JAMCJ Appl Math ComputDOI 10.1007/s12190-012-0597-1

O R I G I NA L R E S E A R C H

Stability analysis and optimal controlof a hand-foot-mouth disease (HFMD) model

Jun-Yuan Yang · Yuming Chen · Feng-Qin Zhang

Received: 15 March 2012© Korean Society for Computational and Applied Mathematics 2012

Abstract In this paper, we propose a system of ordinary differential equations tomodel the hand-foot-mouth disease (HFMD). We derive the expression of the basicreproduction number R0. When R0 < 1, the system only has the disease free equi-librium, which is globally asymptotically stable; otherwise, the system is persistent.By sensitivity analysis, we identify the control parameters. Then we formulate an op-timal control problem to find the optimal control strategy. These results are appliedto the spread of HFMD in Mainland China. The basic reproduction number tells usthat it is outbreak in China.

Keywords Hand-foot-mouth disease · Optimal control · Stability · Persistence ·Sensitivity analysis · Least-squares approach

Mathematics Subject Classification (2010) 34D20 · 92D30 · 49J15

1 Introduction

Hand-foot-mouth disease (HFMD) is a common febrile illness of early childhood,which is caused by viruses that belong to the enterovirus genus (group). This group of

J.-Y. Yang · Y. Chen · F.-Q. ZhangDepartment of Applied Mathematics, Yuncheng University, Yuncheng 044000, Shanxi, P.R. China

J.-Y. Yange-mail: yangjunyuan00@126.com

F.-Q. Zhange-mail: zhafq@263.com

J.-Y. Yang · Y. Chen (�)Department of Mathematics, Wilfrid Laurier University, Waterloo, ON, N2L 3C5 Canadae-mail: ychen@wlu.ca

J.-Y. Yang et al.

viruses includes polioviruses, coxsackieviruses, echoviruses, and enteroviruses. Themost common viruses causing HFMD are coxsackievirus A16 (COX A16) and en-terovirus 71 (EV71) [26]. Sometimes, HFMD may be caused by other enteroviruses.But most patients with fatal complications are infected by EV71.

Usually the enterovirus exhibits strong transitivity, the latent infection is high andthe transmission paths are complicated. These lead to epidemic outbreaks in shorttimes. The disease causes fever, tetter and ulceration on hand, foot and mouth, andmay further develop into myocarditis, pulmonary edema, aseptic meningoencephali-tis, and other complications. HFMD is spread through direct contact with nose dis-charge, saliva, feces and fluid from the blisters.

Since the first HFMD case was reported in New Zealand in 1957, subsequent out-breaks have been reported worldwide. For example, now the disease has high infec-tion in Mainland China. In 2008, there are 488,955 cases reported, with a morbidityof 37/100,000, mortality of 0.0095/100,000 and ill-death rate of 0.26/1000; while in2009, there are 1,155,525 cases reported [24]. Due to the severity of this disease, theMinistry of Health of the People’s Republic of China upgraded HFMD to a Class Ccommunicable disease on May 2, 2008. Since then, HFMD has been included intothe Communicable Disease Surveillance Network [26].

Unfortunately, there is no specific treatment for HFMD. Owing to its global spreadand the associated morbidity and mortality it inflicts, much attention has been focusedon devising methods for controlling the spread of HFMD [15]. In the absence ofeffective anti-HFMD therapeutic treatment and vaccine, HFMD control strategies arebased on taking appropriate preventive measures. These measures include quarantinemechanisms and personal protection against exposure to infected persons.

It is obvious that HFMD not only causes health problems but also has great socialand economical impacts. Therefore, it is important to understand the dynamics ofHFMD spread among the susceptible populations and enable policy makers to curbthe disease spread and reduce the adverse impact of the disease. In the literature, sta-tistical models have been applied to understand HFMD’s spatiotemporal transmissionand discover the relationship between HFMD occurrence and climate [6, 14, 22, 24,26]. However, mathematical modeling is scarcely exploited.

To the best of our knowledge, Chuo et al. [3] were the first to propose a simpledeterministic model to predict the number of infected and the duration of an outbreakwhen it occurs. Their model is based on the SIR (Susceptible-Infected-Recovered)model and is as follows,

dS

dt= αS(t) − βI (t)S(t) − μ0S(t) + δR(t),

dI

dt= βI (t)S(t) − γ I (t) − (μ0 + μ1)I (t), (1)

dR

dt= γ I (t) − δR(t) − μ0R(t).

