# Mathematical Modeling and Short-Term Forecasting of the COVID-19 Epidemic in Bulgaria: SEIRS Model with Vaccination

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## Abstract

**:**

## 1. Introduction

## 2. Time-Dependent SEIRS-Based Model with Vaccination and Vital Dynamics

#### 2.1. The SEIRS-VB Model: Formulation

- $S\left(t\right)$—Susceptible individuals. These are the people who may be infected and can become virus carriers. In this group, we include all individuals without immunity (unvaccinated, not fully vaccinated, vaccinated people for whom the vaccine is ineffective, fully vaccinated or recovered individuals who have lost their immunity). Usually at the beginning of a pandemic, as in the case of COVID-19, the whole host population is susceptible.
- $E\left(t\right)$—Exposed individuals. These are virus carrier individuals in the latent stage, during which they are not virus spreaders. They usually have no symptoms.
- $I\left(t\right)$—Infectious individuals. These are virus carriers and virus spreaders of extremely high infectivity. The former are likely to transmit the virus in case of contact.
- $R\left(t\right)$—Recovered individuals with immunity. These individuals have disease acquired immunity. They have recovered, and thus are protected from the disease.
- $V\left(t\right)$—Vaccinated susceptible individuals. These are fully vaccinated persons for whom the vaccine is effective. However, they have not developed antibodies. They can do so after a certain period of time or else they will become exposed individuals before that. It is worth pointing out that, due to the vaccine imperfection, some of the vaccinated individuals can not develop antibodies, and they can not pass from group $S\left(t\right)$ to group $V\left(t\right)$.
- $B\left(t\right)$—Individuals with vaccination-acquired immunity. These are vaccinated individuals who are well protected from future infection because they have antibodies.

- (1)
- The S-E-I-R-S chain (horizontal red line in Figure 1) describes infection transmission among unvaccinated individuals. Each susceptible individual from (S) can be infected in case of contact with infectious persons and is transferred to group (E) with transmission rate $\beta \left(t\right).$ Furthermore, the individual goes in group (I) with latency rate $\omega \left(t\right)$ and later to group (R) with recovery rate $\gamma \left(t\right)$. Later, this person loses disease-acquired immunity and moves again to group (S) with re-infection rate $\lambda \left(t\right)$.
- (2)
- The chains S-V-B-S and S-V-E-I-R-S describe the change in the proportion of the numbers of the groups that contain vaccinated persons. Each vaccinated person in (S) can move to group (V) with vaccination parameter $\alpha \left(t\right)$ or stay in group (S) due to imperfect vaccines which means that this does not provide 100 % safety (with effectiveness $\sigma <1$). Each individual in group (V) can develop antibodies and go to group (B) with antibodies rate $\mu \left(t\right)$ or be infected before that and move to group (E) with the transmission rate $\beta \left(t\right)$. Individuals in group (B) lose vaccination-acquired immunity and move again to group (S) with re-infection rate $\nu \left(t\right)$.
- (3)
- The SEIRS-VB model describes vital dynamics as well. We assume that all new born individuals are susceptible and they come to group (S) with birth rate $\Lambda \left(t\right)$. At the same time, individuals in each model’s group can move out from the model, due to natural mortality with rate $\theta \left(t\right)$. Of course, infectious individuals (I) can leave the model due to COVID-19 mortality with rate $\tau \left(t\right)$. Since we use a model with re-susceptibility, it is reasonable to assume that all individuals having received booster doses of vaccines are new fully vaccinated persons, i.e., belong to group (S) who can possibly move to (V) due to these doses.

**SEIRS-VB model**is described by the following Cauchy problem for a system of nonlinear ordinary differential equations

#### 2.2. Properties of the Analytic Solution of the SEIRS-VB Model

**Remark**

**1.**

**Remark**

**2.**

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**(i) Existence.**Denote

**(ii) Uniqueness.**Let

## 3. Discretization of the SEIRS-VB Model

**Theorem**

**3.**

**Proof.**

**Remark**

**3.**

## 4. Parameter Identification

- (1)
- $Rtotal\left(t\right)$—The cumulative number of the individuals recovered from the disease to the time t. Unlike $R\left(t\right)$, individuals who have already lost disease-acquired immunity are counted in $Rtotal\left(t\right)$.
- (2)
- $Dtotal\left(t\right)$—The cumulative number of COVID-19 deaths;
- (3)
- $Vtotal\left(t\right)$—The cumulative number of the fully vaccinated individuals.

