Simulation Analysis of the Optimal Vaccination Strategy in theStochastic Infectious Model
Masaaki Ishikawa1
11

Graduate School of Science and Engineering, Yamaguchi University, Japan

Introduction

A prevention of prevalence of infectious diseases is oneof important problems in epidemiology. In the past, thepublic health system has prepared for some strategies suchas antibiotics and vaccines to control the infectious diseasedevelopment. In order to build up more eﬀective strategies, we need a precise theoretical analysis of infectious
diseases. From a theoretical viewpoint, the mathematicalmodel which describes the spread of the infectious disease
has a very important role. In this paper, we study thestochastic modeling of the infectious disease in populations
consisting of four populations: susceptible, infected, recovered and vaccinated ones. First, we propose the stochastic
infectious model with vaccination in Section 2. In Section 3, we construct the optimal vaccination strategy forthe stochastic infectious model using the stochastic maximum principle and the 4-step scheme. Finally, we show
the eﬃciency of the optimal vaccination strategy by thenumerical simulations in Section 4.
2

Stochastic Infectious Model with Vaccination

Denoting the ratio of the population size of susceptible(individuals susceptible to the disease), infected, recovered and vaccinated to the total population at time t beS(t), I(t), R(t) and V (t), consider the interaction between
each population as shown in Fig. 1.
νRµSµSusceptible

β SI

S(t)
uS

φV(Vaccination) Vaccinated
V(t)

µI
γI
InfectedI(t)µR
Recovered

Using the relation between the Gaussian white noise η(t)and the Wiener process w(t) such that dw(t) = η(t)dt, wehave the stochastic infectious model with vaccination:dS(t) = {µ − (µ + u(t))S(t) − βS(t)I(t) + ϕV (t)+νR(t)}dt + εR(t)dw(t),dI(t) = {βS(t)I(t) + σβV (t)I(t) − (µ + γ)I(t)}dt,

(2)
(3)

dR(t) = {γI(t) − (µ + ν)R(t)}dt − εR(t)dw(t),(4)

dV (t) = {u(t)S(t) − (µ + ϕ)V (t) − σβV (t)I(t)}dt,(5)

with the initial conditions
S(0) = S0 , I(0) = I0 , R(0) = R0 , V (0) = V0 .

(6)

Since the death occurs with the same rate µ as the birthrate, we haveS(t) + I(t) + R(t) + V (t) = 1.
3(7)Stochastic Optimal Vaccination Strategy
First, we consider the cost function J(u) such that{∫ T
J(u) = E}ℓ(x(t), u(t))dt,(8)0where ℓ : R3 ×R1 → R1 and is assumed to be continuously
diﬀerentiable.
Then, we study the optimal control problem of ﬁnding
the optimal vaccination rate u∗ ∈ U such thatJ(u∗ ) ≤ J(u),∀ u ∈ U ,(9)

R(t)

σβVI
µVFig. 1. Interaction between each population [1]In Fig. 1, β presents the transmission rate, µ the deathrate(= the birth rate), γ the recovery rate, ν the rate ofloss of immunity, u the vaccination rate, ϕ the rate ofvaccination waning. A vaccination eﬃcacy is denoted by
1 − σ with σ ∈ [0, 1].
Noting that the real infectious disease contains somekinds of random ﬂuctuations caused by changes in the environment, we consider the rate of loss of immunity witha random ﬂuctuation. We replace the rate of loss of immunity ν by
ν −→ ν + εη(t),(1)where the parameter ε is a constant and η(t) denotes theGaussian white noise with zero mean and unit covariance.

where U is an admissible control set deﬁned by
U ≡ {u|0 ≤ u(t) ≤ Cp ,∀ t ∈ Θ ≡ (0, T )},(10)

and where Cp ∈ (0 1] is a constant.
For simplicity of descriptions, we deﬁne the vector:
x(t) = [x1 (t) x2 (t) x3 (t)]′ ≡ [S(t) I(t) R(t)]′ .

(11)

We introduce the Hamiltonian H in such a way that
H(x, u, p, q) = ⟨f (x, u), p⟩ − ℓ(x, u) + ⟨g(x), q⟩,(12)where ⟨·, ·⟩ denotes a Euclidean inner product, p and q are
adjoint vectors, f and g are vectors with components:
f1 = µ − µx1 − βx1 x2 − ux1 + ϕv + νx3 ,(13)

f2 = βx1 x2 +σβvx2 −(µ+γ)x2 , f3 = γx2 −(µ+ν)x3 , (14)g1 = εx3 , g2 = 0, g3 = −εx3 .
and where v = (1 −∑3

j=1xj ).

