# A Hand-Foot-and-Mouth Disease Model with Periodic Transmission Rate in Wenzhou, China

Authors: Yeting Zhu; Boyang Xu; Xinze Lian; Wang Lin; Zumu Zhou; Weiming Wang
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

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 numberR0; if R0<1, the disease-free equilibrium of the model is globally asymptotically stable, which means that the disease will vanish after some period of time; while if R0>1, the model has at least one positive periodic solution and is uniformly persistent, which indicates that HFMD becomes an endemic disease. Epidemiologically, based on the statistical data of HFMD in Wenzhou, we find that the HFMD becomes an endemic disease and will break out in Wenzhou. One of the most interesting findings is that, for controlling the HFMD spread, we must increase the quarantined rate or decrease the treatment cycle.


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## Body

## 1. Introduction

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 [1, 2].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 [3–10]. In 2008, there were 488,955 cases reported, with a morbidity of 37/100,000, mortality of 0.0095/100,000, and ill-death rate of 0.26/1000; while, in 2009, there were 1,155,525 cases reported [11]. Due to the severity of this disease, the Ministry of Health of the People’s Republic of China upgraded HFMD to a Class C communicable disease on May 2, 2008.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 [12, 13].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 [2–11, 14, 15]. The other is compartmental differential equation model [12, 13, 16–20]. Of them, Tiing and Labadin [16] were the first to propose a simple SIR epidemic model to predict the number of the infected and the duration of an outbreak when HFMD occurs and found that the disease spread quite rapidly and the parameter that may be able to be controlled would be the number of susceptible persons.  Roy and Haider [12] and Roy [13] established a simple SEIR model to understand the dynamics of HFMD among young children and found that disease transmission depended more on the number of actively infected people in the population at the initial time and could be controlled by quarantine of more actively infected individuals. Wang and Xue [18] and Yang et al. [19] established SEIQR (susceptible-exposed-infectious-quarantined-recovered) model, respectively, and estimated the basic reproduction number for the HFMD transmission in Mainland China and predicted that HFMD was an endemic disease in China.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 [1]. The occurrence of HFMD has seasonal characteristics and is often associated with climate changes [11, 21]. Periodic changing in the birth rate [22] and seasonally changing in the contact rate [23] are often regarded as sources of periodicity.In the sense of seasonally changing in the contact rate, Liu [17] and Ma et al. [20] established HFMD with periodic transmission rate, respectively. In [17], Liu constructed a periodic SEIQR epidemic model to simulate the dynamics of HFMD transmission and showed that quarantine in the children population had a positive impact on controlling the spread of HFMD. And Ma et al. [20] proposed a SEIIeQR HFMD model, and found that the recessive subpopulation played an important role in the spread of HFMD, and only quarantining the infected is not an effective measure in controlling the disease.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 [24]. Since Wenzhou has a humid subtropical climate zone with an annual average 18.08°C (64.5°F), it is of particular public health significance to update molecular epidemiology of HFMD in Wenzhou. It is reported that, in 2012, there were 147,941 HFMD cases and 17 deaths in Zhejiang province, and it is the first of the “ten legal infectious diseases”; there were 41,438 HFMD cases and 4 deaths in Wenzhou [25]. The numbers of weekly reported HFMD cases in Wenzhou from March 5, 2010, to December 27, 2013, are listed in Table 1.Table 1
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
995The 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.(1)
What is the disease dynamics of the HFMD in Wenzhou?(2)
How can we control the HFMD spread in Wenzhou?The paper is organized as follows. In Section2, we derive a SEIQRS HFMD model. In Section 3, we give the main theoretical results of the HFMD model. In Section 4, based on the HFMD data of the Wenzhou Center for Disease Control and Prevention, we perform some simulation results of the model and sensitivity analysis. In Section 5, we give a brief discussion.


## 2. Model Derivation

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 byS(t),E(t),I(t),Q(t), and R(t), respectively. We denote the total population by N(t), that is N(t)=S(t)+E(t)+I(t)+Q(t)+R(t). And the nonautonomous differential equations for HFMD model are
(1)dSdt=Λ-dS-β(t)SIS+αI+ηR,dEdt=β(t)SIS+αI-(d+σ)E,dIdt=σE-(d+δ1+k+γ1)I,dQdt=kI-(d+δ2+γ2)Q,dRdt=γ1I+γ2Q-(d+η)R,
and the meaning and units of each variable and constant in model (1) are as follows:Λ: the recruitment rate of susceptible (week-1),d: the per capita natural mortality rate (week-1),β(t): the periodic transmission rate coefficient, which is a continuous, positive ω-periodic function,α (0<α≤1): the “psychological” effect, which can be interpreted as follows: when an infectious disease appears and spreads in a region, people deter from risky behavior or from taking precautionary measures in relation to the disease outbreak; human behavior change consequently leads to reduction in number of real susceptible individuals or effective contact rates [26],1/σ: the mean incubation period (week),k: the quarantined rate (week-1); this means that some of the infected people will be hospitalized for treatment, and, thus, these quarantined Q(t) are isolated from other subpopulations,γ1: the infectious recover rate, which means that the infectious individuals recover and return to recovered compartment R from compartments I (week),γ2: the quarantined rate, which means that the quarantined individuals recover and return to recovered compartment R from compartments Q (week),δ1: the disease-induced mortality for the infective individuals (week-1),δ2: the disease-induced mortality for the quarantined individuals (week-1),η: the progression rate of the recovered individual R.