**Problem IDP:**Using the given data ${g}_{1}$, ${\left\{{\tilde{p}}_{k}\right\}}_{k=1}^{K-1}$, ${\left\{{m}_{k}\right\}}_{k=1}^{K}$, find the values

**Algorithm for solving IDP ( with $\mathbf{h}=\mathbf{1}$):**

- Replacing derivatives in (23) and (24) by finite difference with step $h=1$, it leads to the following relations and notations$$\begin{array}{cc}\hfill {I}_{\gamma ,k-1}& :={\gamma}_{k-1}{I}_{k-1}=Rtota{l}_{k}-Rtota{l}_{k-1},\hfill \\ \hfill {I}_{\tau ,k-1}& :={\tau}_{k-1}{I}_{k-1}=Dtota{l}_{k}-Dtota{l}_{k-1},\hfill \end{array}$$$k=2,3,\cdots ,K$. Since the values ${\left\{Dtotal{}_{k}\right\}}_{k=1}^{K}$ and ${\left\{Rtotal{}_{k}\right\}}_{k=1}^{K}$ are given and nondecreasing in respect to k, we find via (25) the nonnegative values ${\left\{{I}_{\gamma ,k-1}\right\}}_{k=2}^{K}$, ${\left\{{I}_{\tau ,k-1}\right\}}_{k=2}^{K}$.
- Since ${N}_{1}={S}_{1}+{A}_{1}+{R}_{1}+{V}_{1}+{B}_{1}$ is given, the relations (20), (24) and (25) imply$${N}_{k}=(1+{\Lambda}_{k-1}-{\theta}_{k-1}){N}_{k-1}-{I}_{\tau ,k-1},\phantom{\rule{0.166667em}{0ex}}k=2,3,\cdots ,K.$$Thus the values of population size can be calculated.
- The values of the vaccination parameter are$${\alpha}_{k-1}=\sigma \frac{Vtota{l}_{k}-Vtota{l}_{k-1}}{{N}_{k-1}},\phantom{\rule{0.166667em}{0ex}}k=2,3,\cdots ,K,$$
- Obviously, ${N}_{k-1}\ne 0$ and we introduce the notations$${I}_{\beta ,k-1}:={\beta}_{k-1}\frac{{I}_{k}}{{N}_{k-1}},\phantom{\rule{0.166667em}{0ex}}k=2,3,\cdots ,K.$$Now, going step by step from $k-1$ to k, using (25) and (28), we rewrite the relations (19) in the form$$\left(\right)$$Starting with the nonnegative initial data ${A}_{1}$ and ${g}_{1}=({S}_{1},{I}_{1},{R}_{1},{V}_{1},{B}_{1}),$ where ${S}_{1}>0$ and ${A}_{1}\ge {I}_{1}>0$, we calculate
- 4.1.
- From the second equation in (29), we obtain$${I}_{\beta ,1}=\frac{1}{{S}_{1}+{V}_{1}}[{A}_{2}+({\theta}_{1}-1){A}_{1}+{I}_{\gamma ,1}+{I}_{\tau ,1}].$$We note that ${I}_{\beta ,1}\ge 0$, because ${A}_{2}-{A}_{1}+{I}_{\gamma ,1}+{I}_{\tau ,1}\ge 0$ are the new cases for day ${t}_{2}$.
- 4.2.
- Now, using (29), we calculate consistently the values ${\left\{{S}_{k}\right\}}_{k=2}^{K}$, ${\left\{{I}_{k}\right\}}_{k=2}^{K}$, ${\left\{{R}_{k}\right\}}_{k=2}^{K}$, ${\left\{{V}_{k}\right\}}_{k=2}^{K}$, ${\left\{{B}_{k}\right\}}_{k=2}^{K}$ and, if ${S}_{k-1}+{V}_{k-1}\ne 0$$${I}_{\beta ,k-1}=\frac{1}{{S}_{k-1}+{V}_{k-1}}[{A}_{k}+({\theta}_{k-1}-1){A}_{k-1}+{I}_{\gamma ,k-1}+{I}_{\tau ,k-1}],\phantom{\rule{0.166667em}{0ex}}k=3,4,\cdots ,K.$$
- 4.3.
- Finally, if ${I}_{k-1}\ne 0$ and ${I}_{k}\ne 0$, we calculate$${\beta}_{k-1}={N}_{k-1}\frac{{I}_{\beta ,k-1}}{{I}_{k}},\phantom{\rule{0.277778em}{0ex}}{\gamma}_{k-1}=\frac{{I}_{\gamma ,k-1}}{{I}_{k-1}},\phantom{\rule{0.277778em}{0ex}}{\tau}_{k-1}=\frac{{I}_{\tau ,k-1}}{{I}_{k-1}},\phantom{\rule{0.166667em}{0ex}}k=2,3,\cdots ,K.$$
- 4.4.
- If one or more of the values ${S}_{k-1}+{V}_{k-1}$, ${I}_{k-1}$ is equal to zero for some k, and the algorithm must stop at the first such k. Otherwise, the algorithm continues and all values ${\left\{{g}_{k}\right\}}_{k=1}^{K}$ and ${\left\{{\widehat{p}}_{k-1}\right\}}_{k=2}^{K}$ can be uniquely determined. Actually, since we are looking for a biologically reasonable solution of the problem $IDP$, i.e., for nonnegative solution of this problem, we should also stop the algorithm if some of the calculated values in the step 4.2 are negative.

