Modeling and Preventive Measures of Hand, Foot and Mouth Disease (HFMD) in China

This paper concentrates on the HFMD data of China from March 2009 to December 2012. We set up a mathematical model to fit those data with the goodness of fit and obtain the optimal parameter values of the model. By the Chi-square test of statistical inference, the optimal parameter values of the model are reasonable. We obtained the basic reproductive number of the disease for each year, and it is larger than 1. Thus, we conclude that HFMD will persist in China under the current conditions, so we investigate the preventive measures to control the HFMD. If the preventive measures proposed in our paper were implemented, HFMD would be controlled quickly and the number of infections would decline rapidly over a period of time.


Introduction
Hand, foot and mouth disease (HFMD), an infectious disease caused by enterovirus and Coxsackievirus, usually happens to children under age of five, with an especially high incidence being observed for those under three. It can result in herpes in such body parts as hands, feet and mouth and even other complications such as myocarditis, pulmonary edema, and aseptic meningoencephalitis in some children. Some severely affected patients may die due to the quick progress of the disease [1]. There are over twenty types of enterovirus leading to HFMD. The common pathogens of HFMD are OPEN ACCESS model with isolation to analyze the conditions of the existence of the endemic equilibrium [20]. Ma et al. proposed a dynamic model with periodic transmission rates to investigate the seasonal HFMD [2]. This paper is organized as follows: Section 2 gives the data of HFMD in China and studies how to build a suitable epidemic model. Section 3 shows the results. Section 4 presents the corresponding conclusions and possible preventive measures to control HFMD.

Data
The Ministry of Health of the People's Republic of China declared that HFMD was ranked as a Class C Infectious Disease on May 2nd, 2008 In general, medical staffs easily determine HFMD infections just by relying on symptoms such as a slight fever followed by blisters and ulcers in the mouth and rashes on the hands and feet. In fact, it is easy to distinguish between chicken pox, dental ulcer, foot-and-mouth disease, herpangina, scarlet fever and so on. Furthermore, the final result of the diagnosis will be made in the laboratory according to samples of throat swabs or feces. So the data from the China CDC comprises the reported data of HFMD from one place or another, and clinically confirmed cases. Once an outpatient is examined to confirm HFMD infection, he or she would be necessarily be hospitalized, so the data that China CDC provides covers hospitalization. Certainly, we have to ignore the rare cases where the patients' conditions were relatively mild and they returned home for treatment with the doctors' permission.

Model Analysis
The population associated with HFMD is divided into five compartments: The susceptible (S), exposed (E), infectious and not hospitalized (I), infectious and hospitalized ( Q ), and recovered ( R) individuals. The total population is N = S + E + I + Q + R. We use β, , ρ, γ 1 , γ 2 , γ to denote the transmission rate, the rate of progression to the infectious and not hospitalized, the proportion of the infectious and hospitalized, the recovery rate of the infectious and not hospitalized, the recovery rate of the infectious and hospitalized individuals, the ratios of the recovered individuals becoming susceptible individuals ( Figure 1).

Figure 1.
Flow chart of compartments of the HFMD model.

Parameters and Model Hypothesis
(1) Given the short course of HFMD, it is reasonable that we ignore the mobility of the patients, the space structure and the environmental climate of the model to collect all these data.
(2) According to the annual data from the China CDC, we can assume the initial numbers Q(0) (see Table 2.).
(4) By simulation, we find that the mortality rate  of HFMD is very small and almost equal to 0, that is  = 0.
(5) According to the biological significance, we set the range of each parameter reasonably in Table 2.

Model Formulation
The differential equations for HFMD model are: , with initial conditions: Following van den Driessche and Watmough [22], we can compute the basic reproduction number:

Parameter Estimation
We estimate parameters of system (1) with the MATLAB (The Mathworks, Inc., Natick, MA, USA) tool fminsearch, which is a part of optimization toolbox. All optimal parameter values are obtained only when the results of fminsearch are convergent. Those values are shown in Table 2. Here, we also calculate the basic reproduction number. It is shown that the same parameters in the model (1) change relatively little in every year in Table 2, so we make predictions about the prevalence trends in subsequent years with the help of the data of HFMD from 2009 to 2012 in China, which means that our research work possesses a certain reference value. More specifically, by analyzing the parameters  , 2  , 0 R in Table 2 we draw up the following conclusions conformed to reality that the incubation period 1/  is approximately equal to 1-2.5595 days, and the course of treatment 2 1/  is about 2.4786-4.9677 days. Moreover, 0 R , indicating the numbers of persons that one patient infects in an average sick period, is basically stable at around 1.1, which explains why HFMD has yet not broken out on a large scale despite the fact that it continues to be prevalent in China.