## 3. Dynamics Analysis of the HFMD Model

### 3.1. Preliminaries

Let(ℝn,ℝ+n) be the standard ordered n-dimensional Euclidean space with a norm ∥·∥. For u,v∈ℝn, we denote u≥v, if u-v∈ℝ+n; u>v, if u-v∈ℝ+n0; and u≫v, if u-v∈Int(ℝ+n).We briefly present some main results of Floquet theory. Consider the following linear periodic system:(2)dudt=A(t)u,
where u is an n×1 vector and A(t) is an n×n matrix of principal periodic ω>0, such that A(t+ω)=A(t).LetΦA(·)(t) be a fundamental solution matrix of (2); that is, a nonsingular matrix each of the columns of which is a solution of equation such that ΦA(·)(0)=𝕀, the identify matrix. Floquet theory show that the fundamental matrix has the form ΦA(·)(t)=P(t)eBt, where P(t)=P(t+ω) is a periodic matrix with initial values P(0)=𝕀 and B is a constant matrix. The matrix ΦA(·)(T)=eBT is the so-called principle or monodromy or Floquet transition matrix of (2). Let r(ΦA(·)(ω)) be the spectral radius of ΦA(·)(ω). It then follows from [27, 28] that ΦA(·)(ω) is a matrix with all entries positive for each t>0. By the Perron-Frobenius theorem [29], r(ΦA(·)(ω)) is the principal eigenvalue of ΦA(·)(ω) in the sense that it is simple and admits an eigenvector u*≫0. The following result is useful for our subsequent comparison arguments.Lemma 1 (see [30, Lemma 2.1]).
Letp=(1/ω)ln⁡r(ΦA(·)(ω)). Then there exists a positive, ω-periodic function v(t) such that eptv(t) is a solution of (2).


### 3.2. Positively Invariant Sets

The study of the dynamics of model (1) requires the introduction of the following important sets:
(3)X={(S,E,I,Q,R)∈ℝ+5},Γ={(S,E,I,Q,R)∈X,0<S+E+I+Q+R≤Λd}.And we can easily obtain the following theorem of the positively invariant sets.Theorem 2 (positively invariant sets).
X and Γ are positive invariant sets for model (1).Proof.
It shows thatX is positively invariant. From model (1), the total population N(t) satisfies the following equation:
(4)Λ-(d+δ1+δ2)N≤dNdtΛ-(d+δ1+δ2)N=Λ-dN-δ1I-δ2Q≤Λ-dN.
Hence, by integration, we check
(5)Λd+δ1+δ2+(N(0)-Λd+δ1+δ2)e-(d+δ1+δ2)t≤N(t)≤Λd+(N(0)-Λd)e-dt.
Hence,
(6)Λd+δ1+δ2≤lim⁡t→∞inf⁡N(t)≤lim⁡t→∞sup⁡N(t)≤Λd,
which implies that Γ is positively invariant with respect to model (1).


### 3.3. The Basic Reproduction NumberR0

It is easy to see that model (1) always has one disease-free equilibrium P0=(Λ/d,0,0,0,0). By the definition of Bacaër and Guernaoui [31] and the general calculation procedure in Wang and Zhao [32], we have
(7)ℱ=(β(t)SIS+αI0000),𝒱=((d+σ)E(d+δ1+k+γ1)I-σE(d+δ2+γ2)Q-kIβ(t)SIS+αI+dS-Λ-ηR(d+η)R-γ1I-γ2Q),𝒱-=((d+σ)E(d+δ1+k+γ1)I(d+δ2+γ2)Qβ(t)SIS+αI+dS(d+η)R),𝒱-=(0σEkIΛ+ηRγ1I+γ2Q).
It follows that
(8)F(t)=(0β(t)0000000),M=(-dη0-(d+η)),V(t)=(d+σ00-σd+δ1+k+γ100-kd+δ2+γ2).Furthermore,F(t) is nonnegative, and -V(t) is cooperative in the sense that the off-diagonal elements of -V(t) are nonnegative.It is obvious that-d and -(d+η) are the eigenvalues of M and are negative. Thus M is stable; namely, r(ϕM(ω))<1. We can see that the eigenvalues of -V are the diagonal elements and are negative. So -V is stable; namely, r(ϕ-V(ω))<1.LetY(t,s),t≥s, be the evolution operator of the linear ω-periodic system. Consider
(9)dydt=-V(t)y.
That is, for each s∈ℝ, the 3×3 matrix Y(t,s) satisfies
(10)dY(t,s)dt=-V(t)Y(t,s),∀t≥s,Y(s,s)=𝕀,
where 𝕀 is the 3×3 identity matrix. Thus, the monodromy matrix Φ-V(t) of (9) equals Y(t,0),t≥0.Following the method in Wang and Zhao [32], we let ϕ(s) be ω-periodic in s and the initial distribution of infectious individuals. So F(s)ϕ(s) is the rate of new infections produced by the infected individuals who are introduced at time s. When t≥s, Y(t,s)F(s)ϕ(s) gives the distribution of those infected individuals who are newly infected by ϕ(s) and remain in the infected compartments at time t. Naturally,
(11)∫-∞t‍Y(t,s)F(s)ϕ(s)ds=∫0∞‍Y(t,t-a)F(t-a)ϕ(t-a)da
is the distribution of accumulative new infections at time t produced by all those infected individuals ϕ(s) introduced at time previous to t.LetCω be the ordered Banach space of all ω-periodic functions from ℝ to ℝ3, which is equipped with the maximum norm ∥·∥ and the positive cone Cω+:={ϕ∈Cω:ϕ(t)≥0,∀t∈ℝ}. Then we can define a linear operator. Consider
(12)(Lϕ)(t)=∫0∞‍Y(t,t-a)F(t-a)ϕ(t-a)da,∀t∈ℝ,ϕ∈Cω.L is called the next infection operator and the spectral radius of L is defined as the basic reproduction number. Consider
(13)R0:=r(L),
for the periodic epidemic model.Lemma 3 (see [32]).
The following statements are valid:(i)
R0=1 if and only if r(ΦF-V(ω))=1;(ii)
R0>1 if and only if r(ΦF-V(ω))>1;(iii)
R0<1 if and only if r(ΦF-V(ω))<1.Thus, the disease-free equilibrium P0=(Λ/d,0,0,0,0) of model (1) is asymptotically stable if R0<1 and unstable if R0>1.Now we introduce the linearω-periodic system
(14)dwdt=(-V(t)+F(t)λ)w,t∈ℝ+
with parameter λ∈ℝ. Let W(t,s,λ),t≥s, be the evolution operator of system (14) on ℝ4. Clearly, ΦF-V(t)=W(t,0,1),t≥0. Moreover,
(15)-V(t)+F(t)λ=(-(d+σ)β(t)λ0σ-(d+δ1+k+γ1)00k-(d+δ2+γ2)).
Hence, we derive
(16)ΦF/λ-V(t)=W(t,0,λ).
Following the general calculation procedure in Wang and Zhao [32], the basic reproduction number R0 is the unique solution of r(W(ω,0,λ))=1.