## 5. Identification of Parameters in the Discrete Problem

- ${\Lambda}_{k}=\Lambda $ is the average birth rate for 2015–2020;
- ${\theta}_{k}=\theta $ is the average natural mortality rate for 2015–2020;
- ${\omega}_{k}=1/{T}_{e}$, where ${T}_{e}$ is the incubation (latency) period for the dominant variant of COVID-19;
- ${\mu}_{k}=1/{T}_{a}$, where ${T}_{a}$ is the average time taken for antibodies to develop;
- ${\nu}_{k}=1/{T}_{b}$, where ${T}_{b}$ is the duration of the immune responses in individuals with vaccination-acquired immunity;
- ${\lambda}_{k}=1/{T}_{r}$, where ${T}_{r}$ is the duration of the immune responses in recovered individuals.

**Remark**

**4.**

## 6. Numerical Solution of the Differential Problem

## 7. Short-Term Forecasting

**Prediction step 1.**It is easy to discover cases/weekdays dependency in the official data. For example, the officially reported cases for Saturday and Sunday (reported on Sunday and Monday, respectively) are much fewer than the cases for Monday (reported on Tuesday). The same is true if we compare a holiday with a working day of the week. For this reason, we assume dependency of the parameters’ values on the weekdays. To verify this assumption, we examine the ratios between any two consecutive daily parameter values over different 7-day periods before the forecast period. For example, we study the behavior of the ratios

**Prediction step 2.**The first predicted day is ${t}_{n}$ and the values of parameters ${I}_{\beta ,n-2}$${I}_{\gamma ,n-2},\phantom{\rule{0.166667em}{0ex}}{I}_{\tau ,n-2},\phantom{\rule{0.166667em}{0ex}}{\alpha}_{n-2}$ for the day ${t}_{n-1}$ are known. Then, our forecast of parameters’ values for the day ${t}_{n}$ is

