# A Mathematical Model for COVID-19 with Variable Transmissibility and Hospitalizations: A Case Study in Paraguay

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## Abstract

**:**

## 1. Introduction

- the amount of available data and the quality of such data, that vary from one country to another, and
- the containment strategies applied in each country.

## 2. Materials and Methods

#### 2.1. SEIR-H Model

#### 2.1.1. Dynamics of The Spread

#### 2.1.2. Reproduction Number

#### 2.1.3. Dynamics of Required Hospital Resources

#### 2.2. Containment Measures in Paraguay

#### 2.3. Estimation of Model Parameters

- 1.
- diagnosed with positive confirmed cases or daily reported cases ${D}^{R}\left({t}_{j}\right)$, excluding the count of confirmed cases from travellers,
- 2.
- diagnosed with positive confirmed cases in the isolated groups who were travellers from abroad ${R}_{im}\left({t}_{j}\right)$,
- 3.
- occupation in general bed in the hospitals by the patients of COVID-19 ${D}^{H}\left({t}_{j}\right)$,
- 4.
- occupation in ICU by patients of COVID-19 ${D}^{U}\left({t}_{j}\right)$,
- 5.
- confirmed daily deaths by COVID-19 ${D}^{F}\left({t}_{j}\right)$.

Algorithm 1: Parameter estimation |

#### 2.3.1. Estimation of Initial Condition

`1`). The initial susceptible population is $S\left({t}_{0}\right)=N-E\left({t}_{0}\right)-I\left({t}_{0}\right)-R\left({t}_{0}\right)$.

`stan`[31] in the platform

`R`[32] is used for estimating the posterior distributions of the parameters. In the estimation,

`stan`uses the Hamiltonian Monte Carlo (HCM) algorithm of Markov chain Monte Carlo method (MCMC) [33]. The log probability ${l}_{p}$ (the logarithm of the right-hand side of the Equation (10)) is evaluated internally in

`stan`, and the parameter set from the posterior distribution samplings that gives the maximum log probability is chosen for the optimum values: $\{{e}_{0}^{\mathrm{opt}},{i}_{0}^{\mathrm{opt}},{\beta}_{0}^{\mathrm{opt}}\}=\{{e}_{{0}_{m}},{i}_{{0}_{m}},{\beta}_{{0}_{m}}|{\mathrm{max}}_{m\in [1,{N}_{\mathrm{iter}}]}\phantom{\rule{4pt}{0ex}}{l}_{p}\}$, where ${N}_{\mathrm{iter}}$ is the number of samplings of the MCMC method. This number of samplings with the amount of Markov chains are maintained large enough to ensure the convergence of the parameters obtained: 30 Markov-chains with 5000 iterations for each chain including, 1000 warm-up iterations, and the average proposal acceptance probability

`adapt_delta`[31] is set to $0.94$. These optimum values of the parameters are used to advance one day, and to obtain the state of the variables: $(S({t}_{0}+1),E({t}_{0}+1),I({t}_{0}+1),R({t}_{0}+1))$. To solve the system of differential equations,

`stan`uses the fourth and fifth-order Runge-Kutta method.

#### 2.3.2. Estimation of Transmissibility

`3`). The estimated transmissibility of a day before, ${\beta}_{j-1}^{\mathrm{opt}}$, is used in order to obtain the state of the variables at time ${t}_{j}$: $(S\left({t}_{j}\right),E\left({t}_{j}\right),I\left({t}_{j}\right),R\left({t}_{j}\right))$. This state is adopted as the initial condition for estimating the transmissibility ${\beta}_{j}$ by performing the Bayesian method as follows:

`adapt_delta`[31] is set initially to $0.94$. This average proposal acceptance probability is dynamically augmented by $0.01$ when divergent transitions after warm-up are observed. In most cases, this value is remained in $0.94$, and there are a few cases when it is augmented up to $0.97$.