We refer the readers to the article for the meanings of the biological parameters andalso for the coming two models. Then Roy and Halder [18] subdivided the infectious

Stability and control of HFMD model

class into asymptomatic infectious individuals E(t) and symptomatic infectious in-dividuals I (t) to obtain an SEIR model as follows,

d

dtS(t) = αN − (βa + βs)S(t)I (t)

N− δS(t),

d

dtE(t) = βaS(t)I (t)

N− (λ + δ)E(t),

d

dtI (t) = βsS(t)I (t)

N− (λ + δ)I (t),

d

dtR(t) = λ

(E(t) + I (t)

) − δR(t).

(2)

For the above two models, only numerical simulations are carried out. Recently, Liu[15] used the following model to take into account of the quarantine measure,

S = Λ − β(t)SI

N− μS − αS + δR,

E = β(t)SI

N− (μ + k)E,

I = kE − (μ + q + ε1 + γ1)I,

Q = qI − (μ + ε2 + γ2)Q,

R = γ1I + γ2Q − (μ + δ + α)R.

(3)

A threshold dynamics is obtained: the disease free equilibrium is globally asymp-totically stable if the threshold value is less than unity; otherwise, the system has apositive periodic solution and the disease persists.

Let us make some observations about the above three models.

(i) Since HFMD is only a common illness of infant and children, it is reasonable toassume a constant recruitment rate of susceptible Λ in (3) rather than a recruit-ment rate proportional to the susceptible in (1) or to the total number of infantand child in (2),

(ii) In (2) and (3), the standard incidence rate is used, which is a saturated incidencerate. An incidence rate is more reasonable for including the crowding effect ofthe infective individuals. However, in the case of HFMD, the number of infectiveindividuals is relatively small. Therefore, it is better to use the bilinear incidencerate as in (1). Mathematically, models with saturated incidence rate is easier toanalyze than those with bilinear incidence rate.

(iii) A person who is exposed to HFMD viruses will exhibit the symptoms afterthree to seven days. This is considered as the incubation period of HFMD. So,as in (3), it is more realistic to assume that after latency some of the exposedbecome infectious.

(iv) An individual will attain a short immunity from HFMD after recovery. Once theimmunity is loss, the individuals return to the susceptible class and is capable ofbeing infected again.

J.-Y. Yang et al.

Table 1 The biologicalmeanings of parameters in (4) Parameter Biological meaning Value

Λ recruitment rate estimated [19, 20]

β transmission rate unknown

μ natural death rate 0.01077 % [15]

1/k latent duration estimated [19, 20]

q quarantine rate unknown

γ1, γ2 recovery rates unknown

ε1, ε2 disease-related death rates estimated [19, 20]

δ loss of immunity rate 0.07 [3]

Motivated by the above observations, in this paper, we propose the followingHFMD model,

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

dS

dt= Λ − βSI − μS + δR,

dE

dt= βSI − (μ + k)E,

dI

dt= kE − (μ + q + ε1 + γ1)I,

dQ

dt= qI − (μ + ε2 + γ2)Q,

dR

dt= γ1I + γ2Q − (μ + δ)R.

(4)

Here, we have five compartments, namely, the susceptible (S), the exposed (E), theinfective (I ), the quarantined (Q), and the recovered (R). The biological meaningsof the parameters are summarized in Table 1. We mention that, in (3), it is assumedthat only the susceptible and recovered individuals are moved out of the group of agebelow 14. This is a little bit controversial. So in our model, the rate of moving out ofthe age group is included in the natural death rate.

It is easy to see that all solutions of (4) with nonnegative initial values are nonneg-ative forever. Moreover, we have

d(S + E + I + Q + R)

dt= Λ − μ(S + E + I + Q + R) − ε1I − ε2Q

≤ Λ − μ(S + E + I + Q + R).

It follows immediately that lim supt→∞[S(t) + E(t) + I (t) + Q(t) + R(t)] ≤ Λ/μ

and hence the set

Ω = {(S,E, I,Q,R) ∈ R

5+∣∣S + E + I + Q + R ≤ Λ/μ

}

is a positively invariant set of (4) and it attracts all solutions of (4). Therefore, weonly need to consider initial values in set Ω for (4).

The focus in [3, 15, 18] is the prevalence of disease. The control strategies are notconsidered. Optimal control theory is a powerful mathematical tool to make decision

Stability and control of HFMD model

involving complex dynamical systems. The Maximum Principle of Pontryagin andan embedded Newton algorithm are used to find an optimal control strategy in manyepidemic models [13]. Although optimal control methods have been employed tostudy the dynamics of some diseases [1, 8, 9, 11, 12, 17, 25], to the best of ourknowledge, they have not been applied to determine optimal control measures forHFMD.

