^{1}

^{1}

^{1}

^{1}

^{2}

^{1}

^{1}

^{2}

We establish an SEIQRS epidemic model with periodic transmission rate to investigate the spread of seasonal HFMD in Wenzhou. The value of this study lies in two aspects. Mathematically, we show that the global dynamics of the HFMD model can be governed by its reproduction number

Hand, foot, and mouth disease (HFMD) was first reported in New Zealand in 1957 and is caused by Coxsackievirus A16 (CVA16) and human enterovirus 71 (HEV71) and occasionally by Coxsackie virus A4-A7, A9, A10, B1-B3, and B5. HFMD is an acute viral illness that usually affects infants and children younger than 5 years old, and, however, it can sometimes occur in adults. HFMD is characterized by fever, intraoral vesicles and erosions, and papulovesicles that favor the palms and soles. HFMD is spread from person to person through nose and throat secretions (such as saliva, sputum, or nasal mucus), blister fluid, or stool of infected persons [

HFMD is moderately contagious and usually not a serious illness among the infected population; however, recent outbreak of HFMD in countries such as China, Singapore, and Finland has brought the world attention to HFMD due to complications of death related cases [

It is obvious that HFMD not only causes health problems but also has great social and economic impacts which are not easily quantifiable. So it is important to understand the spread dynamics of HFMD among the susceptible populations and to enable policy makers to take effective measures to curb the disease spread and reduce the adverse impact of the disease [

There are several types of analytical models that are valuable to understand and predict transmission of HFMD. One is the statistical models which can help us find novel information concerning pathogen detection and some probable coinfection factors in HFMD and have been applied to understand HFMD’s spatiotemporal transmission and discover the relationship between HFMD occurrence and climate [

After a susceptible individual is infected, he/she firstly enter the incubation period of HFMD, which period is about 3 to 7 days. After the incubation period, the infected will show some clinical symptoms, such as having a fever, poor appetite, malaise, and sore throat, and few people may develop dehydration, febrile seizures, encephalitis, meningitis, cardiomyopathy, and so forth. And the infected people will fully recover after 7 to 10 days [

In the sense of seasonally changing in the contact rate, Liu [

Wenzhou is a prefecture-level city in southeastern Zhejiang, province in China. At the time of the 2010 Chinese census, 9,122,100 people lived in Wenzhou [

The numbers of weekly reported HFMD cases in Wenzhou.

Week | Data | Week | Data | Week | Data | Week | Data | Week | Data | Week | Data | Week | Data | Week | Data |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 368 | 26 | 331 | 51 | 46 | 76 | 366 | 101 | 164 | 126 | 469 | 151 | 238 | 176 | 749 |

2 | 380 | 27 | 320 | 52 | 47 | 77 | 371 | 102 | 193 | 127 | 413 | 152 | 189 | 177 | 794 |

3 | 457 | 28 | 475 | 53 | 136 | 78 | 317 | 103 | 234 | 128 | 413 | 153 | 145 | 178 | 708 |

4 | 615 | 29 | 415 | 54 | 153 | 79 | 291 | 104 | 226 | 129 | 467 | 154 | 141 | 179 | 600 |

5 | 581 | 30 | 339 | 55 | 197 | 80 | 441 | 105 | 264 | 130 | 429 | 155 | 131 | 180 | 518 |

6 | 694 | 31 | 352 | 56 | 366 | 81 | 549 | 106 | 456 | 131 | 439 | 156 | 118 | 181 | 505 |

7 | 924 | 32 | 201 | 57 | 322 | 82 | 395 | 107 | 478 | 132 | 676 | 157 | 223 | 182 | 523 |

8 | 1124 | 33 | 254 | 58 | 401 | 83 | 478 | 108 | 743 | 133 | 861 | 158 | 268 | 183 | 520 |

9 | 1200 | 34 | 226 | 59 | 460 | 84 | 410 | 109 | 876 | 134 | 793 | 159 | 351 | 184 | 446 |

10 | 1539 | 35 | 243 | 60 | 549 | 85 | 565 | 110 | 1035 | 135 | 765 | 160 | 455 | 185 | 510 |

11 | 1744 | 36 | 182 | 61 | 574 | 86 | 508 | 111 | 1403 | 136 | 645 | 161 | 517 | 186 | 714 |

12 | 1706 | 37 | 173 | 62 | 767 | 87 | 543 | 112 | 1660 | 137 | 491 | 162 | 596 | 187 | 917 |

13 | 1528 | 38 | 187 | 63 | 1165 | 88 | 538 | 113 | 1908 | 138 | 542 | 163 | 577 | 188 | 672 |

14 | 1329 | 39 | 214 | 64 | 1247 | 89 | 787 | 114 | 2240 | 139 | 600 | 164 | 517 | 189 | 589 |

