# On a Discrete SEIR Epidemic Model with Two-Doses Delayed Feedback Vaccination Control on the Susceptible

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Discrete SEIR Epidemic Model Subject to Two Vaccination Doses

- –
- ${a}_{k}$ is the average recruitment rate proportional to the susceptible at the $kth$ sampling instant related; for instance, to the rates of births and a is a constant reference value for the above sequence; for instance, its average over the whole time period under study and typically it might be unity.
- –
- ${\beta}_{k}$ and ${\beta}_{k}^{e}$ are, respectively, the average transmission rates of the infectious and exposed subpopulations at the $k-th$ sampling instant.
- –
- $\gamma $ is the average recovery rate.
- –
- $\mu $ is the average incubation rate.
- –
- ${K}_{k}$ is the vaccination rate (a feedback control gain) which can be eventually depending on the sampling instants. It is assumed in the sequel that ${\left\{{K}_{k}\right\}}_{0}^{\infty}\subset \left[0,1\right]$.
- –
- ${\rho}_{1}$, ${\rho}_{1}+{\rho}_{2}$ are parameters in $\left(0,1\right]$ which quantify the average effectiveness (or efficiency) of the respective doses. In particular, ${\rho}_{2}$ gives the extra effectiveness obtained from the injection of the second dose. In this context, ${\rho}_{1}+{\rho}_{2}\in \left[0,1\right]$ and ${\rho}_{1}+{\rho}_{2}=1$ refer to the ideal situation, unattainable in practice, of 100 percent effectiveness of the combined injection of the two doses. Note that ${\rho}_{1}={\rho}_{2}={\rho}_{1}+{\rho}_{2}=0$ refers to the worst case where the vaccination is fully superfluous.

## 3. Non-Negativity, Stability and Disease-Free Equilibrium Point

**Theorem**

**1**.

- (i)
- Assume that${N}_{0}={S}_{0}+{E}_{0}+{I}_{0}+{R}_{0}=1$. Then, the total population${N}_{k}={S}_{k}+{E}_{k}+{I}_{k}+{R}_{k}=1$; $\forall k\in {\mathit{Z}}_{0+}$if${a}_{k}\equiv 1$.$$(\mathrm{ii}){N}_{k}=1+{{\displaystyle \sum}}_{j=0}^{k-1}\left({a}_{j}-1\right){S}_{j};\forall k\in {\mathit{Z}}_{0+}$$
- (iii)
- ${\left\{{S}_{k}\right\}}_{k=0}^{\infty}$is non-increasing, and then bounded and convergent, if$${V}_{2,k-{d}_{2}}+{V}_{1,k-{d}_{1}-{d}_{2}}={K}_{k-{d}_{2}}{\rho}_{2}{S}_{k-{d}_{2}}+{K}_{k-{d}_{1}-{d}_{2}}{\rho}_{1}{S}_{k-{d}_{1}-{d}_{2}}\ge \left({a}_{k}-{\beta}_{k}\left({I}_{k}+{\lambda}_{k}^{e}{E}_{k}\right)-1\right){S}_{k};\forall k\in {\mathit{Z}}_{0+}$$$$1+{S}_{k-1}{{\displaystyle \sum}}_{j=0}^{k-1}\left({a}_{j}-1\right)\le {N}_{k}\le 1+{S}_{0}{{\displaystyle \sum}}_{j=0}^{k-1}\left({a}_{j}-1\right);\forall k\in {\mathit{Z}}_{0+}$$
- (iv)
- ${\left\{{S}_{k}\right\}}_{k=0}^{\infty}$is non-negative if and only if$${V}_{2,k-{d}_{2}}+{V}_{1,k-{d}_{1}-{d}_{2}}={K}_{k-{d}_{2}}{\rho}_{2}{S}_{k-{d}_{2}}+{K}_{k-{d}_{1}-{d}_{2}}{\rho}_{1}{S}_{k-{d}_{1}-{d}_{2}}\le \left({a}_{k}-{\beta}_{k}\left({I}_{k}+{\lambda}_{k}^{e}{E}_{k}\right)\right){S}_{k};\forall k\in {\mathit{Z}}_{0+}$$
- (v)
- Assume that$\mu \in \left[0,1\right)$, $\gamma \in \left[0,1\right)$and that (15) holds. Then, any sequence trajectory solution of (1)–(4) subject to (5)–(6) is non-negative.