The purpose of this paper is to understand the spreading dynamics of HFMD anddetermine better control strategies through sensitivity analysis.

The remaining part of this paper is organized as follows. In Sect. 2, we study thestability of the disease free equilibrium and the persistence of (4). In Sect. 3, we de-termine the control parameters based on sensitivity analysis of the basic reproductionnumber. The optimal control strategy is carried out in Sect. 4. Then a case study isdone for a situation in Mainland China in Sect. 5.

2 Stability of the disease free equilibrium and persistence

It is obvious that (4) always has the disease free equilibrium P0 = (Λμ

,0,0,0,0).Moreover, one can easily see that the endemic equilibrium exists if

R0 ≡ Λβk

μ(μ + k)(μ + q + ε1 + γ1)> 1

and the endemic equilibrium is given by P ∗ = (S∗,E∗, I ∗,Q∗,R∗), where

S∗ = Λ

μR0,

E∗ = μ + q + ε1 + γ1

kI ∗,

I ∗ = μ(1 − 1R0

)

β( 1R0

− 1R1

),

Q∗ = q

μ + ε2 + γ2I ∗,

R∗ = γ2q + γ1(μ + ε2 + γ2)

(μ + ε2 + γ2)(μ + δ)I ∗,

R1 = Λβ(μ + δ)(μ + ε2 + γ2)

μδ[γ1(μ + ε2 + γ2) + γ2q] .

Note that R1 is always larger than R0. R0 is the basic reproduction number, whichis defined to be the expected number of secondary cases produced in a completelysusceptible population by a typical infected individual during its entire period of in-fection. R0 can also be calculated by the recipe in van den Driessche and Watmough[23]. Moreover, Diekmann et al. [4] have shown that R0 is the spectral radius of thenext generation matrix.

J.-Y. Yang et al.

We first study the stability of the disease free equilibrium P0. The Jacobian matrixat P0 is

⎛

⎜⎜⎜⎜⎜⎝

−μ 0 −βΛμ

0 δ

0 −(μ + k)βΛμ

0 00 k −(μ + q + ε1 + γ1) 0 00 0 q −(μ + ε2 + γ2) 00 0 γ1 γ2 −(δ + μ)

⎞

⎟⎟⎟⎟⎟⎠

.

The eigenvalues are −μ, −(μ + ε2 + γ2), −(μ + δ), and the roots of

λ2 + (2μ + k + q + ε1 + γ1)λ + (μ + k)(μ + q + ε1 + γ1)(1 − R0) = 0.

It follows that

Theorem 1 The disease free equilibrium P0 is locally asymptotically stable ifR0 < 1 and it is unstable if R0 > 1.

In fact, we have the following further result on the stability of P0.

Theorem 2 Assume R0 < 1. Then the disease free equilibrium P0 is globally asymp-totically stable.

Proof By Theorem 1, it suffices to show that P0 is globally attractive. This is achievedby employing the fluctuation lemma [7]. For this purpose, we introduce a couple ofnotations. For a continuous and bounded function f : [0,∞) → R, let

f∞ = lim inft→∞ f (t) and f ∞ = lim sup

t→∞f (t).

Recall that every solution of (4) with nonnegative initial values is nonnegative for-ever and is bounded. By the fluctuation lemma, there exists a sequence {tn} such thattn → ∞, S(tn) → S∞, and dS(tn)

dt→ 0 as n → ∞. It follows from the first equation

of (4) that

dS(tn)

dt+ βS(tn)I (tn) + μS(tn) = Λ + δR(tn).

Letting n → ∞ gives us

μS∞ ≤ (μ + I∞)S∞ ≤ Λ + δR∞. (5)

Similarly, we can use the other equations in (4) to obtain the following inequalities,

(μ + k)E∞ ≤ βS∞I∞,

(μ + q + ε1 + γ1)I∞ ≤ kE∞,

(μ + ε2 + γ2)Q∞ ≤ qI∞,

(μ + δ)R∞ ≤ γ1I∞ + γ2Q

∞.

(6)

We claim that I∞ = 0. Otherwise, it follows from the first two inequalities of (6) that

(μ + q + ε1 + γ1)I∞ ≤ kE∞ ≤ kβS∞I∞

μ + k,

Stability and control of HFMD model

which leads to

S∞ ≥ (μ + q + ε1 + γ1)(μ + k)

βk= Λ

μR0,

a contradiction since S∞ ≤ Λ/μ and R0 < 1. This proves the claim. Then it fol-lows from (6) again that E∞ = I∞ = Q∞ = R∞ = 0. Therefore, limt→∞ E(t) =limt→∞ I (t) = limt→∞ Q(t) = limt→∞ R(t) = 0. Moreover, using the fluctuationlemma one more time, we find a sequence {sn} such that sn → ∞, S(sn) → S∞, anddS(sn)

dt→ 0 as n → ∞. Again, by the first equation of (4), we get

dS(sn)

dt= Λ − βS(sn)I (sn) − μS(sn) + δR(sn).