#### 2.3.3. Estimation of the Parameters for the Dynamics in the Hospitals

`5`). In order to estimate the proportions ${\lambda}_{*}\left({t}_{{j}_{h}}\right)$, the initial condition for the SEIR part is obtained using the optimal estimated transmissibility at time ${t}_{{j}_{h}}-1$ (${\beta}_{{j}_{h}-1}^{\mathrm{opt}}$), and the initial state of the compartments H, U and F are set using the 7-days average (from ${t}_{{j}_{h}}-3$ to ${t}_{{j}_{h}}+3$) of the data of hospitalisation (normal bed), ICU and daily death, respectively.

`stan`. 60 Markov-chains with 3000 iterations for each chain including 1000 warm-up iterations are used, and the average proposal acceptance probability

`adapt_delta`[31] is set initially to 0.94. Similar to the transmissibility estimation, this average proposal acceptance probability is dynamically augmented by 0.01 when divergent transitions after warm-up are observed. However, this value remains at 0.94 at almost all times, and there are a few cases when it is augmented up to 0.95.

## 3. Results

#### 3.1. Estimated Parameters

#### Estimated Parameters for Hospital Resources

#### 3.2. Model Assessment

#### 3.2.1. Comparisons

#### 3.2.2. Projection Error under the Assumption of Nearly Constant Parameter

## 4. Discussion

## 5. Relevance of the Work

## 6. Conclusions and Future Work

- Integration of health system dynamics into the SEIR model, without losing the simplicity of the SEIR model.
- The methodology of parameter estimation for the time-varying parameters using a moving time-window, which allows us to determine the variability of the parameters according to the spread dynamics and social behaviour.
- The inclusion of the reported cases that were travellers from abroad in the inflow of reported cases, which is important at the beginning of an epidemic.
- Model assessment and statistical analysis of the error as a function of the projection horizon.
- Discussion of spread dynamics in Paraguay using the estimated parameters and trajectories obtained.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Detailed Restriction Policies In Paraguay

- Paraguarí and Concepción in Phase 2 until 12 July 2020, representing at most 7% of the population.
- Alto Paraguay in Phase 3 until 29 July 2020, representing at most 0.25 % of the population.
- Central in Phase 3 until 22 August 2020, in addition to an intermediate phase between 2 and 3, from 23 August to 4 October 2020. That represents at most 37.5% of the population.
- Boquerón and Carmelo Peralta (Alto Paraguay) in Phase 3 from 31 August to 4 October 2020. That represents at most 0.98% of the population.
- Alto Paraná in an intermediate phase between 0 and 1, from 30 July to 20 September 2020, in addition to an intermediate phase between 2 and 3, from 21 September to 4 October 2020. That represents at most 11.4% of the population.
- Caaguazú and Concepción in an intermediate phase between 2 and 3, from 13 September to 4 October 2020. That represents at most 11.3% of the population.
- Greater restrictions in the "red zones” from 18 March to 26 March 2021 and from 24 April to 10 May 2021.
- Greater restrictions nationwide for Easter from 27 March to 4 April 2021.

## Appendix B. Performance Comparison

**Figure A2.**Statistical performance comparison of the daily deaths forecasting between the proposed SEIR-H model with the publicly available IHME COVID-19 projections: The values on the coordinate axes are scaled with the z-score (or standard score defined as: $\left[\varphi -\mathrm{ave}\left(\varphi \right)\right]/\mathrm{std}\left(\varphi \right)$); the solid lines represent the fit to the line equation with zero intercept and estimated slope; the dashed lines show the line equation for the hypothesis zero intercept and unit slope.

