# Modeling COVID-19 with Uncertainty in Granada, Spain. Intra-Hospitalary Circuit and Expectations over the Next Months

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## Abstract

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## 1. Introduction

^{2}[16]. Compared with the province of Madrid with 829.6 people per km

^{2}, the province of Granada has a low population density. Also, the percentage of accumulated infected between 1 May 2020 and 1 June 2020 was [2.3%–4.8%] in Granada and [10.3%–13.3%] in Madrid [17,18]. Therefore, the province of Granada had a low prevalence at the end of the first wave.

## 2. Materials and Methods

#### 2.1. Model Building

- (S) susceptible, when the individual is healthy;
- (Q) in lockdown, when the individual is at home to avoid the spread of the virus. In our model, lockdown only considers the March’s lockdown time;
- (L) latent or exposed, when the individual has been infected but it is not infectious yet;
- (I) infectious, when the individual is capable of spreading SARS-CoV-2;
- (R) recovered, when the individual recovers from the disease being asymptomatic or having mild symptoms;
- (H) hospitalized at ward, when the individual has severe symptoms and needs to be hospitalized;
- (U) in intensive care unit (ICU), when the individual has severe symptoms and needs to be treated in the intensive care unit;
- (F) deceased, when the individual dies because of the disease;
- (HU) after ICU, when an individual is transferred from ICU to other non-ICU department due to improvement in the evolution but still requires hospitalization;
- (A) discharged, when the individual gets better and is discharged from hospital.

- When the alarm state is decreed and most of the people have to be in lockdown, that is, move from S to Q, it is modeled by the term ${s}_{q}\left(t\right)$, where the model parameter ${s}_{q}\left(t\right)$ determines the transit of people from S to Q. ${s}_{q}\left(t\right)$ takes the value 0 except for 16 March 2020 and 31 March 2020, when the lockdown and the strict lockdown began in Spain, and 700,000 and 150,000 more individuals in Granada, respectively, move from S to Q [19].
- When the lockdown finishes, the transit of individuals from Q to S is modeled by the term ${q}_{s}\left(t\right)$ where the model parameter ${q}_{s}\left(t\right)$ is 0 except for 13 April 2020 when the strict lockdown finishes and 150,000 individuals move from Q to S, and from 5 May to 21 June 2020, leaving the lockdown 8750 people every day due to the gradual end of the confinement.
- An individual moves to latent state (L) if he/she gets infected by contact with an infectious individual. People in the hospital are isolated and controlled and, therefore, discarded for contagions. The transit is modeled by the non-linear term $\beta S\left(t\right)\frac{I\left(t\right)}{{P}_{T}}$, where the transmission rate parameter $\beta $ has to be calibrated. Furthermore, this parameter will change over time due to the global public health interventions.
- A latent individual transits to infectious state after a while and this is modeled by the linear term ${l}_{i}L\left(t\right)$, where the latency period ${t}_{0}=1/{l}_{i}$ is the time from the moment in which an individual is infected until the moment in which is able to transmit the virus, ${t}_{0}$. This period is different to the typical incubation period time from infection to onset, and according to [20], it takes from 2 to 7 days. Even though there is limited evidence about the possibility of infection one or two days before onset [21], let us consider that ${t}_{0}$ may take values from 1 to 6 days, and ${l}_{i}=1/{t}_{0}$.
- An infectious individual may become hospitalized (H), admitted to the Intensive Care Unit (U) or recover (R), and these transits are modeled by the linear terms ${i}_{h}I\left(t\right)$, ${i}_{u}I\left(t\right)$ and ${i}_{r}I\left(t\right)$, respectively. Here, we have three possible statuses for infectious individuals. Every one takes its time and has its probability, that is, ${i}_{h}={p}_{1}/{t}_{1}$, ${i}_{u}={p}_{2}/{t}_{2}$ and ${i}_{r}=(1-{p}_{1}-{p}_{2})/{t}_{3}$.
- ${p}_{1}$ is the percentage of infected who become hospitalized and ${t}_{1}$ days the time it takes; ${p}_{2}$ is the percentage of infected people admitted directly at ICU and ${t}_{2}$ days the time it takes. Then, the parameters of the transition from I to H and from I to U are ${i}_{h}={p}_{1}/{t}_{1}$ and ${i}_{u}={p}_{2}/{t}_{2}$, respectively.
- Those infected but are not hospitalized require around ${t}_{3}=14$ days to recover [22]. Thus, the transition from I to R is governed by the parameter ${i}_{r}=(1-{p}_{1}-{p}_{2})/14$.

