Abstract

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 is defined and endemic equilibrium is given. Under , quasi-stationary distribution of the stochastic process is approximated by Gaussian diffusion. Residual, sensitivity, dynamical, and diffusion analyses of the models are conducted. Further, control variables are introduced to the deterministic model and optimal strategies are provided. Based on empirical results, we suggest that the first and most important thing is to control input, screening, treatment, and isolation.

1. Introduction

On 11 January 2020, 41 cases of pneumonia with unknown causes were reported by Wuhan Municipal Health Commission [1]. The main clinical features initially include fever, cough, shortness of breath, or chest radiographs showing invasive pneumonic infiltrates in both lungs. Some patients subsequently developed pneumonia, acute respiratory distress syndrome, kidney failure, and even death [2, 3]. On 12 January 2020, the World Health Organization (WHO) termed it as 2019 novel coronavirus (2019-nCoV) [4]. The National Health Commission of China announced 2019-nCoV infected pneumonia to be included in the management of statutory infectious diseases on 20 January 2020 [5]. On 11 February 2020, the WHO gave an official name to the novel coronavirus that has killed more than 1,000 people as coronavirus disease (COVID-19) [6].

The 2019-nCoV outbreak has received considerable attention from domestic and international scholars [7–13]. Clinical evidence of 2019-nCoV refers to [14–16]. Chan et al. [17] indicated 2019-nCoV disease can spread from person to person by a study of a family cluster. The authors of [18–20] analyzed transmission risk, the unreported number, and basic reproduction number estimations of 2019-nCoV cases in China, respectively. Mathematical modeling can be an important tool for gaining an understanding of the spread of 2019-nCoV such as stochastic and deterministic models. For stochastic versions of SI, SIS, SIR, and SIRS models, refer to [21, 22]. Concerning the deterministic models, special attention has received those using ordinary differential equations and dynamical systems in their formulation [23, 24].

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 [5], and there were 70% of confirmed cases and 97% of deaths in Hebei province, China. In this paper, we propose a stochastic SEI process and a deterministic model to investigate 2019-nCoV transmission in Hubei province. We introduce an SEI Markovian epidemic process to analyze a density process in Section 2.1. A deterministic proportion model is proposed in Section 2.2, including an endemic equilibrium and local stability. A Gaussian diffusion approximation is considered in Section 2.3. If a scaled density process starts close to a deterministic equilibrium, this diffusion process is an Ornstein–Uhlenbeck (O-U) process. Main results are obtained in Section 3, and discussions are given in Section 4.

2. Methods

Suppose that the host population is partitioned into three compartments: susceptible, exposed (infected but symptom-free), and infectious with symptoms. Let , , and be the numbers of susceptible, exposed, and infectious individuals in the population at time , respectively. is the total infected population. In the case of 2019-nCoV, there have been some clues suggesting that exposed individuals without symptom can cause many infections [25]. After one unit time, a susceptible individual can be infected through contacting with the exposed or infectious individuals and enter the class or is still in the class or dies. An exposed individual may have a symptom and enter the class or still stay in class or die. The dynamical transfer of the SEI process is demonstrated in Figure 1.

Figure 1 

Transfer diagram of the SEI epidemic process.

In Figure 1, the parameter is the input rate in the susceptible group. Parameter denotes the natural death rate and is the rate of disease-caused death. The force of infection is and , where and are defined as the effective contact per capita in the exposed and infectious periods. Thus, the incidence rate is . Parameter is a transfer rate from the exposed to infectious classes.

2.1. SEI Markov Process

For any , we denote the process with by a family of Markov chains that take values in state space and have transition rate matrix with representing the rate of transition from state to for , and , where . For convenience, denote with and . From Figure 1, transition rates of the SEI process are defined by six events given in Table 1.

Consider all states of the SEI process conditioned on nonextinction state at time , denoted by . Let be condition probabilities and given bywhere . Such a distribution is called quasi-stationary distribution of the SEI process. For the SEI process, the set of transient states is finite and irreducible. Thus, the quasi-stationary distribution exists and is unique [26].

Remark 1. According to the results of [21, 27], quasi-stationary distribution has different forms depending on the basic reproductive number and the expected population size in steady state. When is greater than 1, the distribution can be approximated by Gaussian diffusion approximation. In this work, we mainly investigate that quasi-stationary distribution is approximately normal when .
DenoteIt can be interpreted as the population densities of the susceptible, exposed, and infectious individuals at time . Define a function such thatwhere . Therefore, is a density process with transition rate given in Table 1. On the other hand, by the definition of , we haveFor a density process, Ross et al. [28] can identify a deterministic analogue. Thus, the density process should behave more deterministically as becomes larger. The following theorem proves this point.

