# Empirical Modeling of COVID-19 Evolution with High/Direct Impact on Public Health and Risk Assessment

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## Abstract

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## 1. Introduction

## 2. Reported Cases and Deaths Data

#### 2.1. Data Scope

#### 2.2. Delimitation of Phases’ Domains

_{c1},N

_{c1}) and Fd(t

_{d1},N

_{d1}) for the reported daily cases and daily deaths, respectively. We note that the initial times (t

_{c0}) and (t

_{d0}) of phase (I) correspond to the day before the first non-null appearance of a new case (Table 2 and Table S1). On the other hand, the final times of the latent phase (I), which are the initial times (t

_{c1}) and (t

_{d1}) of the accelerated phase (II), are obtained by optimization techniques using least-square methods and nonlinear regression of the proposed model for the accelerated phase (II) presented in Section 3. However, the times (t

_{c1}) and (t

_{d1}) of the inflection points (Fc) and (Fd) in Figure 1 can be determined by two techniques—the derivation method and tangent method—which will be detailed below.

#### Derivation Method

_{c1}) and (t

_{d1}).

_{c1}) and (t

_{d1}) when the daily reported cases reach the maximum of the smoothed peak (Figure 2). Indeed, the inflection points (Fc) and (Fd) occur at the daily highest cases t

_{c1}and the daily highest deaths t

_{d1}. Consequently, corresponding coordinates: N

_{c1}= N

_{c}(t = t

_{c1}) and N

_{d1}= N

_{d}(t = t

_{d1}), respectively, are available (Table 1).

## 3. Accelerated Phase Modeling

_{c1}) and (t

_{d1}), as the second boundary of the accelerated phase (II) are determined, the first limit of times (t

_{c}) and (t

_{d}) can be adequately recognized only by nonlinear regression by optimizing the standard deviation (σ) and the relative error (E

_{rel}) between the experimental values (Table S1) and the values estimated by the proposed model.

_{c}(t) and the recorded cumulative deaths N

_{d}(t) in the accelerated phase (II) were suggested:

_{c}) and (δN

_{d}) are reliant on parameters and can be adjusted to the values of the reported cases N

_{c}(t

_{c}) and the deaths N

_{d}(t

_{d}) at the start of the accelerated phase (II), with optimization being the main preferred mean irrespective of the existence of a slight difference from the experimental values shown in Table S1.

_{c}

_{0}) and (A

_{d}

_{0}) denote case activity and death activity, respectively, expressed as follows:

_{c}(t

_{c1}) occurring at the highest day (t

_{c1}) and an inflection point (Fc) for the N

_{c}(t)-curve (Figure 1)—indicates that containment efforts are ineffective and interventions are minimal [30], and the curve takes on different shapes depending on the virus’s infection rate and the health system’s capacity [31].

_{c}(t

_{c1}) which occurs at the highest day (t

_{c1}) coupled with inflection point (Fc) for the N

_{c}(t)-curve (Figure 1) causes the peak height (Figure 2a, Equation (9)), being an indication of very weak containment policies and negligible interventions [32,33]. The curve assumes diverse shapes, depending on the infection rate of the virus and the health system capacity [34].

## 4. Correlation between Reported Cases and Deaths

_{d}

_{0}< A

_{c}

_{0}

_{c}(t) and the cumulative deaths N

_{d}(t) is to eliminate the time-variable and plot N

_{d}(t) as a function of N

_{c}(t), like in Figure 4. We observe an interesting linear dependence in a domain stretched between the two accelerated (II) and delayed (III) phases. After that, the positive deviation to the linearity (with high slope value) indicates that each reported case’s phase always precedes in time the similar phase related to death cases.

_{dc}) of about 64 days is due mathematically to the sign conflict between the two logarithms lnN

_{d}(t) and lnN

_{c}(t), which is clearly revealed in Figure S5. There is a benefit from this feature by following this variation over time from the beginning of the spread, and, when the mortality rate T(t) reaches the maximum, we can predict that the pandemic is preparing to move from the accelerated phase (I) to the delayed phase (II) (assuming there are no great changes in precautionary measures and the peoples’ behavior towards the pandemic.)

## 5. Prediction of Delayed Phase for Symmetric Behavior

_{c1}). This symmetric behavior occurs when there are no changes in the pandemic environments, such as precautionary measures and peoples’ behavior toward the COVID-19 pandemic, etc. On the other hand, the symmetric behavior is translated mathematically by the fact that the inflection point (Fc) will be a center of symmetry of the curve in Figure 1.

_{c}(t = t

_{c1}), the equation predicting the delayed phase (III) is expressed is as follows:

_{c}= 37 days. The previous values are close to the (τ

_{c}) given in Table 2 for the accelerated phase (II). Therefore, we can conclude that in a reliable approximation, we can simplify the problem and put the value of (τ

_{c}) in Equation (14) in place of (τ′

_{c}) without any net imprecision (Figure 6). The discrepancy between experimental values and estimated ones within 320 days is due to the fact that the process of spread is not symmetric.