## References

- Doms, C.; Kramer, S.C.; Shaman, J. Assessing the Use of Influenza Forecasts and Epidemiological Modeling in Public Health Decision Making in the United States. Sci. Rep.
**2018**, 8, 12406. [Google Scholar] [CrossRef][Green Version] - Anderson, R.M.; Heesterbeek, H.; Klinkenberg, D.; Hollingsworth, T.D. How will country-based mitigation measures influence the course of the COVID-19 epidemic? Lancet
**2020**, 395, 931–934. [Google Scholar] [CrossRef] - Sorci, G.; Faivre, B.; Morand, S. Explaining among-country variation in COVID-19 case fatality rate. Sci. Rep.
**2020**, 10, 1–11. [Google Scholar] [CrossRef] - Zimmer, C.; Corum, J.; Wee, S. Coronavirus Vaccine Tracker. Available online: https://www.nytimes.com/interactive/2020/science/coronavirus-vaccine-tracker.html (accessed on 20 May 2021).
- Cooper, I.; Mondal, A.; Antonopoulos, C.G. A SIR model assumption for the spread of COVID-19 in different communities. Chaos Solitons Fractals
**2020**, 139, 110057. [Google Scholar] [CrossRef] - Ellison, G. Implications of Heterogeneous SIR Models for Analyses of COVID-19; Working Paper 27373; National Bureau of Economic Research: Cambridge, MA, USA, 2020. [Google Scholar] [CrossRef]
- Crokidakis, N. Modeling the early evolution of the COVID-19 in Brazil: Results from a susceptible-infectious-quarantined-recovered (SIQR) model. Int. J. Mod. Phys. C
**2020**, 31, 2050135. [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] - Moghadas, S.M.; Fitzpatrick, M.C.; Sah, P.; Pandey, A.; Shoukat, A.; Singer, B.H.; Galvani, A.P. The implications of silent transmission for the control of COVID-19 outbreaks. Proc. Natl. Acad. Sci. USA
**2020**, 117, 17513–17515. [Google Scholar] [CrossRef] [PubMed] - Qian, G.; Yang, N.; Ma, A.H.Y.; Wang, L.; Li, G.; Chen, X.; Chen, X. COVID-19 transmission within a family cluster by presymptomatic carriers in China. Clin. Infect. Dis.
**2020**, 71, 861–862. [Google Scholar] [CrossRef] [PubMed] - Arino, J.; Portet, S. A simple model for COVID-19. Infect. Dis. Model.
**2020**, 5, 309–315. [Google Scholar] [CrossRef] [PubMed] - 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] - Ramos, A.; Ferrández, M.; Vela-Pérez, M.; Kubik, A.; Ivorra, B. A simple but complex enough θ-SIR type model to be used with COVID-19 real data. Application to the case of Italy. Phys. D Nonlinear Phenom.
**2021**, 421, 132839. [Google Scholar] [CrossRef] - Arenas, A.; Cota, W.; Gómez-Gardeñes, J.; Gómez, S.; Granell, C.; Matamalas, J.T.; Soriano-Paños, D.; Steinegger, B. Modeling the Spatiotemporal Epidemic Spreading of COVID-19 and the Impact of Mobility and Social Distancing Interventions. Phys. Rev. X
**2020**, 10, 041055. [Google Scholar] [CrossRef] - Hethcote, H.W. Mathematics of infectious diseases. SIAM Rev.
**2000**, 42, 599–653. [Google Scholar] [CrossRef][Green Version] - Feng, Z. Final and peak epidemic sizes for SEIR models with quarantine and isolation. Math. Biosci. Eng.
**2007**, 4, 675–686. [Google Scholar] [CrossRef] - Akaike, H. A New Look at the Statistical Model Identification. IEEE Trans. Autom. Control
**1974**, 19, 716–723. [Google Scholar] [CrossRef] - Peirlinck, M.; Linka, K.; Sahli Costabal, F.; Kuhl, E. Outbreak dynamics of COVID-19 in China and the United States. Biomech. Model. Mechanobiol.
**2020**, 19, 2179–2193. [Google Scholar] [CrossRef] [PubMed] - Roda, W.C.; Varughese, M.B.; Han, D.; Li, M.Y. Why is it difficult to accurately predict the COVID-19 epidemic? Infect. Dis. Model.
**2020**, 5, 271–281. [Google Scholar] [CrossRef] [PubMed] - Sjödin, H.; Wilder-Smith, A.; Osman, S.; Farooq, Z.; Rocklöv, J. Only strict quarantine measures can curb the coronavirus disease (COVID-19) outbreak in Italy, 2020. Eurosurveillance
**2020**, 25, 2000280. [Google Scholar] [CrossRef] [PubMed] - Wells, C.R.; Townsend, J.P.; Pandey, A.; Moghadas, S.M.; Krieger, G.; Singer, B.; McDonald, R.H.; Fitzpatrick, M.C.; Galvani, A.P. Optimal COVID-19 quarantine and testing strategies. Nat. Commun.
**2021**, 12, 356. [Google Scholar] [CrossRef] - Anderson, R.M.; May, R.M. Infectious Diseases of Humans: Dynamics and Control; Oxford University Press: Oxford, UK, 1992. [Google Scholar]
- Furukawa, N.W.; Brooks, J.T.; Sobel, J. Evidence Supporting Transmission of Severe Acute Respiratory Syndrome Coronavirus 2 While Presymptomatic or Asymptomatic. Emerg. Infect. Dis.
