# Dynamic Behavior of a Stochastic Tungiasis Model for Public Health Education

Authors: Lili Kong; Luping Li; Shugui Kang; Youjun Liu; Wenying Feng
Journal: Discrete Dynamics in Nature and Society
 (2022)

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## Abstract

In this paper, we study the dynamic behavior of a stochastic tungiasis model for public health education. First, the existence and uniqueness of global positive solution of stochastic models are proved. Secondly, by constructing Lyapunov function and using Ito^ formula, sufficient conditions for disease extinction and persistence in the stochastic model are proved. Thirdly, under the condition of disease persistence, the existence and uniqueness of an ergodic stationary distribution of the model is obtained. Finally, the importance of public health education in preventing the spread of tungiasis is illustrated through the combination of theoretical results and numerical simulation.


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## Body

## 1. Introduction

Tungiasis is a parasitic skin disease caused by female sand fleas invading the human epidermis. It is an endemic zoonosis in the tropics. The disease is mainly manifested in the feet. People walking barefoot are infected with fleas on their skin, and the body of the fleas can penetrate the skin and become parasitic [1–3]. A large number of reported infections indicate that the disease spreads in many countries in economically disadvantaged areas, such as Latin America, the Caribbean, and sub-Saharan Africa [4]. In these areas, many patients cannot walk properly due to the effects of the disease, leading to further poverty. However, it remains a neglected health problem in poor communities in countries affected by the disease [5]. This neglect has been verified due to the lack of epidemiological data in Kenya [6]. About 2.6 million 6.5% Kenyans are known to have been affected by the disease, and 10 million are at risk of tungiasis [7]. Tungiasis is a common disease among people living in the mountains of Kenya, western Kenya, the Great Rift Valley, and coastal areas [2]. In 2010, the prevalence of Tungiasis among children aged 5 to 12 in the endemic area of Southern Muranga reached 57% [8]. Dirty living environment is the main factor causing tungiasis [9]. Therefore, good personal hygiene and wearing shoes are the keys to control the spread of the disease. Thus, the introduction of public health education is essential in the preventive treatment of tungiasis. The public health department can attract people’s attention by reporting the transmission route, mode of transmission, and the change of the number of infected people to achieve the purpose of public health education. At the same time, public health programs such as free insecticide delivery, free shoe distribution, and immunization may affect the spread of the disease [10].Mathematical models have proved to be a powerful tool for studying the spread of infectious diseases and are widely used in biology, medicine, and disease control. To control the spread of infectious diseases, scholars build mathematical models to study the dynamic behavior of infectious diseases. For example, Kahuru et al. [11] conducted stability analysis on the transmission dynamics of tungiasis in endemic areas. In [12], Lv et al. studied the dynamics and optimal control of the transmission of tungiasis. In particular, Nyang’ inja et al. [13] discussed a mathematical model of the impact of public health education on tungiasis(1)dSt=Λ−βStIt+γIt−ε+μStdt,dEt=εSt−αβEtIt−μEtdt,dIt=βStIt−αβEtIt−γ+σ+μItdt,where St, Et, and It, respectively, represent the number of susceptible people, public health educators, and infected people at time t. All parameters here are positive. Λ is the constant input rate of the susceptible population. β is the effective contact rate for disease transmission. γ is the recovery rate of the infected person. ε is the rate at which public health education is disseminated to susceptible populations in accordance with public health intervention strategies. μ is the natural mortality rate of various populations. Since the impact of public health education is not permanent and the control measures are taken gradually disappear, the educated population will be infected at a lower rate αβEI, where 0<α<1. α represents the effect of public health education on reducing infection. σ is the fatality rate of the disease. The asymptotic behavior of the equilibrium point is discussed in [13].However, the spread process of infectious diseases is bound to be affected by random disturbances, which is an essential factor to be considered in the establishment of mathematical models. Compared with deterministic models, the discussion of stochastic models is more consistent with the reality of life [14–16]. So far, the stochastic systems play an important role and have been widely used in many fields such as physics, chemistry, mechanics, economics and finance systems, aerospace engineering systems, and control theory [17–21]. For example, Wang et al. [17] developed a stochastic integral sliding mode control strategy for singularly perturbed Markov jump descriptor systems subject to nonlinear perturbation. The authors provided a solution to the stabilization problem for SPMJDSs by developing a novel stochastic integral SMC strategy. Cheng et al. [18] focused on static output feedback control for fuzzy Markovian switching singularly perturbed systems (FMSSPSs) with deception attacks and asynchronous quantized measurement output. The authors developed a more general scheme, where the partial information of FMSSPS is available in both quantizer and controller. Cheng et al. [19] analyzed the problem of hidden Markov model based control for periodic systems subject to singular perturbations and Lur’e cone-bounded nonlinearity. The highlight of their research lied that the hidden Markov model detector is forwarded to observe the fading channel mode, whose detection probabilities are generalized to be partially recognized. Inspired by [22–25], we assume that the random perturbation under the influence of white noise is proportional to the variables S,E, and I in model (1) to obtain a stochastic tungiasis model:(2)dSt=Λ−βStIt+γIt−ε+μStdt+σ1StdB1t,dEt=εSt−αβEtIt−μEtdt+σ2EtdB2t,dIt=βStIt+αβEtIt−γ+σ+μItdt+σ3ItdB3t,where σi2i=1,2,3 are the intensity of the white noise, Biti=1,2,3 is independent standard Brownian motion defined on a complete probability space Ω,ℱ,ℱtt≥0,P with the filtration ℱtt≥0 satisfying the usual conditions. When the noise intensity σi=0i=1,2,3, model (2) becomes the deterministic model (1). Moreover, we assign R+3=x1,x2,x3∈R3|xi>0,i=1,2,3.This article is organized as follows. In Section2, the existence and uniqueness of the global positive solution for the model (2) are proved. In Section 3, the sufficient conditions for disease extinction and strong persistence of the model (2) are obtained. In Section 4, the existence and uniqueness of an ergodic stationary distribution of the model (2) are obtained under the condition of disease persistence. In Section 5, the influence of random noise on disease transmission is analyzed by numerical simulation. Finally, some conclusions are given in Section 6.