**Proof.**

**Remark**

**1.**

**Remark**

**2.**

**Theorem 2**.

**Proof.**

**Theorem**

**3.**

- (1)
- $\mu \in \left[0,1\right)$,$\gamma \in \left[0,1\right)$,${\rho}_{1}+{\rho}_{2}\le 1$,${K}_{k}\le \frac{{a}_{k}}{{I}_{k}+{\lambda}_{k}^{e}{E}_{k}}$(guaranteed if${\beta}_{k}\le \frac{{a}_{k}}{{\left(1+{\lambda}_{k}^{e}\right)}^{}{N}_{k}}$);$\forall k\in {\mathit{Z}}_{0+}$,${\left\{{a}_{k}\right\}}_{k=0}^{\infty}\to a\left(\le 1\right)$,${\left\{{\beta}_{k}\right\}}_{k=0}^{\infty}\to \beta $,${\left\{{\lambda}_{k}^{e}\right\}}_{k=0}^{\infty}\to {\lambda}^{e}$,${\left\{{K}_{k}\right\}}_{k=0}^{\infty}\to K$.
- (2)
- Either the constraint (13) holds with strict inequality for all$k\in {\mathit{Z}}_{0+}$and$K\ne \frac{a}{{\rho}_{1}+{\rho}_{2}}$.
- (3)
- $K\in (0,\mathit{min}(\stackrel{\u2014}{K},1))$, where:

**Proof.**

**Theorem**

**4.**

**Proof.**

## 4. Nonexistence of Endemic Equilibrium Point

**Case**

**1**$({S}_{end}>0)$. Combining (2) and (3) for stationary limiting model parameters for an assumed to exist endemic equilibrium point leads to provided that ${E}_{end}\ne 0$ and ${S}_{end}\ne 0$:

**Case**

**2**$\left({S}_{end}\ge 0\right)$. Note that (40) holds with, which includes also ${S}_{end}>0$ of Case 1 if

**Remark**

**3**

**.**In biological terms, it is possible to re-interpret the condition of asymptotic stability around the disease-free equilibrium point in terms of the basic reproduction number, which indicates the number of secondary infectious individuals generated from one primary infectious one, defined by

## 5. Simulation Results

^{−1}. The initial conditions are ${\mathrm{S}}_{0}=0.9999,{\mathrm{E}}_{0}=0.0001,{\mathrm{I}}_{0}=0,{\mathrm{R}}_{0}=0$ implying that the total population is normalized to unity, without loss of generality. Notice that almost all the population is susceptible and a small fraction of the population is exposed at the beginning. The parameters of the vaccination are ${\mathrm{d}}_{1}+{\mathrm{d}}_{2}=21$ days and ${\mathrm{d}}_{2}=7$ days; therefore, separation between the two doses is of two weeks. The values of the doses effectiveness are given by ${\mathsf{\rho}}_{1}=0.66$ and ${\rho}_{2}=0.3$ in such a way that the total effectiveness of the two doses $\rho ={\rho}_{1}+{\rho}_{2}$ is 96%, in accordance with the average effectiveness of available vaccines. The natural recruitment rate is a = 1, since the natural growth of the population may be rejected when it comes to the epidemic spreading description due to the small number of children affected. The Figure 1 displays the dynamics of the model without vaccination.

_{2}= 7 days and d

_{1}+ d

_{2}range from 10 to 21 days. It is observed in Figure 7 that if the second dose maintains effectivity regardless the dose sparing, it is better to administer it as soon as possible. However, in the real world, the effectivity of the second dose may depend on when this is applied. There are no data on how the effectivity depends on the dose sparing or these are scarce, [46]. Therefore a more accurate simulation on the effect of dose sparing is not carried out in this work.