**2020**, 26, e201595. [Google Scholar] [CrossRef] [PubMed] - He, X.; Lau, E.H.; Wu, P.; Deng, X.; Wang, J.; Hao, X.; Lau, Y.C.; Zhang, F.; Cowling, B.J.; Li, F.; et al. Temporal dynamics in viral shedding and transmissibility of COVID-19. Nat. Med.
**2020**, 26, 672–675. [Google Scholar] [CrossRef][Green Version] - Wu, J.T.; Leung, K.; Leung, G.M. Nowcasting and forecasting the potential domestic and international spread of the 2019-nCoV outbreak originating in Wuhan, China: A modelling study. Lancet
**2020**, 395, 689–697. [Google Scholar] [CrossRef][Green Version] - Peak, C.M.; Kahn, R.; Grad, Y.H.; Childs, L.M.; Li, R.; Lipsitch, M.; Buckee, C.O. Comparative Impact of Individual Quarantine vs. Active Monitoring of Contacts for the Mitigation of COVID-19: A modelling study. medRxiv
**2020**. [Google Scholar] [CrossRef] - Delamater, P.L.; Street, E.J.; Leslie, T.F.; Yang, Y.T.; Jacobsen, K.H. Complexity of the basic reproduction number (R
_{0}). Emerg. Infect. Dis.**2019**, 25, 1–4. [Google Scholar] [CrossRef] [PubMed][Green Version] - Smith, R.J.; Li, J.; Blakeley, D. The failure of R
_{0}. In Computational and Mathematical Methods in Medicine; Chu, H., Ed.; Hindawi Limited: London, UK, 2011; Volume 2011, p. 527610. [Google Scholar] [CrossRef][Green Version] - Presidencia de la República del Paraguay. Decretos (COVID-19). Available online: https://www.mspbs.gov.py/decretos-covid19.html (accessed on 28 May 2021).
- Presidencia de la República del Paraguay. Resoluciones (COVID-19). Available online: https://www.mspbs.gov.py/resoluciones-covid19.html (accessed on 28 May 2021).
- Stan Development Team. RStan: The R interface to Stan. R Package Version 2.21.2 2020. Available online: http://mc-stan.org/ (accessed on 20 May 2021).
- R Core Team. R: A Language and Environment for Statistical Computing. 2020. Available online: https://www.R-project.org/ (accessed on 20 May 2021).
- Neal, R.M. MCMC using Hamiltonian dynamics. In Handbook of Markov Chain Monte Carlo; Jones, G., Meng, X.L., Gelman, A., Brooks, S., Eds.; Chapman and Hall/CRC: New York, NY, USA, 2011; Chapter 5; pp. 113–162. [Google Scholar] [CrossRef]
- Li, Q.; Guan, X.; Wu, P.; Wang, X.; Zhou, L.; Tong, Y.; Ren, R.; Cowling, B.J.; Yang, B.; Leung, G.M.; et al. Early Transmission Dynamics in Wuhan, China, of Novel Coronavirus–Infected Pneumonia. N. Engl. J. Med.
**2020**, 382, 1199–1207. [Google Scholar] [CrossRef] - Friedman, J.; Liu, P.; Troeger, C.E.; Carter, A.; Reiner, R.C.; Barber, R.M.; Collins, J.; Lim, S.S.; Pigott, D.M.; Vos, T.; et al. Predictive performance of international COVID-19 mortality forecasting models. Nat. Commun.
**2021**, 12, 2609. [Google Scholar] [CrossRef] - Wu, S.L.; Mertens, A.N.; Crider, Y.S.; Nguyen, A.; Pokpongkiat, N.N.; Djajadi, S.; Seth, A.; Hsiang, M.S.; Colford, J.M.; Reingold, A.; et al. Substantial underestimation of SARS-CoV-2 infection in the United States. Nat. Commun.
**2020**, 11, 4507. [Google Scholar] [CrossRef] - Moghadas, S.M.; Shoukat, A.; Fitzpatrick, M.C.; Wells, C.R.; Sah, P.; Pandey, A.; Sachs, J.D.; Wang, Z.; Meyers, L.A.; Singer, B.H.; et al. Projecting hospital utilization during the COVID-19 outbreaks in the United States. Proc. Natl. Acad. Sci. USA
**2020**, 117, 9122–9126. [Google Scholar] [CrossRef][Green Version]