### 3.4. Extinction of the Disease

Theorem 4.
The disease-free equilibriumP0=(Λ/d,0,0,0,0) of model (1) is globally asymptotically stable if R0<1 and unstable if R0>1.Proof.
IfR0<1, according to Lemma 3, we have r(ΦF-V(ω))<1. From model (1), for t≥0, we know that
(17)dEdt≤β(t)I-(d+σ)E,dIdt=σE-(d+δ1+k+γ1)I,dQdt=kI-(d+δ2+γ2)Q.
Consider the following comparison system:
(18)dE~dt=β(t)I~-(d+σ)E~,dI~dt=σE~-(d+δ1+k+γ1)I~,dQ~dt=kI~-(d+δ2+γ2)Q~;
that is,
(19)dgdt=(F(t)-V(t))g(t),g(t)=(E~(t),I~(t),Q~(t))T.
By Lemma 1, there exists a positive ω-periodic function g~(t) such that g(t)=eptg~(t) is a solution of (18), where p=(1/ω)ln⁡r(ΦF-V(ω))<0. Then we conclude that g(t)→0 as t→∞, which implies that the zero solution of model (18) is globally asymptotically stable. Applying the comparison principle, we known that E(t)→0,I(t)→0,  and Q(t)→0 as t→∞. It follows that R(t)→0,  and S(t)→Λ/d as t→∞. Therefore, the disease-free equilibrium P0=(Λ/d,0,0,0,0) is globally asymptotically stable. This completes the proof.Remark 5.
From Theorem4, one can know that if R0<1, then P0=(Λ/d,0,0,0,0) is globally asymptotically stable, which means that the disease will vanish after some period of time. Therefore, it is interesting to study the disease-free equilibrium for controlling infectious disease.