- Hospitalized people may move to ICU (U) if they get worse, may die (F) or may be discharged (A). These transits are modeled by the linear terms ${h}_{u}H\left(t\right)$, ${h}_{f}H\left(t\right)$ and ${h}_{a}H\left(t\right)$, respectively. As before, here we have three possible statuses for hospitalized individuals and each one takes its time and its probability.
- ${p}_{4}$ is the percentage of hospitalized people who need to be admitted to ICU and ${t}_{4}$ days the time it takes. Hence, the people in H move to U governed by the parameter ${h}_{u}={p}_{4}/{t}_{4}$.
- ${p}_{5}$ is the percentage of hospitalized people who die after an average of ${t}_{5}$ days. Then, the transition parameter from H to F is ${h}_{f}={p}_{5}/{t}_{5}$.
- $(1-{p}_{4}-{p}_{5})$ is the percentage of hospitalized people who are discharged after an average of ${t}_{6}$ days in the hospital. Hence, the transition parameter from H to A is ${h}_{a}=(1-{p}_{4}-{p}_{5})/{t}_{6}$.

- People in ICU (U) may decease (F) or may get better and be transferred to other non-ICU department (HU). These transits are modeled by the linear terms ${u}_{f}U\left(t\right)$ and ${u}_{hu}U\left(t\right)$, respectively. Here we have 2 possible ways for individuals in ICU, die or get better and each one takes its time.
- The transition parameter from U to F is ${u}_{f}={p}_{7}/{t}_{7}$, where ${p}_{7}$ is the probability to die after ${t}_{7}$ days (in average) if the individual is in the ICU.
- The parameter that governs the transition from U to $HU$ is ${u}_{hu}=(1-{p}_{7})/{t}_{8}$, where ${t}_{8}$ days is the average time an individual needs to leave the ICU because he/she gets better.

- Finally, an individual in $HU$ may get better and be discharged. This transit is modeled by the linear term $h{u}_{a}HU\left(t\right)$, where ${h}_{ua}=1/{t}_{9}$, where ${t}_{9}$ is the average time to be discharged after leaving ICU.

#### 2.2. Model Calibration with Uncertainty

#### 2.3. Vaccination

- (V) vaccinated, when the individual is vaccinated and the vaccine is effective, protecting the individual.

- (1)
- we keep vaccinating at the same pace and no new population restrictions are applied;
- (2)
- the same as (1) applying stronger restrictions from 1 to 15 April;
- (3)
- the same as (2) and the vaccination pace increases to fulfill $70\%$ coverage the 31 August 2021.

## 3. Results

#### 3.1. Calibration

#### 3.2. Model Validation

#### 3.3. Vaccination Simulation

## 4. Discussion

## 5. Conclusions

#### Limitations of the Model

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

Coefficient | p-Value | Coefficient | p-Value | ||
---|---|---|---|---|---|

${\rho}_{12}$ | 0.693 | 0 | ${\rho}_{23}$ | 0.019 | 0.782 |

${\rho}_{13}$ | −0.213 | 0.00215 | ${\rho}_{24}$ | −0.122 | 0.08 |

${\rho}_{14}$ | −0.115 | 0.00769 | ${\rho}_{34}$ | 0.3164 | 0 |

**Figure A1.**Residuals of ICU. Solid line are the actual residuals and the dashed line is the AR approximation.

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**Figure 1.**Flow diagram of the COVID-19 transmission dynamics. The boxes represent the states of individuals respect to the disease, and the letters next to the arrows represent the transition rates between states.

**Figure 2.**Model calibration. (

**A**) Number of daily hospitalized at ward people. (

**B**) Number of daily people in ICU. (

**C**) Number of accumulated deceased people. (

**D**) Number of accumulated discharged people. (

**E**) Number of accumulated recovered people. (

**F**) Transmission rate $\beta $. The black bands are the mean and the CI95%. The red points are the data from the hospitals. Vertical red lines in (

**E**) show the confidence intervals of the prevalence in Granada given by [17,18].

**Figure 3.**Infection Fatality Ratio (IFR). Evolution of the IFR over time in Granada. Observe that IFR is decreasing in the non-virulence season (May–September).

**Figure 4.**Model validation. (

**A**) Number of daily hospitalized at ward people. (

**B**) Number of daily people in ICU. (

**C**) Number of accumulated deceased people. (

**D**) Number of accumulated discharged people. The vertical red lines (23 September 2020) divide the calibration from the validation. The black bands are the mean and the CI95% of the model outputs. The red points are the data collected from the hospitals. The model captures most of the data red points after 23 September, predicting the evolution of the second and the third wave.