**Figure 1.**Schematic diagrams of the SEIR-H model: the region in yellow indicates the transmission dynamics given by a SEIR with external reported cases, and the region in green represents the dynamics related to hospital resources. A more detailed version of the green region is used to show that the R compartment contains the dynamics related to the health system, represented mainly by H, U and F.

**Figure 2.**The estimated transmissibility ${\beta}^{\mathrm{opt}}$ with 95% credibility interval of the posterior distribution, obtained using the data of Paraguay. The horizontal line at $1/7$ corresponds to the value of $\beta $ that gives the basic reproduction number equivalent to one.

**Figure 3.**Details of the data and simulations between May and June 2020. The results from the simulation are obtained using optimal ${\beta}^{\mathrm{opt}}$. Left vertical axis: the results of simulations and data; right vertical axis: transmissibility ${\beta}^{\mathrm{opt}}$.

**Figure 4.**The proportions for the dynamics in the hospitals: the optimal values with the 95 credibility interval.

**Figure 5.**The trajectories from the simulations using ${\beta}^{\mathrm{opt}}\left(t\right)$ and ${\lambda}_{*}^{\mathrm{opt}}\left(t\right)$ compared to the data. Using the posterior distribution estimated on 1 April 2021, regions for trajectories are shown for one month ahead with percentile 95 credibility intervals.

Parameter | Description | Type * | Value ** |
---|---|---|---|

$\beta $ | Transmissibility. | T-D | Est. |

$\alpha $ | Reciprocal of average latent period. | C | 1/3 ${}^{\left(a\right)}$ |

$\gamma $ | Reciprocal of average infectious period. | C | 1/7 ${}^{\left(b\right)}$ |

${\lambda}_{I\to H}$ | Proportion of infected that are hospitalised. | T-D | Est. |

${\lambda}_{I\to F}$ | Proportion of infected that are death. | T-D | Est. |

${\lambda}_{H\to U}$ | Proportion of hospitalised that are admitted in ICU. | T-D | Est. |

${\lambda}_{H\to F}$ | Proportion of hospitalised that are death. | T-D | Est. |

${\lambda}_{U\to F}$ | Proportion of admitted in ICU that are death. | T-D | Est. |

${\delta}_{H\to U}$ | Reciprocal of average stay in hospitals bed before admittion in ICU. | C | 1/7 ${}^{\left(c\right)}$ |

${\delta}_{H\to F}$ | Reciprocal of average stay in hospital bed before death. | C | 1/9 ${}^{\left(c\right)}$ |

${\delta}_{H\to O}$ | Reciprocal of average recovery period in hospital bed. | C | 1/11 ${}^{\left(c\right)}$ |

${\phi}_{U\to F}$ | Reciprocal of average stay in ICU before death. | C | 1/11 ${}^{\left(c\right)}$ |

${\phi}_{U\to O}$ | Reciprocal of average recovery period in ICU. | C | 1/12 ${}^{\left(c\right)}$ |

Denomination | Time Frame | Description |
---|---|---|

Pre-quarantine | 11 March to 19 March 2020 | - Curfew from 8 p.m. to 4 a.m. - First restrictions taken. |