### 3.5. Uniform Persistence of the Disease

Define(20)Γ0:={(S,E,I,Q,R)∈Γ:E>0,I>0}
and ∂Γ0=Γ∖Γ0={(S,E,I,Q,R)∈Γ:E(t)≡0orI(t)≡0}. Denote u(t,x0) as the unique solution of model (1) with the initial value x0:=(S0,E0,I0,Q0,R0). Let P:Γ→Γ be the Poincaré map associated with model (1); that is,
(21)P(x0)=u(ω,x0),∀x0∈Γ,
where ω is the period. By applying the fundamental existence-uniqueness theorem [33], we know that u(t,x0) is the unique solution of model (1) with u(0,x0)=x0. From Theorem 2, we know that P is dissipative point. We then introduce the following lemma.Lemma 6.
IfR0>1, then there exists a ϵ>0 such that when ∥x0-P0∥≤ϵ, for any x0∈Γ0, one has
(22)limsup⁡m→∞d[Pm(x0)-P0]≥ϵ.Proof.
We proceed by contradiction to prove that(23)limsup⁡m→∞d[Pm(x0)-P0]≥ϵ.
If not, then
(24)limsup⁡m→∞d[Pm(x0)-P0]<ϵ,
for some x0∈Γ0. Without loss of generality, we can assume that d(Pm(x0),P0)<ϵ, for all m≥0. By the continuity of the solutions with respect to the initial values, for all ε∈(0,Λ/d), we obtain
(25)∥u(t,Pm(x0))-u(t,P0)∥≤ε,∀t∈[0,ω].
For any t≥0, let t=mω+t1, where t1∈[0,ω] and m=[t/ω], which is the greatest integer less than or equal to t/ω. Then we have
(26)∥u(t,Pm(x0))-u(t,P0)∥=∥u(t1,Pm(x0))-u(t1,P0)∥≤ε,∀t≥0.
It follows that Λ/d-ε≤S(t)≤Λ/d+ε,0≤I(t)≤ε, for t≥0. Then S/(S+αI)≥1-αε/(Λ/d+(α-1)ε):=1-θ>0. Thus, for ∥x0-P0∥≤ϵ, we have
(27)dEdt≥β(t)(1-θ)I-(d+σ)E,dIdt=σE-(d+δ1+k+γ1)I,dQdt=kI-(d+δ2+γ2)Q.
We consider the following comparison system
(28)dE-dt=β(t)(1-θ)I--(d+σ)E-,dI-dt=σE--(d+δ1+k+γ1)I-,dQ-dt=kI--(d+δ2+γ2)Q-.
Set Mε=(0θ0000000). Then, model (28) can be rewritten as
(29)dgdt=(F(t)-V(t)-Mε)g(t),g(t)=(E-(t),I-(t),Q-(t))T.
If R0>1, we obtain r(ΦF-V(ω))>1 by Lemma 3. Then we can choose ε>0 small enough such that r(ΦF-V-Mε(ω))>1. Once again by Lemma 1, it follows that there exists a positive ω-periodic g-(t) such that g(t)=ep1tg-(t) is a solution of model (28), where p1=(1/ω)ln⁡r(ΦF-V(ω))>0, which implies that g(t)→∞ as t→∞. By applying the comparison principle, we have E(t)→∞,I(t)→∞, and before Q(t)→∞ as t→∞. This is a contraction.By using Lemma6, we can obtain the following theorem.Theorem 7.
IfR0>1, model (1) has at least one positive periodic solution and is uniformly persistent.Proof.
We first prove that{Pm}m≥0 is uniformly persistent with respect to (Γ0,∂Γ0). First of all, we show that Γ0 and ∂Γ0 are positively invariant, for any (S0,E0,I0,Q0,R0)∈Γ0. From the first equation of model (1), we derive that
(30)dSdt≥Λ-(d+β(t)α)S+ηR.
So, we have
(31)S(t)≥e-∫0t‍(d+β(s)α-1)dsS(t)hh×(S0+∫0t‍e∫0τ‍(d+β(s)α-1)ds(Λ+ηR(τ))dτ)S(t)>0,∀t>0.
Similarly,
(32)E(t)=e-(d+σ)t(E0+∫0t‍β(τ)S(τ)I(τ)S(τ)+αI(τ)e(d+σ)τdτ)>0,∀t>0,I(t)=e-(d+δ1+k+γ1)t(I0+∫0t‍σE(τ)e(d+δ1+k+γ1)τdτ)>0,∀t>0,Q(t)=e-(d+δ1+k+γ1)t(Q0+∫0t‍kI(τ)e(d+δ1+k+γ1)τdτ)>0,∀t>0,R(t)=e-(d+η)t(R0+∫0t‍(γ1I(τ)+γ2Q(τ))e(d+η)τdτ)>0,∀t>0.
Thus, Γ0 is positively invariant. Clearly, ∂Γ0 is relatively closed in Γ. Set
(33)M∂={(S0,E0,I0,Q0,R0)∈∂Γ0:Pm(S0,E0,I0,Q0,R0)∈∂Γ0,∀m≥0}.
We claim that
(34)M∂={(S,0,0,Q,R):S≥0,Q≥0,R≥0}.
Note that
(35){(S,0,0,Q,R):S≥0,Q≥0,R≥0}⊆M∂.
We only need to prove that
(36)M∂⊆{(S,0,0,Q,R):S≥0,Q≥0,R≥0}.
In fact, for any (S0,E0,I0,Q0,R0)∈∂Γ0∖{(S,0,0,Q,R):S≥0,Q≥0,R≥0}, in the case where E0=0,I0>0, it is clear that S(t)>0, I(t)>0, for all t>0. From the second equation of model (1), we have
(37)E(t)=e-(d+σ)t(E0+∫0t‍β(τ)S(τ)I(τ)S(τ)+αI(τ)e(d+σ)τdτ)>0,∀t>0.
In the case where E0>0,I0=0, then E(t)>0, for any t>0, I˙(0)=kE0>0. Therefore, (S(t),E(t),I(t),Q(t),R(t))∉∂Γ0 for sufficiently small t>0. That is to say, for any (S0,E0,I0,Q0,R0)∉{(S,0,0,Q,R):S≥0,Q≥0,R≥0}, (S0,E0,I0,Q0,R0)∉M∂. This implies that M∂⊆{(S,0,0,Q,R):S≥0,Q≥0,R≥0} and P0 is the only fixed point of P and acyclic in ∂Γ0. Moreover, Lemma 6 implies that P0 is an isolated invariant set in Γ and Ws(P0)∩Γ0=⌀, where Ws(P0) is the stable set of P0. By the acyclicity theorem on uniform persistence for maps [34], it follows that P is uniformly persistent with respect to (Γ0,∂Γ0).
Furthermore, taking advantage of Theorem 1.3.6 in Zhao [34], P has a fixed point. Consider
(38)((S*(0),E*(0),I*(0),Q*(0),R*(0)))∈Γ0.
Then we see that S*(0)≥0, E*(0)>0, I*(0)>0, Q*(0)≥0,  and R*(0)≥0. We further prove that S*(0)>0, Q*(0)>0, and R*(0)>0. Suppose not, if S*(0)=0, from the first equation of model (1), we derive that
(39)dS*(t)dt≥Λ-(d+β(t)α-1)S*(t)+ηR*(t),
with S*(0)=S*(nω)=0, n=1,2,3…. Then we have
(40)S*(nω)≥e-∫0nω‍(d+β(s)α-1)dsS*(nω)w×(S*(0)+∫0nω‍e∫0τ‍(d+β(s)α-1)ds(Λ+ηR*(τ))dτ)S*(nω)>0.
The periodicity of S*(t) implies S*(t)>0, for all t>0. Similarly, Q*(0)>0 and R*(0)>0. Thus, u(t,(S*(0),E*(0),I*(0),Q*(0),R*(0)))∈Intℝ+5 and (S*(t),E*(t),I*(t),Q*(t),R*(t)) is a positive ω-periodic solution of model (1). This completes the proof.Remark 8.
Theorem7 tells us that when the basic reproduction number R0>1, model (1) admits at least one positive periodic solution, and the infectives are ultimately bounded below by some positive constant ε, which indicates that HFMD becomes an endemic disease.Remark 9.
From Theorems4 and 7, we can conclude that R0 is the threshold parameter for the extinction and the uniform persistence of the disease, which partially answer our proposed question in the introduction: “what is the disease dynamics of the HFMD in Wenzhou?”.


## 3.1. Preliminaries

Let(ℝn,ℝ+n) be the standard ordered n-dimensional Euclidean space with a norm ∥·∥. For u,v∈ℝn, we denote u≥v, if u-v∈ℝ+n; u>v, if u-v∈ℝ+n0; and u≫v, if u-v∈Int(ℝ+n).We briefly present some main results of Floquet theory. Consider the following linear periodic system:(2)dudt=A(t)u,
where u is an n×1 vector and A(t) is an n×n matrix of principal periodic ω>0, such that A(t+ω)=A(t).LetΦA(·)(t) be a fundamental solution matrix of (2); that is, a nonsingular matrix each of the columns of which is a solution of equation such that ΦA(·)(0)=𝕀, the identify matrix. Floquet theory show that the fundamental matrix has the form ΦA(·)(t)=P(t)eBt, where P(t)=P(t+ω) is a periodic matrix with initial values P(0)=𝕀 and B is a constant matrix. The matrix ΦA(·)(T)=eBT is the so-called principle or monodromy or Floquet transition matrix of (2). Let r(ΦA(·)(ω)) be the spectral radius of ΦA(·)(ω). It then follows from [27, 28] that ΦA(·)(ω) is a matrix with all entries positive for each t>0. By the Perron-Frobenius theorem [29], r(ΦA(·)(ω)) is the principal eigenvalue of ΦA(·)(ω) in the sense that it is simple and admits an eigenvector u*≫0. The following result is useful for our subsequent comparison arguments.Lemma 1 (see [30, Lemma 2.1]).
Letp=(1/ω)ln⁡r(ΦA(·)(ω)). Then there exists a positive, ω-periodic function v(t) such that eptv(t) is a solution of (2).


