# On a Coupled Time-Dependent SIR Models Fitting with New York and New-Jersey States COVID-19 Data

^{*}

## Abstract

**:**

## 1. Introduction and Model

## 2. Numerical Simulations, Data and Dynamics

#### 2.1. Fitting the Total Number of Infected People and the Number of Deaths

- The curve ${I}_{1}\left(t\right)$ in red corresponds to the simulation of (2) with $k\left(t\right)={k}_{1}=1.057$ and $d\left(t\right)={d}_{1}=0.0016$ for all time.
- The curve ${I}_{2}\left(t\right)$ in green corresponds to the simulation of (2) with $k\left(t\right)={k}_{1}=1.057$ and $d\left(t\right)={d}_{1}=0.016$ for $0\le t\le 21$, and $k\left(t\right)={k}_{2}=0.9$ and $d\left(t\right)={d}_{2}=0.00232$ for $21\le t<24$.
- The curve ${I}_{3}\left(t\right)$ in pink corresponds to the simulation of (2) with $k\left(t\right)$ as given in (3), i.e., $k\left(t\right)={k}_{i}$ and $d\left(t\right)={d}_{i}$, $t\in [{t}_{i-1},{t}_{i})$, $i\in \{1,\dots ,4\}$ with ${t}_{0}=0,{t}_{1}=21,{t}_{2}=24,{t}_{3}=27,{t}_{4}=32$.

#### 2.2. Fitting the Total Number of People at Hospital

#### 2.3. Dynamics

**Theorem**

**1.**

**Remark**

**1.**

## 3. Two Coupled SIR Systems Fitting COVID-19 for NY and NJ States

**Theorem**

**2.**

**Remark**

**2.**

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**This figure illustrates the simulation of system (2) and how it fits the data. In (

**a**), we have plotted the quantity $0.2\times (I+R)$ as a function of time in red. The blue dots correspond to the data retrieved from [20]. Analogously, in (

**b**), we have plotted the quantity ${\int}_{0}^{t}d\left(s\right)I\left(s\right)ds$, which represents the total number of deaths according with the model, as a function of time in red. The blue dots correspond to the data retrieved from [20]. In (

**c**), we have illustrated the quantity $I\left(t\right)$ corresponding to different values of $k\left(t\right),d\left(t\right)$: the curve ${I}_{1}\left(t\right)$ in red corresponds to the simulation of (2) with $k\left(t\right)={k}_{1}$ and $d\left(t\right)={d}_{1}$ for all time. The curve ${I}_{2}\left(t\right)$ in green corresponds to the simulation of (2) with $k\left(t\right)={k}_{1}$ and $d\left(t\right)={d}_{1}$ for $0\le t\le 21$, and $k\left(t\right)={k}_{2}$ and $d\left(t\right)={d}_{2}$ for $t\ge 21$. The curve ${I}_{3}\left(t\right)$ in pink corresponds to the simulation of (2) with $k\left(t\right)$ as given in (3), i.e., $k\left(t\right)={k}_{i}$ and $d\left(t\right)={d}_{i}$, $t\in [{t}_{i-1},{t}_{i})$, $i\in \{1,\dots ,4\}$ with ${t}_{0}=0,{t}_{1}=21,{t}_{2}=24,{t}_{3}=27,{t}_{4}=32$. It illustrates how the health policies flatten the curve. In (

**d**), we have again plotted the solution $I\left(t\right)$ for $k\left(t\right)$ as in (3), for a longer period.

**Figure 2.**This figure illustrates how to provide an estimation for people needing hospitalization thanks to Equation (2) and statistical methods. Panel (

**a**) illustrates an approximation of $I\left({t}_{i}\right),{t}_{i}\in \{16,17,\dots ,32\}$ by a vector $({a}_{1}H\left({t}_{i}\right)+{b}_{1}),{t}_{i}\in \{16,17,\dots ,23\}$ and another vector $({a}_{2}H\left({t}_{i}\right)+{b}_{2}),{t}_{i}\in \{24,25,\dots ,32\}$ where ${a}_{j}$ and ${b}_{j}$, $j\in \{1,2\}$ are the coefficients obtained thanks to the least-square method. Panel (

**b**) then provides a prediction of people in need of hospitalization by plotting the quantity $H\left(t\right)=\frac{I\left(t\right)-{b}_{2}}{{a}_{2}}$ if $I\left(t\right)\ge 41,475.76$, $H\left(t\right)=\frac{I\left(t\right)-{b}_{1}}{{a}_{1}}$ otherwise.

**Figure 3.**This figure illustrates the simulation of system (6) and how it fits the data. In (

**a**), we have plotted the quantity $0.2({I}_{2}+{R}_{2})$ as a function of time in red. Recall that, in the model ${I}_{2}+{R}_{2}$, represents the number of people in the population which has been infected by the virus and are still alive. The blue dots correspond to the data retrieved from [20] and plots the total number of infected minus the number of total deaths. Analogously, in (

**b**) we have plotted the quantity ${\int}_{0}^{t}d2\left(u\right){I}_{2}\left(u\right)du)$ as a function of time in red. The blue dots correspond to the total number of deaths in NJ according with data retrieved from [20]. Panel (

**c**) illustrates ${I}_{1}\left(t\right)$ and ${I}_{2}\left(t\right)$, which represent respectively the infected in NY and NJ.