Letting n → ∞ and using the results just obtained, we get S∞ = Λ/μ ≥ S∞. Itfollows that limt→∞ S(t) = Λ/μ. Thus limt→∞(S(t),E(t), I (t),Q(t),R(t)) = P0and the proof is complete. �

Next, we consider the local stability of the endemic equilibrium P ∗. After lin-earizing (4) around P ∗, we can obtain the characteristic equation at P ∗,

λ5 + b1λ4 + b2λ

3 + b3λ2 + b4λ + b5 = 0, (7)

where

b1 = βI ∗ + 5μ + δ + k + q + ε1 + ε2 + γ1 + γ2 > 0,

b2 = μ(μ + δ) + (μ + ε2 + γ2)(2μ + k + q + ε1 + γ1)

+ (βI ∗ + 2μ + δ

)(3μ + k + q + ε1 + ε2 + γ1 + γ2) > 0,

b3 = μ(μ + δ)(3μ + k + q + ε1 + ε2 + γ1 + γ2)

+ (2μ + k + q + ε1 + γ1)(μ + ε2 + γ2) + β2kS∗I ∗ > 0,

b4 = μ(μ + δ)(2μ + k + q + ε1 + γ1)

+ β2kS∗I ∗(2μ + δ + ε2 + γ2) − βδγ1I∗ > 0,

b5 = βI ∗[(μ + k)(μ + δ)(μ + q + ε1 + γ1)(μ + ε2 + γ2)

− δk(γ2q + γ1(μ + ε2 + γ2)

)]> 0.

The result below follows from the Routh-Hurwitz criterion.

Theorem 3 Suppose that R0 > 1. Then the endemic equilibrium P ∗ is locallyasymptotically stable if b1b2b3 > b2

3 + b21b4 and (b1b4 − b5)(b1b2b3 − b2

3 − b21b4) >

b5(b1b2 − b3)2 + b1b

25.

We guess that the endemic equilibrium P ∗ is always locally stable and this is con-firmed by fixing some parameters. In particular, P ∗ is locally asymptotically stable inthe case where δ = 0. However, it is very difficult to prove this analytically. Instead,the following result tells us that the system is persistent.

Theorem 4 Suppose that R0 > 1. Then system (4) is uniformly persistent in Ω .

J.-Y. Yang et al.

Proof We apply the approach in Thieme [21] to finish the proof. For this pur-pose, choose X = Ω , X1 = Ω , and X2 = ∂Ω . It is easy to obtain that Υ2 ={(S,0,0,0,0)|0 < S ≤ Λ/μ}, Ω2 = ⋃

y∈Υ2ω(y) = {P0}, and {P0} is an isolated

compact invariant set in X. Furthermore, M = {P0} is an acyclic isolated coveringof Ω2.

Now we only need to show that P0 is a weak repeller for X1. Suppose that thereexists a positive orbit (S,E, I,Q,R) of (4) such that

limt→∞

(S(t),E(t), I (t),Q(t),R(t)

) =(

Λ

μ,0,0,0,0

).

Since R0 > 1, there exists a small enough ε > 0 such that

β

(Λ

μ− ε

)k > (μ + k)(μ + q + ε1 + γ1).

For this ε, there exists a t0 > 0 such that S(t) ≥ Λ/μ − ε for t ≥ t0. Then, for t ≥ t0,we have from (4) that

dE

dt≥ β

(Λ

μ− ε

)I − (μ + k)E,

dI

dt= kE − (μ + q + ε1 + γ1)I.

Let

Mε =(−(μ + k) β(Λ

μ− ε)

k −(μ + q + ε1 + γ1)

),

which is an M-matrix. It follows that Mε only has positive eigenvalues. Let v =(v1, v2) be an eigenvector of Mε associated with one eigenvalue. Consider

du1

dt= β

(Λ

μ− ε

)u2 − (μ + k)u1,

du2

dt= ku1 − (μ + q + ε1 + γ1)u2.

