MODELLING SEASONAL HFMD WITH THE RECESSIVE INFECTION IN SHANDONG , CHINA

Hand, foot and mouth disease (HFMD) is one of the major publichealth problems in China. Based on the HFMD data of the Department of Health of Shandong Province, we propose a dynamic model with periodic transmission rates to investigate the seasonal HFMD. After evaluating the basic reproduction number, we analyze the dynamical behaviors of the model and simulate the HFMD data of Shandong Province. By carrying out the sensitivity analysis of some key parameters, we conclude that the recessive subpopulation plays an important role in the spread of HFMD, and only quarantining the infected is not an effective measure in controlling the disease.

1. Introduction.Hand, foot and mouth disease (HFMD) is a common infectious disease among infants and children.HFMD is caused by a group of enterovirus which mainly include coxsackievirus A16 (CA16) and enterovirus 71 (EV71) [4].It is estimated that there are 500,000∼1,800,000 HFMD cases per year in China [21], and a series of recent HFMD outbreaks in China can be found in [9,10,22,26,27].
In the spread of HFMD, children are more susceptible to be infected than adults, because they are less likely than adults to have antibodies and awareness of selfprotection.HFMD spreads mainly among children under five-years old [5], and may 2. The HFMD Model.We classify the population into six compartments according to their states: susceptible, exposed, infected, recessive, quarantined, recovered, which are denoted by S(t), E(t), I(t), I e (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) + I e (t) + Q(t) + R(t).The transition dynamics associated with these subpopulations are illustrated in Figure 1.People who are infected firstly enter the latent period, during which they do not show symptoms and can not infect others.After about 3 ∼ 7 days, these people become the infectious.The infectious people are classified into the infected (I(t)) and the recessive (I e (t)), who are different in the transmission rate.Some of the infected people will be hospitalized for treatment, and thus they (Q(t)) are isolated from other subpopulations.Because HFMD in Shandong was mainly caused by EV71 [27], thus some of the recovered will be reinfected after they lose the immunity.

Figure 1. Flowchart of HFMD transmission with subpopulations
The standard incidence β(t)SI/N is applied in the model because an infectious individual only can contact a finite number of individuals within a unit time among a large population [14].The transmission rate between S(t) and I(t) is β 1 (t), and the transmission rate between S(t) and I e (t) is β 2 (t).As discussed in the Introduction, many epidemiological models [6,8,12,13,16,23,24,28] were simulated by using sinusoidal function of period 1 year (β(t) = β 0 + β sin(ωt + φ)) for the seasonal varying transmission rate.In this model, we use the periodic functions with period ω (here ω = 52 weeks) as the transmission rates.Here a 1 , b 1 , a 2 , b 2 and φ are constant, which can be determined by the least-square fitting in Section 4.
The interpretations and values of parameters are described in   rate of the infected individuals γ 1 are given by [9].In this model we also assume that the infected, the quarantined and the recessive have the same recover rate The values of other parameters can be obtained by the least-square fitting method.The model is described by the following system of nonautonomous differential equations.
3. Mathematical analysis.In this section, we investigate the global stability of disease-free equilibrium and the existence of the positive periodic solution of model (1).It is easy to see that model (1) always has one disease-free equilibrium P 0 = ( Ŝ, 0, 0, 0, 0, 0), where Ŝ = Λ d .By using model (1), we have therefore is the feasible region for model (1), and we also have Theorem 3.1.The region X is positively invariant set for model (1).
It is not difficult to prove Theorem 3.1, thus we put the proof in the Appendix A. We can derive the basic reproduction number of model ( 1) by the definition of Bacaër and Guernaoui [2], and its calculation is based on Floquet theory introduced in [3] and Wang and Zhao [25].The details are given in the Appendix B. For the globally asymptotically stable of the disease-free equilibrium P 0 , we have the following theorem.
About the proof of Theorem 3.2, we also put it in the Appendix C. The following we consider the existence of the positive periodic solution of model (1).Define and ∂X 0 = X \ X 0 .Denote u(t, x 0 ) as the unique solution of model ( 1) with the initial value x 0 = (S 0 , E 0 , I 0 , I 0 e , Q 0 , R 0 ).Let P : X → X be the Poincaré map associated with model (1), i.e., where ω is the period.Applying the fundamental existence-uniqueness theorem [15], we know that u(t, x 0 ) is the unique solution of model ( 1) with u(0, x 0 ) = x 0 .From Theorem 3.1, we know that X is positively invariant and P is dissipative point.To prove the main result about the existence of positive periodic solution of model ( 1), we need the following lemma.
Because the proof of Lemma 3.3 is similar with the proof of Lemma 2.4 in [28], here we omit it.According to Lemma 3.3, we can get the following theorem about the existence of positive periodic solution of model (1).Similarly, the reader can find the proof in the Appendix D. Theorem 3.4.Model (1) has at least one positive periodic solution when R 0 > 1.
4. Simulations and Sensitivity Analysis.In this section, by using model (1), we simulate the reported data of HFMD in Shandong, China from April 2009 to October 2011, and carry out the sensitivity analysis based on the parameters.
We need to estimate the values of parameters of model ( 1), most of which can be obtained from the literature or assumed on the basis of common sense.From the Department of Health of Shandong Province, we obtained the data of HFMD.By using the least-square fitting of I(t), we estimated the values of parameters p, η, and we also obtained β 1 (t) = 1 + 0.3 sin( π 26 t + 2) and β 2 (t) = 0.878 + 0.3 sin( π 26 t + 2).The values of other parameters are listed in Table 1.We need the initial values to perform the numerical simulations.The number of the initial susceptible population at the end of 2008, S(0) = 9.3 × 10 7 , is obtained from the Statistical Information of Shandong Province [18].The numbers of the initial infected and quarantined population I(t) and Q(t) are obtained from the reported data of HFMD, thus I(0) = 5.775 × 10 3 , Q(0) = 50.Because the numbers of the initial exposed population E(0), the recessive population I e (0) and the recovered population R(0) can not be obtained directly, we derive R(0) = 5×10 4 by the parameter γ 1 and E(0) = 8.5×10 4  and I e (0) = 7 × 10 4 are estimated by a reasonable assumption.The numerical simulation of the model (1) about the number of HFMD infectious cases is shown in Figure 2. It indicates that with these parameter values, there is a good fit between the simulation of the model (1) and the infectious cases in Shandong Province from 2009 to 2011.Moreover, with these parameter values, we can roughly estimate that the basic reproduction number R 0 1.04 > 1, which show that HFMD in Shandong Province persist under current circumstances.Furthermore we notice R 0 is also close to one.This is because that we suppose the whole population of Shandong (almost 100 million people) is homogeneously mixing.If we have try to take for the susceptible population by the children under five-years old and their family (less than 10 million people), the estimate for R 0 would be somewhat higher.Moreover, we demonstrate R 0 is a threshold, which determine the disease extinct or not.HFMD will persist under the condition R 0 1.01, where a 1 = 0.95, a 2 = 0.85 in β 1 (t) and β 2 (t).If let a 1 = 0.93 and a 2 = 0.83, we get R 0 0.98.In this case the disease will extinct (see Figure 3).Next we discover the influence of initial values S(0), E(0), I e (0), Q(0) and R(0) on the number of infected cases I(t).From Figure 4, we can see that the initial value of S(t) has a greater impact on I(t) while other initial values have little or no impact on I(t).In order to perform sensitivity analysis of parameters p and k, we fix all parameters except p and k. Figure 5(a) reflects the relation between the basic reproduction number R 0 and the parameter p.We see that the basic reproduction number R 0 increase with the increasing of p, and R 0 always larger than one even if p = 0. From Figure 5(a), we conclude that the parameter p has great influence on R 0 , and the infected and the recessive subpopulation play the dominant role in the spread of Other parameter values in Table 1 do not change.
HFMD. Figure 5(b) shows that the larger k is, the less R 0 is, that is to say, quarantine has a positive impact on controlling the spread of disease.However, even if the quarantine rate k is larger than 3 the basic reproduction number is also larger than one.