## 6. Concluding Remarks

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Appendix A1. Some Auxiliary Technical Results

**Theorem**

**A1.**

- (i)
- The unforced difference Equation (A1), i.e., if${\left\{{u}_{k}\right\}}_{k=0}^{\infty}\equiv 0$, is globally asymptotically stable, so that${\left\{{x}_{k}\right\}}_{k=0}^{\infty}$is bounded and${\left\{{x}_{k}\right\}}_{k=0}^{\infty}\to 0$for any given finite${x}_{0}\in \mathit{R}$, if$K\in {\left(\mathit{max}\left(\frac{a-1}{{\rho}_{1}+{\rho}_{2}},-\stackrel{\u2014}{K}\right),\mathit{min}\left(\frac{1+a}{{\rho}_{1}+{\rho}_{2}},\stackrel{\u2014}{K}\right)\right)}^{}$and, in particular,$K\in \left[0,\stackrel{\u2014}{K}\right)$under the restriction$K\in {\mathit{R}}_{0+}={\mathit{R}}_{+}\cup \left\{0\right\}$, where:$$\stackrel{\u2014}{K}=\mathit{sup}\left\{y\in \left(\frac{a-1}{{\rho}_{1}+{\rho}_{2}},\frac{1+a}{{\rho}_{1}+{\rho}_{2}}\right):y1/f\left(y\right)\right\}withf:{\mathit{R}}_{0+}\to {\mathit{R}}_{0+}definedby:$$$$f\left(y\right)=\underset{\theta \in \left(0,2\pi \right)}{sup}\sqrt{\frac{{\left({\rho}_{2}\left(\mathit{cos}\left(\theta {d}_{2}\right)-1\right)+{\rho}_{1}\left(\mathit{cos}\left(\theta \left({d}_{1}+{d}_{2}\right)\right)-1\right)\right)}^{2}+{\left({\rho}_{2}\mathit{sin}\left(\theta {d}_{2}\right)+{\rho}_{1}\mathit{sin}\left(\theta \left({d}_{1}+{d}_{2}\right)\right)\right)}^{2}}{1+{\left(a-\left({\rho}_{1}+{\rho}_{2}\right)y\right)}^{2}-2\left(a-\left({\rho}_{1}+{\rho}_{2}\right)y\right)\mathit{cos}\theta}}$$
- (ii)
- The unforced difference Equation (A1) is globally asymptotically stable for any given finite${x}_{0}\in \mathit{R}$, so that${\left\{{x}_{k}\right\}}_{k=0}^{\infty}$is bounded and${\left\{{x}_{k}\right\}}_{k=0}^{\infty}\to 0$for any finite${x}_{0}\in \mathit{R}$, if$\left|a\right|<1$and$K\in \left(-\stackrel{\u2014}{K},\stackrel{\u2014}{K}\right)$, were$\stackrel{\u2014}{K}=1/\underset{\theta \in \left[0,2\pi \right)}{sup}\left|\frac{{\rho}_{2}{e}^{-i\theta {d}_{2}}+{\rho}_{1}{e}^{-i\theta \left({d}_{1}+{d}_{2}\right)}}{{e}^{i\theta}-a}\right|$.
- (iii)
- Assume that either the conditions of Property (i) or those of Property (ii) hold and that${\left\{{u}_{k}\right\}}_{k=0}^{\infty}\to 0$. Then,${\left\{{x}_{k}\right\}}_{k=0}^{\infty}$is bounded and${\left\{{x}_{k}\right\}}_{k=0}^{\infty}\to 0$for any given finite initial condition${x}_{0}$.
- (iv)
- Assume that either the conditions of Property (i) or those of Property (ii) hold and that${\left\{{u}_{k}\right\}}_{k=0}^{\infty}$is bounded. Then,${\left\{{x}_{k}\right\}}_{k=0}^{\infty}$is bounded for any given finite initial conditions.