Theorem 1. Consider the density process defined in (2) and . Then, as , converges uniformly in probability over finite time intervals to a unique deterministic trajectory satisfying for andwhere and is defined in (4). That is to say, for fixed and for all , .

Proof. By definition of and Table 1, it is clear that is continuous for the first variable and therefore bounded over a compact set . Then, . On the other hand, every element of in (4) is continuous on any compact set . Thus, is locally Lipschitz on ; that is, there exists a constant such that . Based on Theorem 8.1 in the study of Krutz [29], the theorem follows.
Theorem 1 shows the relationship of density process and deterministic model (5). When is larger, we can investigate the properties of density process by those of .

2.2. Deterministic Proportion Model

Denote . For model (5), it is equal to the deterministic proportion model:

However, model (6) has no explicit solution. Thus, we discuss the local stability of equilibrium by analyzing its characteristic equations. Model (6) has an endemic equilibrium denoted by , satisfying

And by calculation, the endemic equilibrium is obtained as follows:

Define the basic reproduction number as follows:

Theorem 2. If , then the endemic equilibrium is locally asymptotically stable.

Proof. The Jacobian of model (6) at point is given byIts characteristic equation is given as follows:whereSince and , by (8), we haveDenote . Then,According to Hurwitz criterion, all roots of have negative real parts. Then, the endemic equilibrium is local asymptotic stability.

Remark 2. For and larger number , converges uniformly in probability to the endemic equilibrium ; that is, converges uniformly in probability to in the endemic phase.

2.3. Diffusion Approximation

Deterministic model (6) is clearly an approximation of the density process in the endemic phase for (see Remark 2). However, it cannot reflect the fluctuations of the process around the endemic equilibrium . Gaussian diffusion approximation is used to analyze quasi-stationary distribution based on the work described in [29, 30]. For convenience, define

The process is called a scaled density process. Under certain conditions, converges weakly in the space of all sample paths to a Gaussian diffusion with the initial value.

Theorem 3. Consider the process and . Then, converges weakly to Gaussian diffusion for , satisfyingwith initial value , where is three-dimensional Wiener process andfor and .

Proof. From Theorem 1, trajectory of is contained in some compact set for because of the local asymptotic stability of the endemic equilibrium . Therefore, is continuous and bounded to a compact set. On the other hand, by Table 1 and definition of , equation (17) yields. Further, we know that is continuous in the first variable and bounded to a compact set. Then, for , we have . By Theorem 8.2 given by Kurtz in [29], the result follows.
Theorem 3 reveals that the scaled process can be approximated by Gaussian diffusion for larger . From Theorem 2, we get and as so that follows a stationary O-U process given as a solution towith the initial value . The stationary distribution of is multivariate normal with mean 0 and covariance matrix satisfyingFrom the above results, we know that the O-U process has stationary mean 0 and covariance matrix in the asymptotically stable case; that is, . Then, the density process can be approximated by multivariate normal distribution for larger number and . That is to say,for , which is diffusion approximation of the quasi-stationarity distribution . Further, we have

3. Main Results

In this section, deterministic model (6) and stochastic model (21) are applied to analyze the characteristics of 2019-nCoV epidemic in Hubei province, China. The onset and death data were collected from Hubei from 15 January to 5 February 2020 [31, 32] (see Figure 2). The unknown parameters of models are estimated by least-squares method.

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Figure 2 

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, . For model (6), are the population proportions of susceptible, exposed, and infectious individuals of Hubei province. From Figure 2 and [32], we have and . Obviously, . Thus,

Let , and be the estimated values of unknown parameters , and , respectively. The estimated values are obtained by least-squares method (Table 2) with the initial values . From Table 2, we have and . Since , by Theorem 2, the endemic equilibrium is locally asymptotically stable. Further, it reveals that 2019-nCoV infection is still spreading and will be endemic in Hubei province without effective controls. Under model (6), Figure 3 shows the fitted values of susceptible, exposed, and infectious proportions, comparing the actual values of infectious proportions.

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Figure 3 

(a) Fitted curves of , (b) fitted curves of , and (c) true and fitted curves of in Hubei province, from 15 January 2020 to 5 February 2020.

3.1. Residual Analysis

Model validation is the most important step in the model building process. Residual analysis such as the coefficient of determination and mean squared error (MSE) provides measures of model quality. Figure 4 displays an error bar plot of the confidence intervals on the residuals of true and fitted values. As shown in Figure 4, residuals appear to behave randomly and the errors are relatively small for model (6). Therefore, it is reasonable to suggest that the estimators obtained here are reliable.

Figure 4 

confidence intervals of residuals for infectious proportions.

Let be the fitted value of and be the number of observational data. Definewhere . From Figure 4 and the definitions of and MSE, we have and . Thus, model (6) can reflect the dynamics behavior of field data used in our study.