**Prediction step 3.**Since the values ${S}_{n-1},\phantom{\rule{0.166667em}{0ex}}{A}_{n-1},\phantom{\rule{0.166667em}{0ex}}{I}_{n-1},\phantom{\rule{0.166667em}{0ex}}{R}_{n-1},\phantom{\rule{0.166667em}{0ex}}{V}_{n-1},\phantom{\rule{0.166667em}{0ex}}{B}_{n-1}$ and $Rtota{l}_{n-1}$, $Dtota{l}_{n-1}$ for the last day before the considered forecast period are known, we can predict the number of individuals in each of these compartments during the period ${t}_{n},\cdots ,{t}_{n+13}$. Using (25) and the predicted daily values of parameters, we calculate the values ${\left\{\stackrel{\u02c7}{R}toata{l}_{k}\right\}}_{k=n}^{n+13}$ and ${\left\{\stackrel{\u02c7}{D}toata{l}_{k}\right\}}_{k=n}^{n+13}$ of the recovered individuals and COVID-19 deaths, respectively. Furthermore, using again the predicted daily values of the parameters and (29), we calculate the values ${\left\{{\stackrel{\u02c7}{A}}_{k}\right\}}_{k=n}^{n+13}$ of active cases and the new daily cases ${\{{\stackrel{\u02c7}{A}}_{k}-{\stackrel{\u02c7}{A}}_{k-1}+{\stackrel{\u02c7}{I}}_{\gamma ,k-1}+{\stackrel{\u02c7}{I}}_{\tau ,k-1}\}}_{k=n+1}^{n+13}$. More precisely, for each day of the considered time-frame, using the morning values $({\stackrel{\u02c7}{S}}_{k-1},{\stackrel{\u02c7}{A}}_{k-1},{\stackrel{\u02c7}{I}}_{k-1},{\stackrel{\u02c7}{R}}_{k-1},{\stackrel{\u02c7}{V}}_{k-1},{\stackrel{\u02c7}{B}}_{k-1})$, $(\stackrel{\u02c7}{D}tota{l}_{k-1},\stackrel{\u02c7}{R}tota{l}_{k-1})$, the parameter values ${\tilde{p}}_{k-1}$ and the predicted values ${\stackrel{\u02c7}{u}}_{k-1}$, we calculate $({\stackrel{\u02c7}{S}}_{k},{\stackrel{\u02c7}{A}}_{k},{\stackrel{\u02c7}{I}}_{k},{\stackrel{\u02c7}{R}}_{k},{\stackrel{\u02c7}{V}}_{k},{\stackrel{\u02c7}{B}}_{k})$ (using (29)) and $(\stackrel{\u02c7}{D}tota{l}_{k},\stackrel{\u02c7}{R}tota{l}_{k})$ (using (25)).

#### 7.1. The First Time-Frame 18–31 October 2021

#### 7.2. The Second Time-Frame 7–20 February 2022

## 8. Discussion

**herd immunity**threshold (see [40]).

**highly transmitted delta and omicron variants**. An excellent example in this regard was the decision of the United Kingdom Government to allow the opening of society based on a high percentage of vaccination coverage, which minimizes the risk of severe COVID-19 and death. Indeed, significantly increased mortality was not achieved, but the daily incidence reached over 270,000 new cases per day with Omicron. This decision has demonstrated that the purpose of herd immunity, even in a resource-rich environment, is unattainable [43].

**The conclusion is that, in general, re-infection with different strains is possible and, in some cases, may develop more severe infections with the second episode**[44].

**The question is:**How should we be thinking about “herd immunity” to COVID-19? According to today’s data for the rate of “herd immunity”, we can not eliminate SARS-CoV-2, and the virus continues to circulate. We do not have enough effective vaccines, and due to the high mutation rate of the virus, the vast majority of the population still can be exposed. That is why we should keep working, predicting and monitoring the situation according to this reality!

## 9. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**The weekly average values ${\beta}_{week},\phantom{\rule{0.166667em}{0ex}}{\gamma}_{week}$ of infection and recovery rates.

**Figure 3.**The weekly average values of vaccination parameter ${\alpha}_{week}$ and mortality rate ${\tau}_{week}$.

**Figure 7.**The first time-frame. Official (blue) and predicted (red) data for cumulative number of COVID-19 deaths.

**Figure 8.**The first time-frame. Official (blue) and predicted (red) data for cumulative number of recovered individuals.

**Figure 11.**The second time-frame. Official (blue) and predicted (red) data for the cumulative number of COVID-19 deaths.

**Figure 12.**The second time-frame. Official (blue) and predicted (red) data for cumulative number of recovered individuals.

