^{1}

^{1}

^{2}

^{1}

^{2}

Currently, a novel coronavirus (2019-nCoV) causes an outbreak of viral pneumonia in Hubei province, China. In this paper, stochastic and deterministic models are proposed to investigate the transmission mechanism of 2019-nCoV from 15 January to 5 February 2020 in Hubei province. For the deterministic model, basic reproduction number

On 11 January 2020, 41 cases of pneumonia with unknown causes were reported by Wuhan Municipal Health Commission [

The 2019-nCoV outbreak has received considerable attention from domestic and international scholars [

As of 5 February 2020, more than 28,000 cases with 2019-nCoV and 600 deaths have been reported by the National Health Commission of China [

Suppose that the host population

Transfer diagram of the SEI epidemic process.

In Figure

For any

State transitions of

Event | State transition | |||
---|---|---|---|---|

Input rate | (1, 0, 0) | |||

Death of susceptible | (−1, 0, 0) | |||

Infection of susceptible | (−1, 1, 0) | |||

Death of exposed | (0, −1, 0) | |||

Transfer rate | (0, −1, 1) | |||

Death of infected | (0, 0, −1) |

Note:

Consider all states of the SEI process conditioned on nonextinction state

According to the results of [

Denote

It can be interpreted as the population densities of the susceptible, exposed, and infectious individuals at time

For a density process, Ross et al. [

Consider the density process

By definition of

Theorem

Denote

However, model (

And by calculation, the endemic equilibrium

Define the basic reproduction number as follows:

If

The Jacobian of model (

Its characteristic equation is given as follows:

Since

Denote

According to

For

Deterministic model (

The process

Consider the process

From Theorem

Theorem

From the above results, we know that the O-U process

In this section, deterministic model (

Numbers of confirmed and death cases in Hubei province, from 15 January 2020 to 5 February 2020.

The spread of 2019-nCoV started in December 2019, in which the whole population of Hubei was 59,170,000; that is,

Let

The estimated values of model parameters by least-squares method.

Parameters | Definition | Initial values | Estimated values |
---|---|---|---|

Input rate into the susceptible group | 0.003 | 0.0034 | |

Contact rate in the exposed period | 0.478 | 0.2800 | |

Contact rate in the infectious period | 0.368 | 0.2936 | |

Transfer rate from exposed to the infectious period | 0.500 | 0.5554 | |

Disease-caused death rate in the infectious period | 0.023 | 0.0280 | |

Natural death rate | — |

(a) Fitted curves of

Model validation is the most important step in the model building process. Residual analysis such as the coefficient of determination

Let

Basic reproduction number

Effects of six parameters for

Due to uncertainty associated with the estimation of certain parameter values, it is useful to carry out sensitivity analysis to investigate how sensitive

For instance, take

Sensitivity indexes of

Parameter | Sensitivity index |
---|---|

1.0000 | |

0.0567 | |

0.9433 | |

−0.0442 | |

−0.7546 | |

−1.2011 |

In this section, we mainly investigate dynamical properties of

According to the results of Section

Dynamical behaviors of the susceptible, exposed, and infectious individuals.

On the other hand, from (

Similarly, by (

Then,

From (

Figure

(a) Approximation of the quasi-stationary distribution; (b) marginal density of the susceptible; (c) marginal density of the exposed; (d) marginal density of infectious individuals.

Note that dynamical and diffusion properties of the deterministic model and density process are based on larger

Under different control settings, we calculate the endemic equilibrium

(I) Input control variable

(II) Screening control variable

(III) Isolation and treatment variable

(IV) All control variables are nonzero.

Comparing with

The endemic equilibrium

Case | Control strategy | Control variable | Control variable | Control variable | ||
---|---|---|---|---|---|---|

— | 0 | 0 | 0 | (0.1139, 0.0046, 0.0734) | 4.2656 | |

I | (a) | 0.0002 | 0 | 0 | (0.1139, 0.0043, 0.0678) | 4.0146 |

(b) | 0.0004 | 0 | 0 | (0.1139, 0.0039, 0.0622) | 3.7637 | |

(c) | 0.0014 | 0 | 0 | (0.1139, 0.0021, 0.0339) | 2.5092 | |

(d) | 0.0026 | 0 | 0 | (0.1139, 0.0000, 0.0001) | ||

II | (a) | 0 | 0.0300 | 0 | (0.1146, 0.0046, 0.0733) | 4.2396 |

(b) | 0 | 0.1800 | 0 | (0.1182, 0.0046, 0.0726) | 4.1101 | |

(c) | 0 | 0.2300 | 0 | (0.1194, 0.0046, 0.0723) | 4.0669 | |

(d) | 0 | 0.2800 | 0 | (0.1207, 0.0045, 0.0721) | ||

III | (a) | 0 | 0 | 0.0936 | (0.1628, 0.0040, 0.0638) | 2.9828 |

(b) | 0 | 0 | 0.1636 | (0.2400, 0.0031, 0.0485) | 2.0234 | |

(c) | 0 | 0 | 0.1936 | (0.3013, 0.0023, 0.0364) | 1.6123 | |

(d) | 0 | 0 | 0.2336 | (0.4565, 0.0004, 0.0058) | ||

IV | (a) | 0.0001 | 0.0100 | 0.0436 | (0.1327, 0.0042, 0.0669) | 3.5518 |

(b) | 0.0002 | 0.0200 | 0.0936 | (0.1638, 0.0037, 0.0579) | 2.7911 | |

(c) | 0.0003 | 0.0800 | 0.1936 | (0.3147, 0.0016, 0.0253) | 1.4070 | |

(d) | 0.0004 | 0.0940 | 0.2226 | (0.4284, 0.0000, 0.0000) |

Note: the case

In this paper, we propose an SEI epidemic process and a deterministic proportion model to investigate the outbreak of 2019-nCoV epidemic from 15 January 2020 to 5 February 2020 in Hubei province, China. In order to understand the property of the SEI process in endemic phase, the quasi-stationary distribution is of additional importance. An approximation approach is used to analyze quasi-stationary distribution based on the deterministic model.

Firstly, by scale transformation, the SEI process is equivalent to a density process with transition rate. Theorem

The 2019-nCoV data of Hubei province are conducted to evaluate the performance of the proposed models. Least-squares method is used to estimate unknown parameters, and residual analysis provides measures of model quality. Since

The work in this article bears some limitations and concerns. For instance, the time to disease extinction for 2019-nCoV is worth pursuing. Moreover, when

The data used to support the results of this study are available from the National Health Commission of the People’s Republic of China.

The authors declare that there are no conflicts of interest regarding the publication of this article.

This research was supported by the National Natural Science Foundation of China (Grant no. 11661076) and the Science and Technology Department of Xinjiang Uygur Autonomous Region (Grant no. 2018Q011).