**Proof.**

**Corollary**

**A1.**

**Proof.**

**Corollary**

**A2.**

**Proof.**

**Theorem**

**A2.**

- (i)
- If${x}_{0}\ne 0$then${\left\{{x}_{k}\right\}}_{k=0}^{\infty}$is unbounded. If$b=0$and the remaining above conditions hold, then${\left\{{x}_{k}\right\}}_{k=0}^{\infty}$ is bounded.
- (ii)
- If${x}_{0}\ne 0$,${\left\{{\tilde{a}}_{j}\right\}}_{j=0}^{\infty}\subset \left[-1,0\right]$,$\left|{\tilde{a}}_{k+1}\right|>\frac{\left|{\tilde{a}}_{k}\right|}{{a}_{k}+{b}_{k}/{x}_{k}}$,$b>0$,${\left\{{\tilde{b}}_{j}\right\}}_{j=0}^{\infty}\subset \left[0,+\infty \right)$and$\sum}_{k=0}^{\infty}{\tilde{b}}_{k}<+\infty $then a necessary condition for${\left\{{x}_{k}\right\}}_{k=0}^{\infty}\to 0$is$\sum}_{j=0}^{k}{\tilde{a}}_{j}=-\infty $. The result still holds if${\left\{{x}_{k}\right\}}_{k=0}^{\infty}\to x\left(>0\right)$.

**Proof.**

#### Appendix A2. A More Detailed Expansion of the Squared Numerator of (A9)

- (a)
- If ${d}_{1}={d}_{2}$ then, one gets by using $1-\mathit{cos}\left(2\theta {d}_{2}\right)=2{\mathit{sin}}^{2}\theta {d}_{2}$ that:$$n\left(\theta ,{d}_{1},{d}_{1},{\rho}_{1},{\rho}_{2}\right)=2\left({\rho}_{2}^{2}\left(1-\mathit{cos}\left(\theta {d}_{2}\right)\right)+2{\rho}_{1}\left({\rho}_{1}+{\rho}_{2}\right){\mathit{sin}}^{2}\left(\theta {d}_{1}\right)\right)$$
- (b)
- If ${d}_{1}={d}_{2}$ and ${\rho}_{1}={\rho}_{2}$ then$$n\left(\theta ,{d}_{1},{d}_{1},{\rho}_{1},{\rho}_{1}\right)=2\left({\rho}_{1}^{2}\left(1-\mathit{cos}\left(\theta {d}_{2}\right)\right)+4{\rho}_{1}^{2}{\mathit{sin}}^{2}\left(\theta {d}_{2}\right)\right)$$
- (c)
- If ${d}_{1}={d}_{2}=0$ or ${\rho}_{1}={\rho}_{2}=0$ then$$n\left(\theta ,0,0,{\rho}_{1},{\rho}_{1}\right)=n\left(\theta ,{d}_{1},{d}_{1},0,0\right)=0$$
- (d)
- If $\theta =0$ then $n\left(0,{d}_{1},{d}_{1},{\rho}_{1},{\rho}_{1}\right)=0$ which is not evaluated in the supremum over $\left(0,2\pi \right)$ in (A.33). Note that the constraint $y<1/f\left(y\right)$ to calculate $\stackrel{\u2014}{K}$ in Theorem A1 always holds since $1/f\left(y\right)=+\infty $ for $\theta =0,2\pi $ since $n\left(0,{d}_{1},{d}_{1},{\rho}_{1},{\rho}_{1}\right)=0$ so that it has not to be accounted for in the supremum evaluation as Theorem A1 formally establishes.

#### Appendix A3. A Fast and Simple Delay-Dependent Stability Test Based on Rouché′s Theorem of Zeros Within Open Circles Contained in the Unit Circle

**Theorem**

**A3.**

- (i)
- Let$a\in \left(0,1\right)$and consider a circle$\left|z\right|\le \sigma $of center z = 0 and of radius$\sigma \in \left(a,a+\epsilon \right]$for some real$\epsilon >0$. Assume that$0<K<\frac{\left(\sigma -a\right){\sigma}^{{d}_{1}+{d}_{2}}}{{\rho}_{1}+{\rho}_{2}{\sigma}^{{d}_{1}}}$. Then, the unforced difference Equation (A1) has all its zeros in$a\left|z\right|\sigma $. As a result, if$a<1$and$0<\epsilon \le 1-a$then the unforced difference Equation (A1) has all its zeros in$a<\left|z\right|<1$so that it is stable.
- (ii)
- Let${c}_{i}={c}_{i}\left({d}_{1},{d}_{2}\right)\in {\mathit{R}}_{+}$for$i=1,2$, be chosen such that${a}^{{d}_{1}}<{c}_{1}\le \le {\sigma}^{{d}_{1}}\le {c}_{2}$(implying that$\frac{\mathit{ln}{c}_{1}}{\mathit{ln}\sigma}-{d}_{2}<{d}_{1}\le \frac{\mathit{ln}{c}_{2}}{\mathit{ln}\sigma}$. Then, the unforced difference Equation (A1) has its zeros in$\left|z\right|<\sigma $if$$0\le K<\frac{\left({c}_{1}^{-{d}_{1}}-a\right){c}_{1}}{{\rho}_{1}+{\rho}_{2}{c}_{2}}.$$

