**, 11, 761–784. [Google Scholar] [CrossRef] - Parsamanesh, M.; Erfanian, M. Global dynamics of an epidemic model with standard incidence rate and vaccination strategy. Chaos Solitons Fractals
**2018**, 117, 192–199. [Google Scholar] [CrossRef] - Bacaër, N. The model of Kermack and McKendrick for the plague epidemic in Bombay and the type reproduction number with seasonality. J. Math. Biol.
**2012**, 64, 403–422. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Giordano, G.; Blanchini, F.; Bruno, R.; Colaneri, P.; Di Filippo, A.; Di Matteo, A.; Colaneri, M. Modelling the COVID-19 epidemic and implementation of population-wide interventions in Italy. Nat. Med.
**2020**, 26, 855–860. [Google Scholar] [CrossRef] - Yang, Z.; Zeng, Z.; Wang, K.; Wong, S.-S.; Liang, W.; Zanin, M.; Liu, P.; Cao, X.; Gao, Z.; Mai, Z.; et al. Modified SEIR and AI prediction of the epidemics trend of COVID-19 in China under public health interventions. J. Thorac. Dis.
**2020**, 12, 165–174. [Google Scholar] [CrossRef] - Yang, C.Y.; Wang, J. A mathematical model for the novel coronavirus epidemic in Wuhan, China. Math. Biosci. Eng.
**2020**, 17, 2708–2724. [Google Scholar] [CrossRef] - Nokes, D.J.; Swinton, J. Vaccination in pulses: A strategy for global eradication of measles and polio? Trends Microbiol.
**1997**, 5, 14–19. [Google Scholar] [CrossRef] - Shulgin, B. Pulse vaccination strategy in the SIR epidemic model. Bull. Math. Biol.
**1998**, 60, 1123–1148. [Google Scholar] [CrossRef] - Sisodiya, O.S.; Misra, O.P.; Dhar, J. Dynamics of cholera epidemics with impulsive vaccination and disinfection. Math. Biosci.
**2018**, 298, 46–57. [Google Scholar] [CrossRef] - Last, J.M. Dictionary of Epidemiology; Oxford University Press: New York, NY, USA, 2001. [Google Scholar]
- Brauer, F.; Feng, Z.; Castillo-Chavez, C. Discrete epidemic models. Math. Biosci. Eng.
**2010**, 7, 1–15. [Google Scholar] [CrossRef] [PubMed] - Allen, L.J. A primer on stochastic epidemic models: Formulation, numerical simulation, and analysis. Infect. Dis. Model.
**2017**, 2, 128–142. [Google Scholar] [CrossRef] [PubMed] - Baxter, D. Active and passive immunity, vaccine types, excipients and licensing. Occup. Med.
**2007**, 57, 552–556. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Heffernan, J.M.; Smith, R.J.; Wahl, L.M. Perspectives on the basic reproductive ratio. J. R. Soc. Interface
**2005**, 2, 281–293. [Google Scholar] [CrossRef] - Asociacioón Espanñola de Pediatría; Comité Asesor de Vacunas. Calendario de Vacunaciones de la Asociación Espanola de Pediatrńa: Recomendaciones. 2020. Available online: https://vacunasaep.org/profesionales/calendario-de-vacunaciones-de-la-aep-2020 (accessed on 1 April 2020).
- Organizacioón Mundial de la Salud. Cobertura Vacunal. 2020. Available online: https://www.who.int/es/news-room/fact-sheets/detail/immunization-coverage (accessed on 1 April 2020).
- John, T.J.; Vashishtha, V.M. Eradicating poliomyelitis: India’s journey from hyperendemic to polio-free status. Indian J. Med. Res.
**2013**, 137, 881–894. [Google Scholar] - Noticias ONU. La Falta de Pruebas Para Detectar el Coronavirus Oculta Casos y Muertes en América Latina. 2020. Available online: https://news.un.org/es/story/2020/04/1473512 (accessed on 15 April 2020).
- Organización Mundial de la Salud. Rastreo de los Contactos en Situaciones de Brotes Epidémicos. 2017. Available online: https://www.who.int/features/qa/contact-tracing/es/ (accessed on 15 April 2020).
- Zastrow, M. South Korea is reporting intimate details of COVID-19 cases: Has it helped? Nature
**2020**. [Google Scholar] [CrossRef] - Sun, K.; Viboud, C. Impact of contact tracing on SARS-CoV-2 transmission. Lancet Infect. Dis.
**2020**, 20, 876–877. [Google Scholar] [CrossRef] - Jonker, W. The European Struggle with COVID-19 Contact Tracing Apps. European Institute of Innovation & Technology, 2020. Available online: https://eit.europa.eu/news-events/news/european-struggle-covid-19-contact-tracing-apps (accessed on 15 April 2020).
- Transparencia Sobre el Nuevo Coronavirus (COVID-19). euskadi.eus. 2020. Available online: https://www.euskadi.eus/preguntas-sobre-el-confinamiento-cuarentena/web01-a3korona/es/ (accessed on 15 April 2020).