Phase 0 | 20 March to 3 May 2020 | - 24 h curfew. - Confinement/isolation. |

Phase 1 | 4 May to 24 May 2020 | - Curfew from 9 p.m. to 5 a.m. - Flexibility: marginal. |

Phase 2 | 25 May to 14 June 2020 | - Curfew from 9 p.m. to 5 a.m. - Flexibility: slight. |

Phase 3 | 15 June to 19 July 2020 | - Curfew from 23 p.m. to 5 a.m. (Sunday to Thursday) and midnight to 5 a.m. (Friday to Saturday). - Flexibility: moderate. - Greater local restrictions in two departments, representing at most 7% of the population. |

Phase 4 | 20 July to 4 October 2020 | - Curfew from 23 p.m. to 5 a.m. (Sunday to Thursday) andmidnight to 5 a.m. (Friday to Saturday). - Flexibility: high. - Greater local restrictions in six departments, representing at most 61% of the population. |

New normality | From 5 October 2020 | - Curfew from midnight to 5 a.m. (but from 8 p.m. to 5 a.m. in red areas). - Only specific restrictions, until the appearance of the vaccine or cure. |

Parameter | Search Range | Prior Distribution |
---|---|---|

$E\left({t}_{0}\right)$ | $[0,100]$ | Uniform |

$I\left({t}_{0}\right)$ | $[0,100]$ | Uniform |

$\beta $ | $[0,0.8]$ | Uniform |

${\lambda}_{I\to H}$ | $[0,0.15]$ | Uniform |

${\lambda}_{I\to F}$ | $[0,0.004]$ | Uniform |

${\lambda}_{H\to U}$ | $[0,0.3]$ | Uniform |

${\lambda}_{H\to F}$ | $[0,0.25]$ | Uniform |

${\lambda}_{U\to F}$ | $[0.2,1]$ | Normal: $\mu =0.4$, $\sigma =0.1$ |

Variable | $\mathrm{rel}\_\mathrm{dev}$ | $\mathrm{dev}\_\mathrm{rel}\_\mathrm{error}$ |
---|---|---|

Daily Reported | $\approx 0.413$ | $\approx 0.668$ |

Hospitalised | $\approx 0.039$ | $\approx 0.047$ |

UCI | $\approx 0.036$ | $\approx 0.050$ |

Daily Death | $\approx 0.204$ | $\approx 0.234$ |

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## Share and Cite

**MDPI and ACS Style**

Shin, H.H.; Sauer Ayala, C.; Pérez-Estigarribia, P.; Grillo, S.; Segovia-Cabrera, L.; García-Torres, M.; Gaona, C.; Irala, S.; Pedrozo, M.E.; Sequera, G.; Vázquez Noguera, J.L.; De Los Santos, E. A Mathematical Model for COVID-19 with Variable Transmissibility and Hospitalizations: A Case Study in Paraguay. *Appl. Sci.* **2021**, *11*, 9726.
https://doi.org/10.3390/app11209726

**AMA Style**

Shin HH, Sauer Ayala C, Pérez-Estigarribia P, Grillo S, Segovia-Cabrera L, García-Torres M, Gaona C, Irala S, Pedrozo ME, Sequera G, Vázquez Noguera JL, De Los Santos E. A Mathematical Model for COVID-19 with Variable Transmissibility and Hospitalizations: A Case Study in Paraguay. *Applied Sciences*. 2021; 11(20):9726.
https://doi.org/10.3390/app11209726

**Chicago/Turabian Style**

Shin, Hyun Ho, Carlos Sauer Ayala, Pastor Pérez-Estigarribia, Sebastián Grillo, Leticia Segovia-Cabrera, Miguel García-Torres, Carlos Gaona, Sandra Irala, María Esther Pedrozo, Guillermo Sequera, José Luis Vázquez Noguera, and Eduardo De Los Santos. 2021. "A Mathematical Model for COVID-19 with Variable Transmissibility and Hospitalizations: A Case Study in Paraguay" *Applied Sciences* 11, no. 20: 9726.
https://doi.org/10.3390/app11209726