**Proof.**

- Case a)
- If ${d}_{1}$ is even then for $\theta \in \left[0,2\pi \right)$:$$\begin{array}{ll}{\sigma}^{2\left({d}_{1}+{d}_{2}\right)}\left({\sigma}^{2}+{a}^{2}\right)& >{K}^{2}\underset{\theta \in \left[0,2\pi \right)}{max}\left({\rho}_{1}^{2}+{\rho}_{2}^{2}{\sigma}^{2{d}_{1}}+2\left({\rho}_{1}{\rho}_{2}{\sigma}^{{d}_{1}}+a{\sigma}^{2\left({d}_{1}+{d}_{2}\right)}\right)\mathit{cos}\left(\theta \right)\right)\\ & ={K}^{2}\left({\rho}_{1}^{2}+{\rho}_{2}^{2}{\sigma}^{2{d}_{1}}+2\left({\rho}_{1}{\rho}_{2}{\sigma}^{{d}_{1}}+a{\sigma}^{2\left({d}_{1}+{d}_{2}\right)}\right)\right)\\ & ={K}^{2}{\left({\rho}_{1}+{\rho}_{2}{\sigma}^{{d}_{1}}\right)}^{2}+2a{\sigma}^{2\left({d}_{1}+{d}_{2}\right)}\end{array}$$$$K\left({\rho}_{1}+{\rho}_{2}{\sigma}^{{d}_{1}}\right)<{\sigma}^{{d}_{1}+{d}_{2}}\left(\sigma -a\right)$$
- Case b)
- If ${d}_{1}$ is odd then for $\theta \in \left[0,2\pi \right)$:$$\begin{array}{ll}{\sigma}^{2\left({d}_{1}+{d}_{2}\right)}\left({\sigma}^{2}+{a}^{2}\right)& >{K}^{2}\underset{\theta \in \left[0,2\pi \right)}{max}\left({\rho}_{1}^{2}+{\rho}_{2}^{2}{\sigma}^{2{d}_{1}}+2\left({\rho}_{1}{\rho}_{2}{\sigma}^{{d}_{1}}-a{\sigma}^{2\left({d}_{1}+{d}_{2}\right)}\right)\mathit{cos}\left({d}_{1}\theta \right)\right)\\ & ={K}^{2}\left({\rho}_{1}^{2}+{\rho}_{2}^{2}{\sigma}^{2{d}_{1}}+2\left|{\rho}_{1}{\rho}_{2}{\sigma}^{{d}_{1}}-a{\sigma}^{2\left({d}_{1}+{d}_{2}\right)}\right|\right)\end{array}$$

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**Figure 8.**Comparison in the evolution of the number of infectious between the administration of a double dose (K = 0.01) or a single dose to a broader population (K = 0.02).

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De la Sen, M.; Alonso-Quesada, S.; Ibeas, A.; Nistal, R. On a Discrete SEIR Epidemic Model with Two-Doses Delayed Feedback Vaccination Control on the Susceptible. *Vaccines* **2021**, *9*, 398.
https://doi.org/10.3390/vaccines9040398

**AMA Style**

De la Sen M, Alonso-Quesada S, Ibeas A, Nistal R. On a Discrete SEIR Epidemic Model with Two-Doses Delayed Feedback Vaccination Control on the Susceptible. *Vaccines*. 2021; 9(4):398.
https://doi.org/10.3390/vaccines9040398

**Chicago/Turabian Style**

De la Sen, Manuel, Santiago Alonso-Quesada, Asier Ibeas, and Raul Nistal. 2021. "On a Discrete SEIR Epidemic Model with Two-Doses Delayed Feedback Vaccination Control on the Susceptible" *Vaccines* 9, no. 4: 398.
https://doi.org/10.3390/vaccines9040398