15 | 1252 | 40 | 186 | 65 | 1168 | 90 | 689 | 115 | 1974 | 140 | 644 | 165 | 552 | 190 | 482 |

16 | 1447 | 41 | 166 | 66 | 1060 | 91 | 854 | 116 | 2163 | 141 | 579 | 166 | 546 | 191 | 369 |

17 | 1663 | 42 | 192 | 67 | 1388 | 92 | 785 | 117 | 1545 | 142 | 691 | 167 | 668 | 192 | 326 |

18 | 1346 | 43 | 190 | 68 | 1686 | 93 | 618 | 118 | 1611 | 143 | 777 | 168 | 874 | 193 | 366 |

19 | 1247 | 44 | 181 | 69 | 1532 | 94 | 476 | 119 | 1868 | 144 | 783 | 169 | 1068 | 194 | 491 |

20 | 1094 | 45 | 171 | 70 | 1323 | 95 | 414 | 120 | 1683 | 145 | 732 | 170 | 1266 | 195 | 402 |

21 | 794 | 46 | 95 | 71 | 819 | 96 | 365 | 121 | 1631 | 146 | 561 | 171 | 1001 | 196 | 439 |

22 | 673 | 47 | 66 | 72 | 743 | 97 | 308 | 122 | 1228 | 147 | 636 | 172 | 1037 | 197 | 371 |

23 | 584 | 48 | 55 | 73 | 636 | 98 | 206 | 123 | 758 | 148 | 441 | 173 | 1363 | 198 | 358 |

24 | 448 | 49 | 37 | 74 | 602 | 99 | 162 | 124 | 502 | 149 | 345 | 174 | 1051 | 199 | 382 |

25 | 420 | 50 | 42 | 75 | 478 | 100 | 161 | 125 | 519 | 150 | 225 | 175 | 995 |

The aim of this study is to use mathematical modeling to gain some insights into the transmission dynamics of HFMD in Wenzhou and to assess the aforementioned preventive strategies. In particular, we aim to answer the following questions through our analytic and numerical results of the HFMD model.

What is the disease dynamics of the HFMD in Wenzhou?

How can we control the HFMD spread in Wenzhou?

The paper is organized as follows. In Section

In order to establish an HFMD model in Wenzhou, we classify the population into five compartments according to their states: susceptible, exposed, infected, quarantined, and recovered, which are denoted by

Let

We briefly present some main results of Floquet theory. Consider the following linear periodic system:

Let

Let

The study of the dynamics of model (

And we can easily obtain the following theorem of the positively invariant sets.

It shows that

It is easy to see that model (

Furthermore,

It is obvious that

Let

Following the method in Wang and Zhao [

Let

The following statements are valid:

Now we introduce the linear

The disease-free equilibrium

If

From Theorem

Define

If

We proceed by contradiction to prove that

By using Lemma

If

We first prove that

Furthermore, taking advantage of Theorem 1.3.6 in Zhao [

Theorem

From Theorems

In this section, by using model (

Thanks to the insightful work of Liu [

the recruitment rate of susceptible

the natural mortality rate

the transmission rate

the “psychological” effect

the mean incubation period

the quarantined rate

the infected recover rate

the quarantined recover rate

the disease-induced mortality

the progression rate

And the number of the initial susceptible population at the end of 2010,

The numerical simulation results of the model (

In Figure

The solid curve represents the simulation results and the stars are the weekly data from 2010 to 2013 reported by the Wenzhou CDC.

The solid curve represents the simulation results and the stars are the weekly data from 2011 to 2012 reported by the Wenzhou CDC.

The population dynamics of model (

The solid curve represents the simulation results and the stars are the weekly data from 2012 to 2013 reported by the Wenzhou CDC.

For the sake of learning the spread dynamics of HFMD in Wenzhou further, we display the simulation results via the statistical data 2011-2012 (Figure

In Figure

Comparing Figure

For the sake of learning the effects of the quarantined rate (

The influence of the quarantined rate

In this paper, we construct an SEIQRS epidemic model with periodic transmission rates to investigate the spread of seasonal HFMD in Wenzhou. We show that the global dynamics of the HFMD model (

Epidemiologically, we partially provide answers to the two questions proposed in the introduction: what is the spread dynamics of the HFMD in Wenzhou? How can we control the HFMD spread in Wenzhou?

From Figures

From Figure

In [

Furthermore, from Figures

It is well known that no vaccine currently exists to protect individuals from infection with the viruses that cause HFMD, but such vaccines are being developed [

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors would like to thank Jing-An Cui and Meng Ye (Beijing University of Civil Engineering and Architecture) and Maoxing Liu and Yangjun Ma (North University of China) for very helpful suggestions and comments which led to the improvement of the original paper. This research is supported by the National Science Foundation of China (61373005) and Zhejiang Provincial Natural Science Foundation (LY12A01014).