**Figure 5.**Average COVID-19 evolution in the three proposed scenarios with $70\%$ of vaccine effectiveness. Scenario 1, continuous line. Scenario 2, dotted line. Scenario 3, dashed line. (

**A**) Number of infected people. (

**B**) Number of daily hospitalized at ward people. (

**C**) Number of daily people in ICU. (

**D**) Number of accumulated deaths. (

**E**) Number of recovered people. (

**F**) Number of vaccinated people. The wave in October 2021 in hospitalized and ICU appears because we are in the virulence season and the probability to be hospitalized or at ICU increases and the time to enter decreases.

Parameter | Transition | Time | Value |
---|---|---|---|

${s}_{q}$ | $S\u27f6Q$ | 16 March | 700,000 |

31 March | 150,000 | ||

${q}_{s}$ | $Q\u27f6S$ | 13 April | 150,000 |

5 May to 21 June | 8750 | ||

${t}_{3}$ | $I\u27f6R$ | All simulation | 14 |

**Table 2.**Summary of the model parameters to be calibrate and their calibration ranges. The calibration ranges are obtained from the hospital’s data.

Parameter | Transition | Time Range | Calibration Range |
---|---|---|---|

$I\left(0\right)$ | − | 1 March | 100–5000 |

$L\left(0\right)$ | − | 1 March | 100–5000 |

$\beta $ | $S\u27f6L$ | 1 March to 15 March | 0.07–0.7 |

15 March to 4 May | 0.03–0.4 | ||

virulence season | 0.03–0.4 | ||

non-virulence season | 0.03–0.4 | ||

${t}_{0}$ | $L\u27f6I$ | All simulation | 1–6 |

${p}_{1}$ | $I\u27f6H$ | virulence season | 0.01– 0.04 |

non-virulence season | 0.01–0.04 | ||

${t}_{1}$ | $I\u27f6H$ | All simulation | 9–12 |

${p}_{2}$ | $I\u27f6U$ | virulence season | 0.001– 0.004 |

non-virulence season | 0.001–0.004 | ||

${t}_{2}$ | $I\u27f6U$ | All simulation | 9–12 |

${p}_{4}$ | $H\u27f6U$ | virulence season | 0.000 –0.139 |

non-virulence season | 0.000–0.119 | ||

${t}_{4}$ | $H\u27f6U$ | virulence season | 1–8.6 |

non-virulence season | 3.2–13.6 | ||

${p}_{5}$ | $H\u27f6F$ | virulence season | 0.097–0.297 |

non-virulence season | 0.018–0.218 | ||

${t}_{5}$ | $H\u27f6F$ | virulence season | 1–15 |

non-virulence season | 1–15.2 | ||

${t}_{6}$ | $H\u27f6A$ | virulence season | 3–15 |

non-virulence season | 4–29 | ||

${p}_{7}$ | $U\u27f6F$ | virulence season | 0.245–0.445 |

non-virulence season | No deaths, ${p}_{7}=0$ | ||

${t}_{7}$ | $U\u27f6F$ | virulence season | 1–23.3 |

non-virulence season | - | ||

${t}_{8}$ | $U\u27f6HU$ | virulence season | 6–36.2 |

non-virulence season | 4.7–80.1 | ||

${t}_{9}$ | $HU\u27f6A$ | virulence season | 4–25 |

non-virulence season | 2.7–18.9 |

Parameter | Transition | Time | Calibrated Value Mean (2.5%CI–97.5%CI) |
---|---|---|---|