Parameter | Description | Units |
---|---|---|

$\Lambda \left(t\right)$ | birth rate | $\frac{births}{population}$$/day$ |

$a\left(t\right)$ | vaccination rate | $\frac{vaccinated}{population}/day$ |

$\sigma $ | vaccine effectiveness | $\frac{excess\phantom{\rule{0.277778em}{0ex}}risk}{risk\phantom{\rule{0.277778em}{0ex}}among\phantom{\rule{0.277778em}{0ex}}vaccinated}$ |

$\alpha \left(t\right)$ | vaccination parameter | $\sigma a\left(t\right)$ |

$\beta \left(t\right)$ | transmission rate | 1/days |

$\gamma \left(t\right)$ | recovery rate | 1/ days |

$\omega \left(t\right)$ | latency rate | 1/days |

$\theta \left(t\right)$ | natural mortality rate | $\frac{deaths}{population}/day$ |

$\tau \left(t\right)$ | mortality rate of infectious people | $\frac{deaths}{infectious}/day$ |

$\lambda \left(t\right)$ | reinfection rate of recovered individuals | 1/days |

$\nu \left(t\right)$ | reinfection rate of vaccinated individuals | 1/days |

$\mu \left(t\right)$ | antibody rate | 1/days |

Parameter | Description | Values |
---|---|---|

$\Lambda $ | birth rate | $2.4095\times {10}^{-5}$ |

$\theta $ | natural mortality rate | $4.1904\times {10}^{-5}$ |

${T}_{e}$ | latency period | 7 days ^{1},
6 days ^{2}, 5 days ^{3}, 4 days ^{4} |

${T}_{a}$ | time taken for antibodies to develop | 14 days |

${T}_{b}$ | duration of vaccine-based immunity | 180 days |

${T}_{r}$ | duration of disease-based immunity | 180 days |

$\sigma $ | vaccine effectiveness | ${0.85}^{2},{0.70}^{3},{0.45}^{4}$ |

^{1}Wuhan variant,

^{2}Alpha variant,

^{3}Delta variant,

^{4}Omicron variant.

Mon/Sun | Tue/Mon | Wed/Tue | Thur/Wed | Fri/Thur | Sat/Fri | Sun/Sat | |
---|---|---|---|---|---|---|---|

k | ${\mathit{\delta}}_{\mathit{\beta},\mathit{k}}^{1}$ | ${\mathit{\delta}}_{\mathit{\beta},\mathit{k}}^{2}$ | ${\mathit{\delta}}_{\mathit{\beta},\mathit{k}}^{3}$ | ${\mathit{\delta}}_{\mathit{\beta},\mathit{k}}^{4}$ | ${\mathit{\delta}}_{\mathit{\beta},\mathit{k}}^{5}$ | ${\mathit{\delta}}_{\mathit{\beta},\mathit{k}}^{6}$ | ${\mathit{\delta}}_{\mathit{\beta},\mathit{k}}^{7}$ |

$n-8$ | 5.573 | 0.999 | 0.959 | 0.957 | 1.081 | 0.543 | 0.536 |

$n-15$ | 5.452 | 1.021 | 0.887 | 0.905 | 1.079 | 0.502 | 0.499 |

$n-22$ | 5.098 | 1.014 | 0.832 | 0.957 | 1.020 | 0.587 | 0.418 |

$n-29$ | 5.138 | 0.929 | 0.388 | 2.156 | 1.037 | 0.545 | 0.478 |

$n-36$ | 4.559 | 1.015 | 0.918 | 0.926 | 1.097 | 0.503 | 0.482 |

$n-43$ | 1.386 | 2.957 | 0.975 | 0.833 | 0.998 | 0.559 | 0.481 |

$n-50$ | 4.624 | 0.982 | 0.850 | 0.896 | 1.048 | 0.521 | 0.572 |

**Table 4.**Relative errors of the prediction for week 1 (18–24 October 2021) and week 2 (25–31 October 2021).

Compartment | Error (${\mathit{l}}_{2}$, Week${}_{1}$) | Error (${\mathit{l}}_{2}$, Week${}_{2}$) | Error (${\mathit{l}}_{\mathit{\infty}}$, Week${}_{1}$) | Error (${\mathit{l}}_{\mathit{\infty}}$, Week${}_{2}$) |
---|---|---|---|---|