## 3.2. Positively Invariant Sets

The study of the dynamics of model (1) requires the introduction of the following important sets:
(3)X={(S,E,I,Q,R)∈ℝ+5},Γ={(S,E,I,Q,R)∈X,0<S+E+I+Q+R≤Λd}.And we can easily obtain the following theorem of the positively invariant sets.Theorem 2 (positively invariant sets).
X and Γ are positive invariant sets for model (1).Proof.
It shows thatX is positively invariant. From model (1), the total population N(t) satisfies the following equation:
(4)Λ-(d+δ1+δ2)N≤dNdtΛ-(d+δ1+δ2)N=Λ-dN-δ1I-δ2Q≤Λ-dN.
Hence, by integration, we check
(5)Λd+δ1+δ2+(N(0)-Λd+δ1+δ2)e-(d+δ1+δ2)t≤N(t)≤Λd+(N(0)-Λd)e-dt.
Hence,
(6)Λd+δ1+δ2≤lim⁡t→∞inf⁡N(t)≤lim⁡t→∞sup⁡N(t)≤Λd,
which implies that Γ is positively invariant with respect to model (1).


## 3.3. The Basic Reproduction NumberR0

It is easy to see that model (1) always has one disease-free equilibrium P0=(Λ/d,0,0,0,0). By the definition of Bacaër and Guernaoui [31] and the general calculation procedure in Wang and Zhao [32], we have
(7)ℱ=(β(t)SIS+αI0000),𝒱=((d+σ)E(d+δ1+k+γ1)I-σE(d+δ2+γ2)Q-kIβ(t)SIS+αI+dS-Λ-ηR(d+η)R-γ1I-γ2Q),𝒱-=((d+σ)E(d+δ1+k+γ1)I(d+δ2+γ2)Qβ(t)SIS+αI+dS(d+η)R),𝒱-=(0σEkIΛ+ηRγ1I+γ2Q).
It follows that
(8)F(t)=(0β(t)0000000),M=(-dη0-(d+η)),V(t)=(d+σ00-σd+δ1+k+γ100-kd+δ2+γ2).Furthermore,F(t) is nonnegative, and -V(t) is cooperative in the sense that the off-diagonal elements of -V(t) are nonnegative.It is obvious that-d and -(d+η) are the eigenvalues of M and are negative. Thus M is stable; namely, r(ϕM(ω))<1. We can see that the eigenvalues of -V are the diagonal elements and are negative. So -V is stable; namely, r(ϕ-V(ω))<1.LetY(t,s),t≥s, be the evolution operator of the linear ω-periodic system. Consider
(9)dydt=-V(t)y.
That is, for each s∈ℝ, the 3×3 matrix Y(t,s) satisfies
(10)dY(t,s)dt=-V(t)Y(t,s),∀t≥s,Y(s,s)=𝕀,
where 𝕀 is the 3×3 identity matrix. Thus, the monodromy matrix Φ-V(t) of (9) equals Y(t,0),t≥0.Following the method in Wang and Zhao [32], we let ϕ(s) be ω-periodic in s and the initial distribution of infectious individuals. So F(s)ϕ(s) is the rate of new infections produced by the infected individuals who are introduced at time s. When t≥s, Y(t,s)F(s)ϕ(s) gives the distribution of those infected individuals who are newly infected by ϕ(s) and remain in the infected compartments at time t. Naturally,
(11)∫-∞t‍Y(t,s)F(s)ϕ(s)ds=∫0∞‍Y(t,t-a)F(t-a)ϕ(t-a)da
is the distribution of accumulative new infections at time t produced by all those infected individuals ϕ(s) introduced at time previous to t.LetCω be the ordered Banach space of all ω-periodic functions from ℝ to ℝ3, which is equipped with the maximum norm ∥·∥ and the positive cone Cω+:={ϕ∈Cω:ϕ(t)≥0,∀t∈ℝ}. Then we can define a linear operator. Consider
(12)(Lϕ)(t)=∫0∞‍Y(t,t-a)F(t-a)ϕ(t-a)da,∀t∈ℝ,ϕ∈Cω.L is called the next infection operator and the spectral radius of L is defined as the basic reproduction number. Consider
(13)R0:=r(L),
for the periodic epidemic model.Lemma 3 (see [32]).
The following statements are valid:(i)
R0=1 if and only if r(ΦF-V(ω))=1;(ii)
R0>1 if and only if r(ΦF-V(ω))>1;(iii)
R0<1 if and only if r(ΦF-V(ω))<1.Thus, the disease-free equilibrium P0=(Λ/d,0,0,0,0) of model (1) is asymptotically stable if R0<1 and unstable if R0>1.Now we introduce the linearω-periodic system
(14)dwdt=(-V(t)+F(t)λ)w,t∈ℝ+
with parameter λ∈ℝ. Let W(t,s,λ),t≥s, be the evolution operator of system (14) on ℝ4. Clearly, ΦF-V(t)=W(t,0,1),t≥0. Moreover,
(15)-V(t)+F(t)λ=(-(d+σ)β(t)λ0σ-(d+δ1+k+γ1)00k-(d+δ2+γ2)).
Hence, we derive
(16)ΦF/λ-V(t)=W(t,0,λ).
Following the general calculation procedure in Wang and Zhao [32], the basic reproduction number R0 is the unique solution of r(W(ω,0,λ))=1.