## 6. Prediction of Delayed Phase for Asymmetric Behavior

_{c1}) occurring at the inflection point Fc (t = t

_{c1}), the equation predicting the delayed phase (III) becomes expressed as follows:

_{c}) is needed to be estimated using optimizations techniques. The downside of this situation is that we cannot apply any nonlinear regression if we do not have enough data points after the highest day (t = t

_{c1}). However, a successful prediction should also be in agreement with the limiting value (Nc

_{∞}) of the reported cumulative case at the end of the COVID-19 pandemic (Equation (17)).

_{c}= 56 days) determined by the least-square method of nonlinear regression.

## 7. Results and Discussion

## 8. Conclusions

_{c}(t), is computed. Afterward, development and comparison of numerical simulations with data were performed. To give physical meaning to the three parameters in our suggested model for future work, probable causal correlation with factors such as infected, recovered, hospitalized, serious cases, etc. will be investigated.

## Supplementary Materials

_{c1},N

_{c}

_{1}) related the total reported cases N

_{c}(t) for the first 350 days in Romania. (b) Graphical determination of the inflection point Fd(t

_{d}

_{1},N

_{d}

_{1}) related the total death cases N

_{d}(t) for the first 350 days in Romania. Figure S4. Second derivative function (d2Nc.dt2) related the total death cases N

_{c}(t) for the first 400 days in Romania. Figure S5. Natural logarithm of total reported cases lnN

_{c}(t) with time in the accelerated phase (II). Figure S6. Difference between logarithms of the derivative functions of the total reported cases and deaths; to confirm the global maximum occurring InNc.Nd-Curve (Figure 5).

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**New daily reported cases (

**a**) and new daily reported deaths (

**b**) for the first 380 days of the pandemic in Romania.

**Figure 3.**(

**a**) The total reported cases N

_{c}(t) for the first 200 days of the pandemic in Romania, the accelerated phase (II) using Equation (3). (

**b**) The total reported deaths N

_{d}(t) for the first 200 days of the pandemic in Romania, the accelerated phase (II) using Equation (4).

**Figure 4.**Cumulative deaths N

_{d}(t) versus the total reported cases Nc(t) for the first 350 days of the pandemic in Romania.

**Figure 5.**Mortality rate T(t) as a function of time for the first 350 days of the pandemic in Romania.

**Figure 6.**Total reported cases Nc(t) for the first 400 days for delayed phase (III) in symmetric behavior using Equation (14) and τ

_{c}’ = 46 days.

**Figure 7.**Total reported cases Nc(t) of the first 400 days for delayed phase (III) in asymmetric behavior using Equation (16) with τ

_{c}′ = 56 days and N

_{c∞}= 753,500.

**Table 1.**Different spread phases and identification of the accelerated phase for the reported cases and deaths.

Phase 0 | Phase I | Phase II | Phase III | |||
---|---|---|---|---|---|---|

Reported cases | ||||||

Absence | t = t_{c0} | Latent | t = t_{c} | Accelerated | t = t_{c1} | Delayed |

Deaths | ||||||

Absence | t = t_{d0} | Latent | t = t_{d} | Accelerated | t = t_{d1} | Delayed |

t_{c0} | t_{c} | t_{c}_{1} | ${\tau}_{c}$ | ${N}_{c0}$ | Erel | σ | N_{c}_{1} | A_{c}_{0} |

0 | 7 | 266 | 67.92 | 6745 | 7.85% | 13,693 | 373,474 | 99.31 |

t_{d0} | t_{d} | t_{d}_{1} | ${\tau}_{d}$ | ${N}_{d0}$ | Erel | σ | N_{d}_{1} | A_{d}_{0} |

25 | 15 | 279 | 121.88 | 1135 | 9.78% | 404.3 | 11,331 | 9.312 |

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**MDPI and ACS Style**

Ouerfelli, N.; Vrinceanu, N.; Coman, D.; Cioca, A.L.
Empirical Modeling of COVID-19 Evolution with High/Direct Impact on Public Health and Risk Assessment. *Int. J. Environ. Res. Public Health* **2022**, *19*, 3707.
https://doi.org/10.3390/ijerph19063707

**AMA Style**

Ouerfelli N, Vrinceanu N, Coman D, Cioca AL.
Empirical Modeling of COVID-19 Evolution with High/Direct Impact on Public Health and Risk Assessment. *International Journal of Environmental Research and Public Health*. 2022; 19(6):3707.
https://doi.org/10.3390/ijerph19063707

**Chicago/Turabian Style**

Ouerfelli, Noureddine, Narcisa Vrinceanu, Diana Coman, and Adriana Lavinia Cioca.
2022. "Empirical Modeling of COVID-19 Evolution with High/Direct Impact on Public Health and Risk Assessment" *International Journal of Environmental Research and Public Health* 19, no. 6: 3707.
https://doi.org/10.3390/ijerph19063707