Active cases | 0.018 | 0.034 | 0.029 | 0.042 |

New daily cases | 0.090 | 0.138 | 0.132 | 0.168 |

Deaths | 0.005 | 0.013 | 0.007 | 0.016 |

Recovered | 0.001 | 0.001 | 0.002 | 0.002 |

Mon/Sun | Tue/Mon | Wed/Tue | Thur/Wed | Fri/Thur | Sat/Fri | Sun/Sat | |
---|---|---|---|---|---|---|---|

k | ${\mathit{\delta}}_{\mathit{\gamma},\mathit{k}}^{1}$ | ${\mathit{\delta}}_{\mathit{\gamma},\mathit{k}}^{2}$ | ${\mathit{\delta}}_{\mathit{\gamma},\mathit{k}}^{3}$ | ${\mathit{\delta}}_{\mathit{\gamma},\mathit{k}}^{4}$ | ${\mathit{\delta}}_{\mathit{\gamma},\mathit{k}}^{5}$ | ${\mathit{\delta}}_{\mathit{\gamma},\mathit{k}}^{6}$ | ${\mathit{\delta}}_{\mathit{\gamma},\mathit{k}}^{7}$ |

$n-8$ | 2.311 | 0.857 | 1.051 | 1.306 | 1.135 | 0.557 | 0.593 |

$n-15$ | 4.249 | 0.459 | 1.790 | 0.971 | 0.656 | 0.292 | 2.444 |

$n-22$ | 0.447 | 3.186 | 0.490 | 0.442 | 5.049 | 0.649 | 1.101 |

$n-29$ | 10.917 | 1.425 | 4.729 | 0.650 | 0.562 | 0.408 | 1.326 |

$n-36$ | 1.229 | 2.100 | 0.565 | 2.221 | 0.401 | 2.655 | 0.047 |

$n-43$ | 4.265 | 1.381 | 1.485 | 0.485 | 0.512 | 0.201 | 4.522 |

$n-50$ | 8.402 | 0.878 | 0.734 | 0.950 | 0.237 | 0.681 | 1.293 |

**Table 6.**Relative errors of the prediction for week 1 (7–13 February 2022) and week 2 (14–20 February 2022).

Compartment | Error (${\mathit{l}}_{2}$, Week${}_{1}$) | Error (${\mathit{l}}_{2}$, Week${}_{2}$) | Error (${\mathit{l}}_{\mathit{\infty}}$, Week${}_{1}$) | Error (${\mathit{l}}_{\mathit{\infty}}$, Week${}_{2}$) |
---|---|---|---|---|

Active cases | 0.015 | 0.048 | 0.023 | 0.077 |

New daily cases | 0.107 | 0.230 | 0.137 | 0.210 |

Deaths | 0.001 | 0.005 | 0.001 | 0.007 |

Recovered | 0.001 | 0.028 | 0.002 | 0.041 |

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**MDPI and ACS Style**

Margenov, S.; Popivanov, N.; Ugrinova, I.; Hristov, T.
Mathematical Modeling and Short-Term Forecasting of the COVID-19 Epidemic in Bulgaria: SEIRS Model with Vaccination. *Mathematics* **2022**, *10*, 2570.
https://doi.org/10.3390/math10152570

**AMA Style**

Margenov S, Popivanov N, Ugrinova I, Hristov T.
Mathematical Modeling and Short-Term Forecasting of the COVID-19 Epidemic in Bulgaria: SEIRS Model with Vaccination. *Mathematics*. 2022; 10(15):2570.
https://doi.org/10.3390/math10152570

**Chicago/Turabian Style**

Margenov, Svetozar, Nedyu Popivanov, Iva Ugrinova, and Tsvetan Hristov.
2022. "Mathematical Modeling and Short-Term Forecasting of the COVID-19 Epidemic in Bulgaria: SEIRS Model with Vaccination" *Mathematics* 10, no. 15: 2570.
https://doi.org/10.3390/math10152570