Chi-Square Test of Goodness of Fit
In order to test how well our model reflects the data actually, we consider the following hypotheses: Null hypothesis, 0 H : The estimated parameters are equal to actual values.
Alternative hypothesis, 1 H : The estimated parameters are not equal to actual values.
The Chi-square values and degrees of freedom for each year are shown in Table 3. Therefore, we cannot reject the null hypothesis at the 5% significant level by Pearson's criterion of Chi-square test [24].  The numerical simulation of the number of the infectious and hospitalized individuals of HFMD by model (1) is shown in Figure 2. The simulation provides a good match with the data of HFMD in China from March 2009 to December 2012.

Sensitivity Analysis
Next, we consider the impact of parameters on 0 R . Considering  ,  , 1  and 2  as an independent variable and the other parameters as constants, respectively. One obtains implicit differentiation of 0 R with respect to  ,  , 1  and 2  , respectively:  In theory, if we can control some parameters such that 0 1 R  , then the disease will die out. On this basis we shall put forward detail preventive strategies of HFMD in the following section.

Discussion and Conclusions
There have been plenty of papers on how to control and prevent HFMD from a public health and statistical model perspective. However, there are few works [2] constructing differential equations models to simulate data of HFMD. All the parameters of the model proposed in [1] and [2] are estimated without using statistical methods to verify the rationality of the parameters. Consequently, we demonstrate rationality of the parameters in our model as well as consistence with reality by applying the Chi-square test.
In this paper, we consider the HFMD data reported by China CDC, and construct a SEIQRS model to fit the HFMD data. From the last column in Table 2, the basic reproductive number in each year is larger than 1. Thus, we conclude that HFMD will persist in China under the current conditions. As a matter of fact, there is no effective vaccine or antiviral treatment specifically for HFMD, but if we can provide some preventive measures to control the HFMD, it will be very meaningful.
Next, we select the year of 2012 as an example (The simulation of the sum of not hospitalized infectious ( ) I t and hospitalized infectious ( ) Q t is presented in Figure 3a.), and propose some preventive measures as follows: Strategy 1: Reducing the transmission rate  for the susceptible can effectively control the spread of HFMD (see Figure 3b). Therefore, health-care education such as washing hands before meals and after using the toilet, and making air fresh indoors and so on, should be carried out in kindergardens, schools, hospitals and other places to popularize health knowledge and advocate good personal hygiene habits. Kindergardens should clean and disinfect toys and appliances every day. In addition, hospitals should strengthen infection control practices to avoid nosocomial cross infection. Strategy 2: Reducing the rate of progression to infective individuals  can control the spread of HFMD at a lower level (see Figure 3c). For example, more fruits, vegetables, and regular exercise will increase their immunity. Even if they carry the enterovirus, they may not be infected.
Strategy 3: Increasing the recovery rate of non-hospitalized infectious individuals 1  can effectively control the spread of HFMD (see Figure 3d), so we suggest they should see a doctor in a timely fashion and thus reduce the chance of contact with other people when the adults appear to have symptoms of fever, rash and so on. However, changing the recovery rate of hospitalized infectious individuals 2  , the control effect is relatively small (see Figure 3e).
In a word, if we use the above preventive measures, the HFMD will be controlled quickly and the number of infections will decline rapidly in a period of time. Those measures can effectively prevent the large-scale diffusion of the disease.
We can see that  and 1  are the most sensitive parameters comparing Figure 3b,d with the others because just slight changes can achieve the goal of control. These existing measures to control and prevent HFMD can be essentially attributed to how to reduce  . Based on the discussion in this paper, it is vitally important not only to reduce  but also to increase 1  . In addition, it is more effectively to increase 1  than to increase 2  precisely because persons with latent infection are more than apparent infection. So far, there are few papers using ordinary differential equations models to simulate the real data of HFMD and make measures to control and prevent it. Not only that, a powerful theoretical basis is provided for more detail control countermeasures in our paper.  Table 2.
(e) Simulation of the sum of ( ) I t and ( ) Q t with 2  = 0.8052 (=0.2013 × 4), other parameters from the sixth column of Table 2.

Acknowledgments
We would like to thank anonymous reviewers for their helpful comments which improved the presentation of this work. This research is supported by the National Natural Science Foundation of China (No.11071275 and No.11228104).

Author Contributions
Yong Li and Jinhui Zhang contributed equally to the work. Yong Li conceptualized and designed the study, drafted the initial manuscript, and approved the final manuscript as submitted. Xinan Zhang carried out the initial analyses, reviewed and revised the manuscript, and approved the final manuscript as submitted.