## 3.4. Extinction of the Disease

Theorem 4.
The disease-free equilibriumP0=(Λ/d,0,0,0,0) of model (1) is globally asymptotically stable if R0<1 and unstable if R0>1.Proof.
IfR0<1, according to Lemma 3, we have r(ΦF-V(ω))<1. From model (1), for t≥0, we know that
(17)dEdt≤β(t)I-(d+σ)E,dIdt=σE-(d+δ1+k+γ1)I,dQdt=kI-(d+δ2+γ2)Q.
Consider the following comparison system:
(18)dE~dt=β(t)I~-(d+σ)E~,dI~dt=σE~-(d+δ1+k+γ1)I~,dQ~dt=kI~-(d+δ2+γ2)Q~;
that is,
(19)dgdt=(F(t)-V(t))g(t),g(t)=(E~(t),I~(t),Q~(t))T.
By Lemma 1, there exists a positive ω-periodic function g~(t) such that g(t)=eptg~(t) is a solution of (18), where p=(1/ω)ln⁡r(ΦF-V(ω))<0. Then we conclude that g(t)→0 as t→∞, which implies that the zero solution of model (18) is globally asymptotically stable. Applying the comparison principle, we known that E(t)→0,I(t)→0,  and Q(t)→0 as t→∞. It follows that R(t)→0,  and S(t)→Λ/d as t→∞. Therefore, the disease-free equilibrium P0=(Λ/d,0,0,0,0) is globally asymptotically stable. This completes the proof.Remark 5.
From Theorem4, one can know that if R0<1, then P0=(Λ/d,0,0,0,0) is globally asymptotically stable, which means that the disease will vanish after some period of time. Therefore, it is interesting to study the disease-free equilibrium for controlling infectious disease.


## 3.5. Uniform Persistence of the Disease

Define(20)Γ0:={(S,E,I,Q,R)∈Γ:E>0,I>0}
and ∂Γ0=Γ∖Γ0={(S,E,I,Q,R)∈Γ:E(t)≡0orI(t)≡0}. Denote u(t,x0) as the unique solution of model (1) with the initial value x0:=(S0,E0,I0,Q0,R0). Let P:Γ→Γ be the Poincaré map associated with model (1); that is,
(21)P(x0)=u(ω,x0),∀x0∈Γ,
where ω is the period. By applying the fundamental existence-uniqueness theorem [33], we know that u(t,x0) is the unique solution of model (1) with u(0,x0)=x0. From Theorem 2, we know that P is dissipative point. We then introduce the following lemma.Lemma 6.
IfR0>1, then there exists a ϵ>0 such that when ∥x0-P0∥≤ϵ, for any x0∈Γ0, one has
(22)limsup⁡m→∞d[Pm(x0)-P0]≥ϵ.Proof.
We proceed by contradiction to prove that(23)limsup⁡m→∞d[Pm(x0)-P0]≥ϵ.
If not, then
(24)limsup⁡m→∞d[Pm(x0)-P0]<ϵ,
for some x0∈Γ0. Without loss of generality, we can assume that d(Pm(x0),P0)<ϵ, for all m≥0. By the continuity of the solutions with respect to the initial values, for all ε∈(0,Λ/d), we obtain
(25)∥u(t,Pm(x0))-u(t,P0)∥≤ε,∀t∈[0,ω].
For any t≥0, let t=mω+t1, where t1∈[0,ω] and m=[t/ω], which is the greatest integer less than or equal to t/ω. Then we have
(26)∥u(t,Pm(x0))-u(t,P0)∥=∥u(t1,Pm(x0))-u(t1,P0)∥≤ε,∀t≥0.
It follows that Λ/d-ε≤S(t)≤Λ/d+ε,0≤I(t)≤ε, for t≥0. Then S/(S+αI)≥1-αε/(Λ/d+(α-1)ε):=1-θ>0. Thus, for ∥x0-P0∥≤ϵ, we have
(27)dEdt≥β(t)(1-θ)I-(d+σ)E,dIdt=σE-(d+δ1+k+γ1)I,dQdt=kI-(d+δ2+γ2)Q.
We consider the following comparison system
(28)dE-dt=β(t)(1-θ)I--(d+σ)E-,dI-dt=σE--(d+δ1+k+γ1)I-,dQ-dt=kI--(d+δ2+γ2)Q-.
Set Mε=(0θ0000000). Then, model (28) can be rewritten as
(29)dgdt=(F(t)-V(t)-Mε)g(t),g(t)=(E-(t),I-(t),Q-(t))T.
If R0>1, we obtain r(ΦF-V(ω))>1 by Lemma 3. Then we can choose ε>0 small enough such that r(ΦF-V-Mε(ω))>1. Once again by Lemma 1, it follows that there exists a positive ω-periodic g-(t) such that g(t)=ep1tg-(t) is a solution of model (28), where p1=(1/ω)ln⁡r(ΦF-V(ω))>0, which implies that g(t)→∞ as t→∞. By applying the comparison principle, we have E(t)→∞,I(t)→∞, and before Q(t)→∞ as t→∞. This is a contraction.By using Lemma6, we can obtain the following theorem.Theorem 7.