$I\left(0\right)$ | − | 1 March | 1192 (1166–1226) |

$L\left(0\right)$ | − | 1 March | 574 (559–589) |

$\beta $ | $S\u27f6L$ | 1 March to 15 March | 0.391 (0.384–0.403) |

15 March to 4 May | 0.039 (0.036–0.043) | ||

virulence season | 0.125 (0.114–0.136) | ||

non-virulence season | 0.100 (0.094–0.104) | ||

${t}_{0}$ | $L\u27f6I$ | All simulation | 3.8 (3.5-4.0) days |

${p}_{1}$ | $I\u27f6H$ | virulence season | 0.027 (0.026–0.029) |

non-virulence season | 0.009 (0.006–0.013) | ||

${t}_{1}$ | $I\u27f6H$ | All simulation | 11.5 (10.8–11.9) days |

${p}_{2}$ | $I\u27f6U$ | virulence season | 0.00239 (0.00206–0.00278) |

non-virulence season | 0.00060 (0.00045–0.00075) | ||

${t}_{2}$ | $I\u27f6U$ | All simulation | 10.5 (9.5–11.6) days |

${p}_{4}$ | $H\u27f6U$ | virulence season | 0.019 (0.018–0.021) |

non-virulence season | 0.018 (0.017–0.020) | ||

${t}_{4}$ | $H\u27f6U$ | virulence season | 4.3 (4.3–4.4) days |

non-virulence season | 8.3 (8.2–8.4) days | ||

${p}_{5}$ | $H\u27f6F$ | virulence season | 0.136 (0.135–0.137) |

non-virulence season | 0.075 (0.071–0.084) | ||

${t}_{5}$ | $H\u27f6F$ | virulence season | 8.6 ( 8.5–8.7) days |

non-virulence season | 7.5 (7.4–7.6) days | ||

${t}_{6}$ | $H\u27f6A$ | virulence season | 13.6 (13.5–13.7) days |

non-virulence season | 13.2 (13.2–13.3) days | ||

${p}_{7}$ | $U\u27f6F$ | virulence season | 0.248 (0.243–0.254) |

non-virulence season | 0 | ||

${t}_{7}$ | $U\u27f6F$ | virulence season | 13.7 (13.6–13.8 ) days |

non-virulence season | - | ||

${t}_{8}$ | $U\u27f6HU$ | virulence season | 15.0 (14.8–15.1) days |

non-virulence season | 19.9 (19.3–20.6) days | ||

${t}_{9}$ | $HU\u27f6A$ | virulence season | 13.7 (13.6–13.7) days |

non-virulence season | 11.0 (10.9-11.1) days |

**Table 4.**Comparison between the required percentage to reach the herd immunity (H) with the percentage of immunized people the 1 May and the 1 September (% immunized). Observe that the percentage of immunized people is very close to the required in Scenarios 1 and 2, and greater in Scenario 3. This explains why, after 1 May, there is a decline in the number of infected, more significant in scenario 3 where the vaccination pace increases.

Scenario 1 | ||
---|---|---|

1 May 2021 | 1 September 2021 | |

${R}_{0}$ | $1.40$, (1.31–1.46) | $1.74$, (1.59–1.89) |

H | $28.47\%$, (23.66%–31.51%) | $42.53\%$, (37.11%–47.09%) |

Recovered | $164,902$, (121916–212431) | $223,362$, (168,933–275,884) |

Vaccinated | $77,948$ | $170,135$ |

Immunized | $242,850$, (199,864–290,379) | $393,497$, (339,068–446,019) |

% immunized | $26.35\%$, (21.69%–31.51%) | $42.70\%$, (36.79%–48.40%) |

Scenario 2 | ||

1 May 2021 | 1 Sep 2021 | |

${R}_{0}$ | $1.40$, (1.31–1.46) | $1.74$, (1.59–1.89) |

H | $28.47\%$, (23.66%–31.51%) | $42.53\%$, (37.11%–47.09%) |

Recovered | 162738, (120,647–209,646) | 204643, (154,719–253,861) |

Vaccinated | $77,948$ | $170,135$ |

Immunized | $240,685$, (198,594–287,593) | $374,779$, (324,855–423,997) |

% immunized | $26.12\%$, (21.55 %–31.21%) | $40.67\%$, (35.25 %–46.01%) |

Scenario 3 | ||

1 May 2021 | 1 Sep 2021 | |

${R}_{0}$ | $1.40$, (1.31–1.46) | $1.74$, (1.59–1.89) |

H | $28.47\%$, (23.66%–31.51%) | $42.53\%$, (37.11%–47.09%) |

Recovered | 162737, (120,641–209,647) | $185,675$, (138,932–235,429) |

Vaccinated | $113,917$ | $648,532$ |

Immunized | $276,654$, (234,558–323,564) | $834,207$, (787,464–883,961) |

% immunized | $30.02\%$, (25.45%–35.11%) | $90.53\%$, (85.45%–95.93%) |

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**MDPI and ACS Style**

Garrido, J.M.; Martínez-Rodríguez, D.; Rodríguez-Serrano, F.; Sferle, S.-M.; Villanueva, R.-J.
Modeling COVID-19 with Uncertainty in Granada, Spain. Intra-Hospitalary Circuit and Expectations over the Next Months. *Mathematics* **2021**, *9*, 1132.
https://doi.org/10.3390/math9101132

**AMA Style**

Garrido JM, Martínez-Rodríguez D, Rodríguez-Serrano F, Sferle S-M, Villanueva R-J.
Modeling COVID-19 with Uncertainty in Granada, Spain. Intra-Hospitalary Circuit and Expectations over the Next Months. *Mathematics*. 2021; 9(10):1132.
https://doi.org/10.3390/math9101132

**Chicago/Turabian Style**

Garrido, José M., David Martínez-Rodríguez, Fernando Rodríguez-Serrano, Sorina-M. Sferle, and Rafael-J. Villanueva.
2021. "Modeling COVID-19 with Uncertainty in Granada, Spain. Intra-Hospitalary Circuit and Expectations over the Next Months" *Mathematics* 9, no. 10: 1132.
https://doi.org/10.3390/math9101132