5.
Discussion.The transmission of HFMD has been a growing concern in China.
In this paper, by using HFMD data of Shandong Province, we constructed an SEII e QR model with periodic transmission rates to investigate the spread of seasonal HFMD in Shandong.From the simulations, we concluded that HFMD will persist in Shandong Province under current circumstances.By carrying out the sensitivity analysis of some key parameters, we found that the recessive subpopulation plays an important role in the spread of HFMD while the quarantine subpopulation has a little effect in controlling the disease.Even if the quarantine rate k is larger, the basic reproduction number is still larger than one, that is to say, HFMD still persist even with a larger quarantine rate.Therefore the quarantine is not an effective measure in many measures of controlling HFMD.According to WHO, there is no an effective vaccine or antiviral treatment specifically for HFMD.However, the risk of infection can be minimized by good hygiene practices, including: (i) washing hands frequently and thoroughly with soap and cleaning dirty surfaces and soiled items; (ii) avoiding close contact with the infective; (iii) not sharing personal items such as spoons, cups and other utensils with other people.In a word, publicity and education on the risk and prevention of HFMD is necessary and should be strengthened especially in endemic areas.
Appendix A: Calculation of the basic reproduction number.We evaluate the basic reproduction number R 0 for system (1) following the definition of Bacaër and Guernaoui [2] and the calculation procedure for ODEs based on Floquet theory introduced in [3].According to Wang and Zhao [25], we have So we derive Now we introduce the following linear ω-periodic equation with parameter z ∈ (0, ∞).Let W (t, s, z), t ≥ s, s ∈ R, be the evolution operator of system (3) on R 4 .Clearly, Φ F −V (t) = W (t, 0, 1), ∀t ≥ 0. To determine the threshold of dynamics, we use Theorems 2.1 and 2.2 in Wang and Zhao [25] which is a generalization of §3.4 in [3].First of all, we can verify the seven assumptions in the theorems.Then we can obtain that all eigenvalues of the matrix W (ω, 0, z).Because the W (ω, 0, z) is more complex, we do not want to show its accurate expression in this paper.Using (ii) in Theorem 2.1 in Wang and Zhao [25], we can calculate the basic reproduction number.
Appendix B: Proof of Theorem 3.1.
Proof.From model (1), the total population N (t) satisfies the following equation, and for any N (t 0 ) ≥ 0, the general solution of ( 4) is that we have lim which implies that X is positively invariant with respect to system (1).
Appendix C: Proof of Theorem 3.2.

Figure 2 .Figure 3 .
Figure 2. The solid curve represents the simulation curve and the stars are the weekly data reported by the Department of Health of Shandong.

Figure 5 .
Figure 5.The influence of parameters on R 0 .(a) Versus p; (b) versus k.Other parameter values inTable 1 do not change.

Table 1 .
. The source [A] is from Shandong Statistical Yearbook 2010, 2011 and 2012 [18].From [A] we obtain the values of the annual average birth rate and natural death rate, then we divide them by 52 and derive the weekly birth population Λ and natural death rate d.The source [B] is the reported data of HFMD in Shandong from April 2009 to October 2011.As similar as the above, we derive the weekly disease-related death δ 1 , δ 2 and the quarantine rate k.The average incubation period 1/σ and the recover Descriptions and values of parameters