IfR0>1, model (1) has at least one positive periodic solution and is uniformly persistent.Proof.
We first prove that{Pm}m≥0 is uniformly persistent with respect to (Γ0,∂Γ0). First of all, we show that Γ0 and ∂Γ0 are positively invariant, for any (S0,E0,I0,Q0,R0)∈Γ0. From the first equation of model (1), we derive that
(30)dSdt≥Λ-(d+β(t)α)S+ηR.
So, we have
(31)S(t)≥e-∫0t‍(d+β(s)α-1)dsS(t)hh×(S0+∫0t‍e∫0τ‍(d+β(s)α-1)ds(Λ+ηR(τ))dτ)S(t)>0,∀t>0.
Similarly,
(32)E(t)=e-(d+σ)t(E0+∫0t‍β(τ)S(τ)I(τ)S(τ)+αI(τ)e(d+σ)τdτ)>0,∀t>0,I(t)=e-(d+δ1+k+γ1)t(I0+∫0t‍σE(τ)e(d+δ1+k+γ1)τdτ)>0,∀t>0,Q(t)=e-(d+δ1+k+γ1)t(Q0+∫0t‍kI(τ)e(d+δ1+k+γ1)τdτ)>0,∀t>0,R(t)=e-(d+η)t(R0+∫0t‍(γ1I(τ)+γ2Q(τ))e(d+η)τdτ)>0,∀t>0.
Thus, Γ0 is positively invariant. Clearly, ∂Γ0 is relatively closed in Γ. Set
(33)M∂={(S0,E0,I0,Q0,R0)∈∂Γ0:Pm(S0,E0,I0,Q0,R0)∈∂Γ0,∀m≥0}.
We claim that
(34)M∂={(S,0,0,Q,R):S≥0,Q≥0,R≥0}.
Note that
(35){(S,0,0,Q,R):S≥0,Q≥0,R≥0}⊆M∂.
We only need to prove that
(36)M∂⊆{(S,0,0,Q,R):S≥0,Q≥0,R≥0}.
In fact, for any (S0,E0,I0,Q0,R0)∈∂Γ0∖{(S,0,0,Q,R):S≥0,Q≥0,R≥0}, in the case where E0=0,I0>0, it is clear that S(t)>0, I(t)>0, for all t>0. From the second equation of model (1), we have
(37)E(t)=e-(d+σ)t(E0+∫0t‍β(τ)S(τ)I(τ)S(τ)+αI(τ)e(d+σ)τdτ)>0,∀t>0.
In the case where E0>0,I0=0, then E(t)>0, for any t>0, I˙(0)=kE0>0. Therefore, (S(t),E(t),I(t),Q(t),R(t))∉∂Γ0 for sufficiently small t>0. That is to say, for any (S0,E0,I0,Q0,R0)∉{(S,0,0,Q,R):S≥0,Q≥0,R≥0}, (S0,E0,I0,Q0,R0)∉M∂. This implies that M∂⊆{(S,0,0,Q,R):S≥0,Q≥0,R≥0} and P0 is the only fixed point of P and acyclic in ∂Γ0. Moreover, Lemma 6 implies that P0 is an isolated invariant set in Γ and Ws(P0)∩Γ0=⌀, where Ws(P0) is the stable set of P0. By the acyclicity theorem on uniform persistence for maps [34], it follows that P is uniformly persistent with respect to (Γ0,∂Γ0).
Furthermore, taking advantage of Theorem 1.3.6 in Zhao [34], P has a fixed point. Consider
(38)((S*(0),E*(0),I*(0),Q*(0),R*(0)))∈Γ0.
Then we see that S*(0)≥0, E*(0)>0, I*(0)>0, Q*(0)≥0,  and R*(0)≥0. We further prove that S*(0)>0, Q*(0)>0, and R*(0)>0. Suppose not, if S*(0)=0, from the first equation of model (1), we derive that
(39)dS*(t)dt≥Λ-(d+β(t)α-1)S*(t)+ηR*(t),
with S*(0)=S*(nω)=0, n=1,2,3…. Then we have
(40)S*(nω)≥e-∫0nω‍(d+β(s)α-1)dsS*(nω)w×(S*(0)+∫0nω‍e∫0τ‍(d+β(s)α-1)ds(Λ+ηR*(τ))dτ)S*(nω)>0.
The periodicity of S*(t) implies S*(t)>0, for all t>0. Similarly, Q*(0)>0 and R*(0)>0. Thus, u(t,(S*(0),E*(0),I*(0),Q*(0),R*(0)))∈Intℝ+5 and (S*(t),E*(t),I*(t),Q*(t),R*(t)) is a positive ω-periodic solution of model (1). This completes the proof.Remark 8.
Theorem7 tells us that when the basic reproduction number R0>1, model (1) admits at least one positive periodic solution, and the infectives are ultimately bounded below by some positive constant ε, which indicates that HFMD becomes an endemic disease.Remark 9.
From Theorems4 and 7, we can conclude that R0 is the threshold parameter for the extinction and the uniform persistence of the disease, which partially answer our proposed question in the introduction: “what is the disease dynamics of the HFMD in Wenzhou?”.