(8)

Let u(t) = (u1(t), u2(t)) be a solution of (8) through (E0, I0) at t = t0. Since thesemiflow of (8) is monotone and Mεv > 0, it follows that ui(t) is strictly increasingand ui(t) → ∞ as t → ∞, i = 1 and 2. Then both E(t) and I (t) tend to infinite ast → ∞, contradicting with the boundedness of solutions to (4). Therefore, P0 is aweak repeller for X1. �

Since (4) is a high dimensional system, it is difficult to analyze the global stabilityof P ∗. However, we can prove the global stability of P ∗ for the case where δ = 0.

Theorem 5 Suppose R0 > 1 and δ = 0. Then the endemic equilibrium P ∗ is globallyasymptotically stable.

Proof Since P ∗ is locally asymptotically stable, it suffices to show that P ∗ is globallyattractive. When δ = 0, system (4) is decoupled. We first consider the subsystem

Stability and control of HFMD model

dS

dt= Λ − βSI − μS,

dE

dt= βSI − (μ + k)E, (9)

dI

dt= kE − (μ + q + ε1 + γ1)I.

We shall apply the Lyapunov function method to prove that every solution withI (0) > 0 tends to (S∗,E∗, I ∗) as t → ∞. For this purpose, it is helpful to note that

μ = Λ

S∗ − βI ∗,

μ + k = βS∗I ∗

E∗ ,

μ + q + ε1 + γ1 = kE∗

I ∗ .

(10)

By Theorem 4, the following Lyapunov function is well defined,

V (t) = S(t) − S∗ − S∗ lnS(t)

S∗ + E(t) − E∗ − E∗ lnE(t)

E∗

+ μ + k

k

[I (t) − I ∗ − I ∗ ln

I (t)

I ∗

].

For simplicity of notation, we let

x = S

S∗ , y = E

E∗ , z = I

I ∗ .

Differentiating V (t) along solutions of (9) and using (10), we obtain

dV (t)

dt= (x − 1)

[Λ

(1

x− 1

)− βS∗I ∗(z − 1)

]+ βS∗I ∗(y − 1)

(xz

y− 1

)

+ kE∗(z − 1)

(y

z− 1

)

= 2Λ + (μ + k)E∗ − (Λ − βS∗I ∗)x − Λ

x− βS∗I ∗ xz

y− (μ + k)E∗ y

z

= (Λ − βS∗I ∗)

(2 − x − 1

x

)+ βS∗I ∗

(3 − 1

x− y

z− xz

y

)

≤ 0.

It is easy to see that dV (t)dt

= 0 holds only when x = 1 and y = z, or S = S∗ andE/E∗ = I/I ∗. It is easy to see that the maximal invariant set of (9) on {(S,E, I)|S =S∗,E/E∗ = I/I ∗} is {(S∗,E∗, I ∗)}. By LaSalle’s Invariable Principle [10], wehave (S(t),E(t), I (t)) → (S∗,E∗, I ∗) as t → ∞. This combined with (4) gives us(S(t),E(t), I (t),Q(t),R(t)) → P ∗ as t → ∞. �

J.-Y. Yang et al.

Fig. 1 The values of correlationcoefficient for outcome R0

Fig. 2 Effect of β and γ1 on thebasic reproduction number R0

3 Sensitivity analysis

We know that the basic reproduction number R0 is an important quantity in charac-terizing the spread of disease as illustrated in Sect. 2. In this section, we investigatethe effect of parameters on R0. We assume that all parameters are centered at thevalues given in Table 1. Using Latin Hypercube sampling techniques with 1000 sam-ples, we obtain Fig. 1, which shows the tornado chart of Partial Rank CorrelationCoefficient (PRCC) for various parameters relative to R0, the output variable.

Tornado plot of partial rank correlation coefficients indicates the importance ofeach parameter’s uncertainty in contributing to the variability in the time to eradicateinfection. Among the parameters, Λ, μ, k, ε1, ε2, and δ are intrinsic characteristics ofHFMD whereas β , q , γ1, and γ2 can be controlled. Note that R0 is independent of γ2.From Fig. 1, we can easily see that β is highly positively correlated to R0 while q

and γ1 are highly negatively correlated to R0. This implies that decreasing the trans-

Stability and control of HFMD model

mission rate and increasing the recovery rate of the infected and the quarantine ratecan reduce the spread of the epidemics. Figure 2 clearly indicates this observation.Here, we fix all parameters except β and γ1.

4 The optimal control

Control theory is used to identify ways of producing maximum performance at a min-imal cost under various sets of assumptions. The aim of this section is to formulatean optimal control problem by generalizing the autonomous model (4) and then tofind the optimal control strategies.