**Table 1.**Total number of cases reported in NY state from 1 March to 1 April. See [20].

Day | 3/1 | 3/2 | 3/3 | 3/4 | 3/5 | 3/6 | 3/7 | 3/8 | 3/9 | 3/10 | 3/11 |

Number of Cases | 1 | 1 | 2 | 11 | 22 | 44 | 89 | 106 | 142 | 173 | 217 |

Day | 3/12 | 3/13 | 3/14 | 3/15 | 3/16 | 3/17 | 3/18 | 3/19 | 3/20 | 3/21 | 3/22 |

Number of Cases | 326 | 421 | 610 | 732 | 950 | 1374 | 2382 | 4152 | 7102 | 10356 | 15168 |

Day | 3/23 | 3/24 | 3/25 | 3/26 | 3/27 | 3/28 | 3/29 | 3/30 | 3/31 | 4/1 | |

Number of Cases | 20875 | 25665 | 33066 | 38987 | 44635 | 53363 | 59568 | 67174 | 75832 | 83804 |

**Table 2.**Total number of deaths reported in NY state from 1 March to 1 April. See [20].

Day | 3/1 | 3/2 | 3/3 | 3/4 | 3/5 | 3/6 | 3/7 | 3/8 | 3/9 | 3/10 | 3/11 |

Number of Deaths | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

Day | 3/12 | 3/13 | 3/14 | 3/15 | 3/16 | 3/17 | 3/18 | 3/19 | 3/20 | 3/21 | 3/22 |

Number of Deaths | 0 | 0 | 2 | 6 | 10 | 17 | 27 | 30 | 57 | 80 | 122 |

Day | 3/23 | 3/24 | 3/25 | 3/26 | 3/27 | 3/28 | 3/29 | 3/30 | 3/31 | 4/1 | |

Number of Deaths | 159 | 218 | 325 | 432 | 535 | 782 | 965 | 1224 | 1550 | 1941 |

**Table 3.**Summary of the values used in Figure 1c to obtain the curves I

_{1}(t), I

_{2}(t) and I

_{3}(t). Recall that k

_{1}= 1.057, k

_{2}= 0.9, k

_{3}= 0.67, k

_{4}= 0.71, d

_{1}= 0.0016, d

_{2}= 0.00232, d

_{3}= 0.00232, d

_{4}= 0.0068, t

_{0}= 0, t

_{1}= 21, t

_{2}= 24, t

_{3}= 27 and t

_{4}= 32.

$k\left(t\right)$ | $d\left(t\right)$ | |

${I}_{1}\left(t\right)$ | $k\left(t\right)=1.057$ | $d\left(t\right)=0.0016$ |

${I}_{2}\left(t\right)$ | $1.057$ if $t<21$; $0.9$ otherwise | $0.0016$ if $t<21$; $0.00232$ otherwise |

${I}_{3}\left(t\right)$ | ${k}_{i}$ if ${t}_{i-1}\le t<{t}_{i},i\in \{1,\dots ,4\}$ | ${d}_{i}$ if ${t}_{i-1}\le t<{t}_{i},i\in \{1,\dots ,4\}$ |

Day | 3/16 | 3/17 | 3/18 | 3/19 | 3/20 | 3/21 | 3/22 | 3/23 | 3/24 | 3/25 |

Total Number of Hospitalized | 326 | 496 | 617 | 1042 | 1496 | 2043 | 2629 | 3343 | 4079 | 5327 |

Day | 3/26 | 3/27 | 3/28 | 3/29 | 3/30 | 3/31 | 4/1 | |||

Total Number of Hospitalized | 6481 | 7328 | 8503 | 9517 | 10929 | 12226 | 13383 |

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**MDPI and ACS Style**

Ambrosio, B.; Aziz-Alaoui, M.A. On a Coupled Time-Dependent SIR Models Fitting with New York and New-Jersey States COVID-19 Data. *Biology* **2020**, *9*, 135.
https://doi.org/10.3390/biology9060135

**AMA Style**

Ambrosio B, Aziz-Alaoui MA. On a Coupled Time-Dependent SIR Models Fitting with New York and New-Jersey States COVID-19 Data. *Biology*. 2020; 9(6):135.
https://doi.org/10.3390/biology9060135

**Chicago/Turabian Style**

Ambrosio, Benjamin, and M. A. Aziz-Alaoui. 2020. "On a Coupled Time-Dependent SIR Models Fitting with New York and New-Jersey States COVID-19 Data" *Biology* 9, no. 6: 135.
https://doi.org/10.3390/biology9060135