## 4. Simulation Results via Statistical Data in Wenzhou

In this section, by using model (1), we simulate the reported data of HFMD in Wenzhou and carry out the sensitivity analysis based on the parameters.Thanks to the insightful work of Liu [17], Ma et al. [20], and Yang et al. [19], we estimate the values of all parameters of model (1) in details:(i)
the recruitment rate of susceptibleΛ: from the statistical data of Wenzhou population [24], one can know that the average birth number of Wenzhou population from 2005 to 2010 is 102729; then we adopt the weekly birth population number Λ=1975;(ii)
the natural mortality rated: in this paper, we take the average death rate of Wenzhou population as the natural mortality rate d≈0.642% [24];(iii)
the transmission rateβ(t): there are some epidemiological models using sinusoidal function (β(t)=β0+β1sin(ωt+ϕ)) for the seasonal varying transmission rate [16, 17, 19, 20]; in this paper, we use the periodic function β(t)=β0+β1sin(ωt+ϕ) with period ω (here ω=52 weeks) as the transmission rates; by using the least-square fitting of the numbers of HFMD I(t), we can estimate the values of β0, β1, and ϕ;(iv)
the “psychological” effectα: in this paper, we adopt α=0.5; it would be likely to note that when α=1, the transmission rate is the classical standard incidence rate;(v)
the mean incubation period1/σ: as discussed in the introduction section, a person who is exposed to HFMD viruses will exhibit the symptoms after 3 to 7 days, for simplicity; in this paper, we assume that the average incubating time is 4 days, then the mean incubation period 1/σ can be determined by 4/7;(vi)
the quarantined ratek: some of the infected people will be hospitalized to isolate for treatment; the ratio of these quarantined in infectious is called the quarantined rate; based on the statistical data, we can determine k=0.007177;(vii)
the infected recover rateγ1: we assume that the average recovery time of infectious is 8 days; then the recovery rate is 7/8=0.875;(viii)
the quarantined recover rateγ2: we assume that the average recovery time of quarantined is 16 days; then the recovery rate is 7/16=0.4375;(ix)
the disease-induced mortalityδ1 and δ2: we assume that the death comes from the quarantined; that is, δ1=0; and by using the least-square fitting of HFMD number I(t), we can determine δ2=0.024887;(x)
the progression rateη: by using the least-square fitting of HFMD number I(t), we can determine η=0.015.And the number of the initial susceptible population at the end of 2010,S(0)=9,122,100, is obtained from the Wenzhou City Population and Family Planning Commission [24]. The numbers of the initial infected I(0) and quarantined Q(0) can be obtained from the reported data of HFMD. Because the numbers of the initial exposed population E(0) and the recovered population R(0) cannot be obtained directly, we derive E(0) by k and R(0) by γ1 and kγ2.The numerical simulation results of the model (1) about the number of HFMD infectious are shown in Figures 2–4.In Figure1, we show the spread dynamics of HFMD in Wenzhou from March 2010 to November 2013, and the parameters in model (1) undefined in the above are taken as
(41)I(0)=368,E(0)=210,Q(0)=322,R(0)=323,β(t)=0.91+0.39sin(πt26+1.75).
In this case, R0≈0.9833<1; the disease-free equilibrium P0=(307792,0,0,0,0) of model (1) is globally asymptotically stable, which means the HFMD disease will vanish after long time. The HFMD will die out, but is it always true?Figure 1
The solid curve represents the simulation results and the stars are the weekly data from 2010 to 2013 reported by the Wenzhou CDC.Figure 2
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 (1) with parameters (43). (a) Time series plots; (b) phase portraits (periodic solution).
(a)
(b)Figure 4
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 (Figure2) and 2012-2013 (Figure 4), respectively. In Figure 2, with parameters
(42)I(0)=136,E(0)=78,Q(0)=119,R(0)=167,β(t)=0.9335+0.5225sin(πt26+1.575);
then R0≈1.0259>1, which means that HFMD becomes an endemic disease in Wenzhou. In this case, from Theorems 4 and 7, one can know that model (1) has at least one positive periodic solution which is uniformly persistent. Figures 3(a) and 3(b), with the same parameters as in Figure 2, confirm this conclusion.In Figure4, with parameters(43)I(0)=234,E(0)=134,Q(0)=205,R(0)=206,β(t)=0.908+0.47sin(πt26+1.575),
and, hence, R0≈0.9723<1, which means the HFMD disease will vanish after long time.Comparing Figure1 with Figures 2 and 4, one can see that with partial or the whole HFMD data in Wenzhou from 2010 to 2013, with different time periodic data, then by using least-square fitting method, we can obtain different periodic transmission rate β(t), and, hence, obtain different R0, so we can obtain different disease spread dynamics and draw different conclusions. This difference is induced by different data and initial values.For the sake of learning the effects of the quarantined rate (k) and infected recovery rate (γ1) on the controlling of disease spreading, we consider the relations between the reproduction number R0 with k and γ1, respectively. Figure 5(a) shows that the larger quarantined rate k is, the less R0 is; and that is to say, quarantine has a positive impact on controlling the spread of disease. Figure 5(b) has the similar tendency with Figure 5(a); that is, the larger quarantined rate γ1 is, the less R0 is. And γ1=7/T (T is the cycle of treatment); hence, we can conclude that the less the cycle of treatment T is, the less R0 is.The influence of the quarantined ratek (a) and infected recover rate γ1 (b) on R0.
(a)
(b)


## 5. Conclusions and Discussions

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 (1) can be governed by its reproduction number R0: if R0<1; the disease-free equilibrium P0=(Λ/d,0,0,0,0) of model (1) is globally asymptotically stable, which means the disease will die out; while R0>1, model (1) has at least one positive periodic solution and is uniformly persistent, which indicates that HFMD becomes an endemic disease; that is, the HFMD will break out.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 Figures1, 2, and 4, we can obtain different disease spread dynamics and draw different conclusions based on the different data and initial values. In fact, a mathematical model may help to explain a system, to study the effects of different components, and to make predictions about disease spreading behaviour. We need to point out that all these three figures are partly explaining and predicting the spread dynamics of the HFMD in Wenzhou. Besides, in the cases of Figures 1, 2, and 4, we obtain different basic reproduction numbers R0, which can mathematically help us to determine whether an infectious disease can spread or not. There are two values of R0 less than 1, and close to 1. It is worthy to note that, in the HFMD model (1), we assume that the whole population of Wenzhou (more than 9 million people) is homogeneously mixing. If we try to take children under five—17 years old and their family as the susceptible population (about 1 million people), the value of R0 would be larger. All in all, we can conclude that the HFMD in Wenzhou persists under current circumstances; that is, the HFMD becomes an endemic disease and will break out in Wenzhou.From Figure5, we find that the quarantined rate (k) and the infected recovery rate (γ1) are very beneficial to control the disease outbreak. That is to say, for controlling the HFMD spread in Wenzhou, we must increase the quarantined rate or decrease the treatment cycle.In [20], based on the statistical data HFMD in Shandong province, China, Ma et al. indicated that the quarantined rate k has little effect on controlling the disease spread. This is very different from our present results. It is reported that the HFMD incidence strongly depends on the climate [19]. We think this difference may be induced by different climate: Shandong province is in north China, and Wenzhou is in south China.Furthermore, from Figures1, 2, and 4, one can see that the fittings have some flaws because of oscillation peak in the data of summer 2011 and 2012. We think this phenomenon may represent an alarm for next year prevalence. In addition, there may be more suitable incidence rate β(t) to fit the spread dynamics of the HFMD in Wenzhou. This is desirable in future studies.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 [35]. HFMD is highly contagious and is transmitted by oropharyngeal secretions such as saliva or nasal mucus, by direct contact or fecal-oral transmission. Preventive measures include avoiding direct contact with infected individuals (including keeping infected children home from school), proper cleaning of shared utensils, disinfecting contaminated surfaces, and proper hand hygiene. These measures have been proved to be effective in decreasing the transmission of the viruses responsible for HFMD [1, 35].



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*Source: 234509-2014-03-25.xml*