In the previous section, we have seen that β , q , and γ1 are highly correlated to thebasic reproduction number R0. This leads to the following model with three controls,

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

dS(t)

dt= Λ − β

(1 − u1(t)

)S(t)I (t) − μS(t) + δR(t),

dE(t)

dt= β

(1 − u1(t)

)S(t)I (t) − (μ + k)E(t),

dI (t)

dt= kE(t) − (

μ + q + qT u3(t) + ε1 + γ1 + γ1T u2(t))I (t),

dQ(t)

dt= (

q + qT u3(t))I (t) − (μ + ε2 + γ2)Q(t),

dR(t)

dt= (

γ1 + γ1T u2(t))I (t) + γ2Q(t) − (μ + δ)R(t).

(11)

Naturally, each control incurs some cost: social distancing generates economic lossesand effective treatment requires the existence and support of a costly public healthinfrastructure. Unfortunately, we do not have good data on the costs associated withthese efforts. Hence, we use the “relative” cost for the controls, that is, the quadraticterm [B1u

21(t)+B2u

22(t)+B3u

23(t)]/2, where Bi represents the weight for the control

ui , i = 1, 2, and 3. The goal is to minimize the exposed and infected individuals, thecost of implementing control. Thus we formulate an optimal control problem withthe objective functional given by

J(u(t)

) =∫ tf

0

[A1E(t) + A2I (t) + B1

2u2

1(t) + B2

2u2

2(t) + B3

2u2

3(t)

]dt, (12)

where u(t) = u1(t), u2(t), u3(t), A1 and A2 represent the weight constants of theexposed and infected individuals, respectively.

We assume that the cost is proportional to the square of the corresponding objec-tive functional. Our goal is to find optimal control functions u∗ ∈ U such that

J(u∗(t)

) = min{J(u(t)

)∣∣u ∈ U}

subject to (11), where

U ={

u(t) = (u1(t), u2(t), u3(t)

)∣∣∣∣ui(t) is Lebesgue measurable on [0, tf ],0 ≤ ui(t) ≤ 0.9, i = 1,2,3

}

is the control set. We prove the existence of an optimal control for (11) and thendevelop the optimality system. Note that, for bounded Lebesgue measurable controls

J.-Y. Yang et al.

and nonnegative initial conditions, the solutions are nonnegative and bounded (see,for example, [13]).

Theorem 6 Consider the objective functional J given by (12) with u ∈ U sub-ject to the constraint state system (11). There exists u∗ ∈ U such that J (u∗(t)) =min{J (u(t))|u ∈ U}.

Proof The integrand of the objective functional J given by (12) is convex on theclosed, convex control set U . The conditions for the existence of optimal control aresatisfied as the model is linear in the control variables and is bounded by a linearsystem in the state variables (see Fleming and Rishel [5, Theorem 4.1]). �

Using Pontryagin’s maximum principle [16], we now derive the necessary con-ditions that a pair of optimal controls and corresponding states must satisfy. To thispurpose, we define the Hamiltonian function for the system,

H = L(E, I,u1, u2, u3) + λ1dS

dt+ λ2

dE

dt+ λ3

dI

dt+ λ4

dQ

dt+ λ5

dR

dt, (13)

where λi , i = 1,2,3,4,5, are the adjoint variables, and

L(E, I,u1, u2, u3) = A1E(t) + A2I (t) + B1

2u2

1(t) + B2

2u2

2(t) + B3

2u2

3(t)

is the Lagrangian of the control problem.

Theorem 7 Given an optimal control u∗ = (u∗1, u

∗2, u

∗3), and corresponding state

solution (S, E, I , Q, R) of the corresponding state system (11), there exist adjointvariables λi , i = 1,2,3,4,5, satisfying

λ′1 = (λ1 − λ2)βI + λ1μ,

λ′2 = −A1 + λ2(μ + k) − λ3k,

λ′3 = −A2 + (λ1 − λ2)βS + λ3

(μ + q + qT u3(t) + ε1 + γ1 + γ1T u2(t)

)

− λ4(q + qT u3(t)

) − λ5(γ1 + γ1T u2(t)

),

λ′4 = (μ + ε1 + γ2)λ4 − λ5γ2,

λ′5 = λ5(μ + δ),

(14)

with transversality conditions (or boundary conditions)

λi(tf ) = 0, i = 1,2,3,4,5. (15)

Furthermore, the optimal controls u∗1, u∗

2, and u∗3 are given by

u∗1 = max

{min

{βSI (λ2 − λ1)

B1,0.9

},0

},

u∗2 = max

{min

{γ1T I (λ3 − λ5)

B2,0.9

},0

}, (16)

u∗3 = max

{min

{γ2T I (λ3 − λ4)

B3,0.9

},0

}.