- De La Sen, M.; Nistal, R.; Alonso-Quesada, S.; Ibeas, A. Some Formal Results on Positivity, Stability, and Endemic Steady-State Attainability Based on Linear Algebraic Tools for a Class of Epidemic Models with Eventual Incommensurate Delays. Discret. Dyn. Nat. Soc.
**2019**, 2019, 8959681. [Google Scholar] [CrossRef] [Green Version] - Strogatz, S.H. Nonlinear Dynamics and Chaos; Perseus Books: Reading, UK, 1994. [Google Scholar]
- Chen, G. Stability of Nonlinear Systems. In Encyclopedia of RF and Microwave Engineering; Chang, K., Ed.; John Wiley & Sons: Hoboken, NJ, USA, 2005; pp. 4881–4896. [Google Scholar]
- Klausmeier, C.A. Floquet theory: A useful tool for understanding nonequilibrium dynamics. Theor. Ecol.
**2008**, 1, 153–161. [Google Scholar] [CrossRef] - Bittani, S.; Colaneri, P. Periodic Systems: Filtering and Control; Springer: London, UK, 2009. [Google Scholar] [CrossRef]
- DeJesus, E.X.; Kaufman, C. Routh-Hurwitz criterion in the examination of eigenvalues of a system of nonlinear ordinary differential equations. Phys. Rev. A
**1987**, 35, 5288–5290. [Google Scholar] [CrossRef] [Green Version] - Khan, M.A.; Ullah, S.; Ullah, S.; Farhan, M. Fractional order SEIR model with generalized incidence rate. AIMS Math.
**2020**, 5, 2843–2857. [Google Scholar] [CrossRef] - He, S.; Peng, Y.; Sun, K. SEIR modeling of the COVID-19 and its dynamics. Nonlinear Dyn.
**2020**, 101, 1667–1680. [Google Scholar] [CrossRef] [PubMed] - Pandey, G.; Chaudhary, P.; Gupta, R.; Pal, S. SEIR and regression model based COVID-19 outbreak predictions in India. arXiv
**2020**, arXiv:2004.00958. [Google Scholar] - Annas, S.; Pratama, M.I.; Rifandi, M.; Sanusi, W.; Side, S. Stability analysis and numerical simulation of SEIR model for pandemic COVID-19 spread in Indonesia. Chaos Solitons Fractals
**2020**, 139, 110072. [Google Scholar] [CrossRef] [PubMed] - Fabiano, N.; Radenović, S. On COVID-19 diffusion in Italy: Data analysis and possible outcome. Vojn. Glas.
**2020**, 68, 216–224. [Google Scholar] [CrossRef] - Wang, W.; Wu, Z.-N.; Wang, C.; Hu, R. Modelling the spreading rate of controlled communicable epidemics through an entropy-based thermodynamic model. Sci. China Ser. G Phys. Mech. Astron.
**2013**, 56, 2143–2150. [Google Scholar] [CrossRef] - De La Sen, M.; Nistal, R.; Ibeas, A.; Garrido, A.J. On the Use of Entropy Issues to Evaluate and Control the Transients in Some Epidemic Models. Entropy
**2020**, 22, 534. [Google Scholar] [CrossRef] - De La Sen, M.; Agarwal, R.P.; Ibeas, A.; Alonso-Quesada, S. On a Generalized Time-Varying SEIR Epidemic Model with Mixed Point and Distributed Time-Varying Delays and Combined Regular and Impulsive Vaccination Controls. Adv. Differ. Equ.
**2010**, 2010, 281612. [Google Scholar] [CrossRef] - Beretta, E.; Kolmanovskii, V.; Shaikhet, L. Stability of epidemic model with time delays influenced by stochastic perturbations. Math. Comput. Simul.
**1998**, 45, 269–277. [Google Scholar] [CrossRef] - Santonja, F.; Shaikhet, L. Probabilistic stability analysis of social obesity epidemic by a delayed stochastic model. Nonlinear Anal. Real. World Appl.
**2014**, 17, 114–125. [Google Scholar] [CrossRef] - D’Onofrio, A. Stability properties of pulse vaccination strategy in SEIR epidemic model. Math. Biosci.
**2002**, 179, 57–72. [Google Scholar] [CrossRef] - De La Sen, M. Adaptive sampling for improving the adaptation transients in hybrid adaptive control. Int. J. Control
**1985**, 41, 1189–1205. [Google Scholar] [CrossRef] - Delasen, M. A method for improving the adaptation transient using adaptive sampling. Int. J. Control
**1984**, 40, 639–665. [Google Scholar] - De Jesus, L.F.; Silva, C.M.; Vilarinho, H. Periodic orbits for periodic eco-epidemiological systems with infected prey. Electron. J. Qual. Theory Differ. Equ.
**2020**, 54, 1–20. [Google Scholar] [CrossRef]