(15)

㻜㻚㻤Then, the stochastic maximum principle [2] yields that㻜㻚㻣∗∗

dp = −H(x , u , p, q)x dt + q(t)dw(t), p(T ) = 0.
∗∗∗

H(x , u , p, q) = max H(x , u, p, q),㻼㼛㼜㼡㼘㼍㼠㼕㼛㼚㻌㻰㼑㼚㼟㼕㼠㼕㼑㼟dx∗ = H(x∗ , u∗ , p, q)p dt + g(x∗ )dw(t), x∗ (0) = x0 , (16)
(17)
(18)

u∈U

where x∗ (t) is an optimal trajectory of x(t).
We solve the forward-backward stochastic diﬀerential
equations (16) and (17) using the four-step scheme [3].
Assume that p(t) and x(t) are related byp(t) = θ(t, x∗ (t)),

(19)

⟩

1
tr[(θxi )′x gg ′ ] + H(x, θ, q)xi = 0.
2

⟨ i⟩qi (t) = θx , g ,㻝㻜㻝㻡㻞㻜㻞㻡㻰㼍㼥㼟㻌㻟㻜㻖㻻㼜㼠㼕㼙㼍㼘㻌㼂㼍㼏㼏㼕㼚㼍㼠㼕㼛㼚㻌㻾㼍㼠㼑㻌㼡㻌㻔㼠㻕㻜㻚㻡㻜㻚㻠㻜㻚㻟㻜㻚㻞㻜㻚㻝㻜㻜㻡㻝㻜㻝㻡㻞㻜

㻞㻡

㻟㻜

㻰㼍㼥㼟㻌

Fig. 4. Time Evolution of Optimal Vaccination Rate(22)Figures 2 and 3 show the trajectory of each population density under the optimal vaccination strategy andno vaccination. Figure 4 is the trajectory of the optimal
vaccination rate u∗ (t). From Figs. 2 and 3, by the optimal
vaccination strategy, we can see that the infected and thesusceptible are decreasing and the recovered is increasing.
From Fig. 4, we can know at what rate we will vaccinate. Comparing Fig. 2 with Fig. 3, trajectories underno vaccination have been more signiﬁcantly impacted by
the noise than ones under vaccination because the noise isproportional to S(t)I(t) as shown in Eqs. (24) and (25).

(23)

Since Eq. (21) is a deterministic partial diﬀerential equation, we can solve Eq. (21) with Eq. (22) under the terminal condition (23).
4㻡(21)

with the terminal condition
θi (T, x) = 0, (i = 1, 2, 3).

㻾㼑㼏㼛㼢㼑㼞㼑㼐㻌㻾㻔㼠㻕㻜㻚㻢

Noting that p(t) = θ(t, x(t)) and u is a function of p, q andx, it follows from Eqs. (17) and (20) that⟨㻜㻚㻞

㻜㻚㻣

⟩
⟨
1
θx , f + tr[(θxi )′x gg ′ ]}dt + θxi , g dw. (20)2θti + θxi , f +㻵㼚㼒㼑㼏㼠㼕㼢㼑㻌㻵㻔㼠㻕㻜㻚㻟

Fig. 3. Trajectory of Each Population Density under No
Vaccination⟩⟨ i㻿㼡㼟㼏㼑㼜㼠㼕㼎㼘㼑㻌㻿㻔㼠㻕㻜㻚㻠㻜㻜㼂㼍㼏㼏㼕㼚㼍㼠㼕㼛㼚㻌㻾㼍㼠㼑㻌dθ= {θti +㻜㻚㻡㻜㻚㻝

where θ is some vector-valued function with components
θi (t, x∗ (t)) (i = 1, 2, 3) to be determined.
In the sequel, the asterisks of x(t) and u(t) are omitted
for simplicity of descriptions.
Using the Itô’s lemma to θi (t, x(t)), we have for i =1, 2, 3,i㻜㻚㻢Simulations

For simplicity of numerical calculations, we assume thatthe vaccination has a complete eﬃciency and recovered
does not become reinfected and the vaccinated is included
in recovered, i.e., ν = σ = ϕ = 0. We consider the random
ﬂuctuation in the infection rate β. Setting as β → β +εη(t), we havedS = {µ − (µ + u)S − βSI}dt − εSIdw(t),(24)dI = {βSI − (µ + γ)I}dt + εSIdw(t),(25)dR = {γI + uS − µR}dt.

(26)

Numerical simulations are performed under the parametervalues: S0 = 0.60, I0 = 0.20, β = 0.0024, µ = 1/73.5, T =30, γ = 1/6570, ε = 0.05, m = 2.0, n = 2.0, r = 0.8, Cp =0.7.
㻜㻚㻤5Conclusions

In this paper, we have considered the stochastic modeling of the infectious disease spreading under vaccinationand have studied the stochastic optimal vaccination problem. In the stochastic optimal vaccination problem, wederived the feasible optimal vaccination strategy using thestochastic maximum principle and the four-step scheme.
In numerical simulations, for the restriction of numericalcalculations, we have considered the optimal control problem of the stochastic SIR model with vaccination. Resultsof numerical simulations have shown that our optimal vaccination strategy is eﬀective. From simulation results obtained in this paper, we can know how much vaccination
coverage is required in order to control the infectious disease with restricted vaccination rate.

㻼㼛㼜㼡㼘㼍㼠㼕㼛㼚㻌㻰㼑㼚㼟㼕㼠㼕㼑㼟㻜㻚㻣㻖㻾㼑㼏㼛㼢㼑㼞㼑㼐㻌㻾㻌㻔㼠㻕㻜㻚㻢

References㻜㻚㻡㻜㻚㻠㻖㻵㼚㼒㼑㼏㼠㼕㼢㼑㻌㻵㻌㻔㼠㻕㻜㻚㻟㻜㻚㻞㻖㻿㼡㼟㼏㼑㼜㼠㼕㼎㼘㼑㻌㻿㻌㻔㼠㻕㻜㻚㻝㻜㻜㻡㻝㻜㻝㻡㻞㻜㻞㻡㻟㻜㻰㼍㼥㼟㻌

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