Stability and control of HFMD model

Proof The adjoint system results from Pontryagin’s Principle [16],

λ1 = −∂H

∂S, λ2 = −∂H

∂E, . . . , λ5 = −∂H

∂R

with zero transversality. To get the characterization of the optimal control givenby (16), we solve the equations on the interior of the control set,

∂H

∂ui

= 0, i = 1,2,3.

Using bounds on the controls, we obtain the desired characterization. �

The optimal control and the state are found by solving the optimality system,which consists of the state system (11), the adjoint system (14), initial conditionsat t = 0, boundary conditions (15), and the characterization of the optimal control(16). To solve the optimality system, we use the initial and transversality conditionstogether with the characterization of the optimal control u∗ = (u∗

1, u∗2, u

∗3) given by

(16). In addition, the second derivatives of the Lagrangian with respect to u∗1, u∗

2, andu∗

3 are positive, respectively, which shows that the optimal problem is minimum atcontrols u∗

1, u∗2, and u∗

3. By substituting the values of u∗1, u∗

2, and u∗3 into the control

system (11), we get the following system⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

dS(t)

dt= Λ − μS(t) + δR(t)

− β

(1 − max

{min

{βS(t)I (t)(λ2(t) − λ1(t))

B1,0.9

},0

})S(t)I (t),

dE(t)

dt= β

(1 − max

{min

{βS(t)I (t)(λ2(t) − λ1(t))

B1,0.9

},0

})S(t)I (t)

− (μ + k)I (t),

dI (t)

dt= kE(t) −

(μ + q + qT max

{min

{qT I (λ3(t) − λ4(t))

B3,0.9

},0

}

+ ε1 + γ1 + γ1T max

{min

{γ1T I (λ5(t) − λ3(t))

B2,0.9

},0

})I (t),

dQ(t)

dt=

(q + qT max

{min

{qT I (t)(λ3(t) − λ4(t))

B3,0.9

},0

})I (t)

− (μ + ε2 + γ2)Q(t),

dR(t)

dt=

(γ1 + γ1T max

{min

{γ1T I (t)(λ5(t) − λ3(t))

B2,0.9

},0

})I (t)

+ γ2Q(t) − (μ + δ)R(t),

(17)

with the Hamiltonian H at (S, E, I , Q, R) being

J.-Y. Yang et al.

H (t, S, E, I , Q, R, λ1, λ2, λ3, λ4, λ5)

= A1E(t) + A2I (t) + B1(max

{min

{βSI (λ2(t)−λ1(t))B1

,0.9},0

})2

2

+ B2(max

{min

{ γ1T I (λ5(t)−λ3(t))B2

,0.9},0

})2

2

+ B3(max

{min

{ γ2T R(λ5(t)−λ4(t))B3

,0.9},0

})2

2

+ λ1dS

dt+ λ2

dE

dt+ λ3

dI

dt+ λ4

dQ

dt+ λ5

dR

dt. (18)

The optimal control and state variables can be found numerically by solving (17)and (18). Take Λ = 2, μ = 0.001731, δ = 0.95, γ1 = 3, γ2 = 0.8235, γ1T = 30, qT =30, β = 0.4, ε1 = 0.001731, ε2 = 0.001731, k = 7/4, q = 5, A1 = 1, A2 = 1, B1 =200, B2 = 100, B3 = 100 and the initial conditions (S(0),E(0), I (0),Q(0),R(0)) =(10,8,2,1,1). The numerical results are depicted in Fig. 3 and Fig. 4. It follows thatthe optimal control time is at the first five days.

5 An application to Mainland China

In this section, we apply the obtained results to the situation in China. From the web-site of China Ministry of Health [2], we have obtained the monthly numbers of newlyreported HFMD cases (infected) from March 2009 to February 2012 in China whichare summarized in Table 2 and depicted in Fig. 5, respectively. From Table 2, it is easyto see that the infected cases have obvious seasonal fluctuation and seasonal peaksoccur in early summers. This may be due to the fact of more skin exposedness, kidscrowded to play outside, and very activeness of EV71 and COX A16 viruses. Now,because of the improved medical environment and high quality medicine for HFMD,most HFMD cases in China are mainly treated in hospitals. There are few cases arequarantined or isolated. As a result, to study the spread of HFMD in China, we takeq = γ2 = ε2 = 0 in model 4. The statistics show that the total number of the wholeChinese population in 2008 is 1328020000 and the birth rate is 12.13 per thousand[19, 20]. Hence Λ = 1349199.6. From Ref. [26], the infected case in February 2009is about 16000. We assume that I (0) = 16000. The exposed period is 4–7 days whichis reported by CDC government and thus 1

μ+k= 1

6 per month. Then k = 5.9998923.