**Figure 7.**Infectious proportion evolution under impulsive vaccination for $T=120\text{}\mathrm{days}$.

**Figure 9.**Model evolution for adapted impulsive vaccination with $A=0.0045$, ${p}_{\mathrm{max}}=0.60$, ${p}_{\mathrm{min}}=0.30$, $T=30\text{}\mathrm{days}$.

**Figure 10.**Detail of the infectious individuals proportion evolution for adapted impulsive vaccination with $A=0.0045$, ${p}_{\mathrm{max}}=0.60$, ${p}_{\mathrm{min}}=0.30$, $T=30\text{}\mathrm{days}$.

**Figure 11.**Model evolution for adapted impulsive vaccination with $A=0.0045$, ${p}_{\mathrm{max}}=0.60$, ${p}_{\mathrm{min}}=0.55$, $T=30\text{}\mathrm{days}$.

**Figure 12.**Detail of the infectious individuals proportion evolution for adapted impulsive vaccination with $A=0.0045$, ${p}_{\mathrm{max}}=0.60$, ${p}_{\mathrm{min}}=0.55$, $T=30\text{}\mathrm{days}$.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Etxeberria-Etxaniz, M.; Alonso-Quesada, S.; De la Sen, M.
On an SEIR Epidemic Model with Vaccination of Newborns and Periodic Impulsive Vaccination with Eventual On-Line Adapted Vaccination Strategies to the Varying Levels of the Susceptible Subpopulation. *Appl. Sci.* **2020**, *10*, 8296.
https://doi.org/10.3390/app10228296

**AMA Style**

Etxeberria-Etxaniz M, Alonso-Quesada S, De la Sen M.
On an SEIR Epidemic Model with Vaccination of Newborns and Periodic Impulsive Vaccination with Eventual On-Line Adapted Vaccination Strategies to the Varying Levels of the Susceptible Subpopulation. *Applied Sciences*. 2020; 10(22):8296.
https://doi.org/10.3390/app10228296

**Chicago/Turabian Style**

Etxeberria-Etxaniz, Malen, Santiago Alonso-Quesada, and Manuel De la Sen.
2020. "On an SEIR Epidemic Model with Vaccination of Newborns and Periodic Impulsive Vaccination with Eventual On-Line Adapted Vaccination Strategies to the Varying Levels of the Susceptible Subpopulation" *Applied Sciences* 10, no. 22: 8296.
https://doi.org/10.3390/app10228296