The initial exposed population is E(0) = I (0)k

= 2667. For the initial recovered pop-ulation, we just give a priori estimate R(0) = 32432. The ratio of the whole youngpopulation aged from 1 to 14 is 18.5 % and thus S(0) = 251608901. Also we canobtain the death rate ε1 = 0.000289. From Table 1, we see that we only need to findthe two important parameters β and γ1, which form the components of the controlvariable x = (β, γ1)

T ∈ R2. We apply the classical least-squares approach to estimate

Stability and control of HFMD model

Fig. 3 Dynamics of HFMD with and without optimal control

the parameters. For a given set of parameters x, we compute the discrete response I

of the model and estimate the error by

f (x) = 1

2

Nmax∑

i=0

δi(Ii − Ii )2,

where

δi ={

1 if there is a data available at time i

0 otherwise

J.-Y. Yang et al.

Fig. 4 The optimal controlvariables u∗

1, u∗2, and u∗

3

Table 2 The numbers ofmonthly reported HFMDcases (4)

Month Year

2009 2010 2011 2012

January – 37567 32179 50758

February – 23862 10609 40505

March 54713 77756 34709 –

April 212435 248609 99819 –

May 169073 354347 230460 –

Jun 178680 343100 303594 –

July 162060 261263 253442 –

August 99897 119096 132154 –

September 85504 101654 120802 –

October 76948 87612 122491 –

November 61918 79591 171805 –

December 47817 60879 126679 –

is a sampling function and Nmax is the number of integration steps. Hence, the param-eter identification problem is then casted into the unconstrained optimization problem

find x∗ = arg minx

f (x),

i.e., minimizing the misfit between model and data is sought.In order to estimate the parameters in our model, we assume the transmission

rate is a periodic function since the HFMD incidence strongly depends on the cli-mate [6, 14, 22, 24]. Using Matlab we find the parameters as shown in Table 3.With the values in Table 1 and Table 2, the reported cases and the simulation re-sults are summarized in Fig. 6. There is almost no discrepancy between the datareported and data simulated. By the next generation operator theory, we can obtainR0 ≈ βk

(μ+k)(μ+q+ε1+γ1)(1 − (μ+k)(μ+ε1+γ1+q)

π2/36+(2μ+k+q+ε1+γ1)

12 ) (see, for example, Liu [15]).

Stability and control of HFMD model

Fig. 5 The reported HFMDcases from March 2009 toFebruary 2012 in China

Fig. 6 The reported data vs thenumerical solution of HFMDmodel in China

Table 3 The optimalparameters Parameter

β γ1

Value 4.3066(1 + 0.2431 sin(π/6 + 1.0364))/N 3.8639

Moreover, the basic reproduction number is R0 ≈ 1.392, which indicates that HFMDbecomes an endemic disease in China.

6 Conclusions

In this paper, a system of ordinary differential equations for HFMD is proposed andanalyzed. We obtain the basic reproduction number R0, which determines whetherthe disease can be eradicated or there is an outbreak. If the basic reproduction numberis less than one, then the disease free equilibrium is globally stable; otherwise, the

J.-Y. Yang et al.

system is persistent. Using the SASAT software for sensitive analysis, we find thecontrol parameters. Then we formulate an optimal control problem by extending theautonomous system through increasing the social distancing measures, decreasingquarantine rate and the treatment efforts. Finally, we apply the obtained results toestimate the basic reproduction number for the case in China. It turns out that thebasic reproduction number is R0 = 1.392, which indicates that HFMD outbreaks inChina. As a result, to control the spread of HFMD in China, strategies mentioned inSect. 3 should be applied, that is, increase personal protection against exposure toinfected persons so that β will be decreased, take quarantine mechanism to increaseq , and improve the medical environment like developing high efficient medicinesagainst HFMD so that γ1 will be increased.

Acknowledgements The authors would like to thank the anonymous referees for their constructive com-ments, which greatly improve the presentation of the paper. This work was done when Yang was a post-doctoral fellow at the Department of Mathematics, Wilfrid Laurier University. He would like to thank theDepartment for the hospitality. Research is supported partially by the NSF of China (11071283), the Sci-ences Foundation of Shanxi (2009011005-3), the Young Sciences Foundation of Shanxi (2011021001-1),the Foundation of University (YQ-2011045, JY-2011036), the Natural Science and Engineering ResearchCouncil of Canada (NSERC), the Early Researcher Award Program of Ontario, the One Hundred TalentsProject of Shanxi Province, and the Program of Key Disciplines in Shanxi Province (20111028).

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