## 2. Existence and Uniqueness of the Positive Solution

In this section, we will prove that there is a unique global positive solution to model (2).Theorem 1.
For any initial value(S0,E0,I0)∈R+3, there exists a unique positive solution (St,Et,It) to model (2) for t≥0, and the solution will remain in R+3 with probability one.Proof.
Because the coefficients of model (2) satisfy the local Lipschitz conditions, there is a unique local positive solution on t∈[0,τe) for any initial value (S0,E0,I0)∈R+3, where τe is the explosion time [26]. And then, we show that τe=∞ a.s.. Let k0>0 be sufficiently large so that (S0,E0,I0) lies within the interval 1/k0,k0×1/k0,k0×1/k0,k0. For ∀k>k0, we define the stopping time as(3)τk=inft∈[0,τe):minSt,Et,It≤1kormaxSt,Et,It≥k.
Then, we setinf∅=+∞. Clearly, τk is increasing when k⟶∞. Let τ∞=limk⟶∞τk, then we have τ∞≤τe a.s.. If τ∞=∞ a.s. is true, then τe=∞ a.s. Next, we declare that τ∞=∞ a.s. If this statement is false, for ∀k>k0, ∃T>0,δ∈0,1 such that Pτ∞≤T>δ. Thus, there is a positive integer k1≥k0 such that Pτk≤T≥δ for ∀k≥k1.
Define aC2-function V: R+3⟶R+ by(4)VS,E,I=S+a−alnSa+E+b−blnEb+I+1−lnI,where positive constants, a and b, are the undetermined coefficients. The nonnegativity of the function V can be derived from u+1−lnu≥0 for u>0. By using It o^ formula, we have(5)dVS,E,I=1−aSdS+1−bEdE+1−1IdI+12aS2dS2+12bE2dE2+121I2dI2=LVdt+σ1S−adB1t+σ2E−bdB2t+σ3I−1dB3t,where(6)LV=1−aSΛ−βSI+γI−ε+μS+1−bEεS−αβEI−μE+1−1IβSI+αβEI−γ+σ+μI+12aσ12+12bσ22+12σ32≤Λ−σ−aβI−μ−bαβI+a+b+1μ+γ+σ+aε+12aσ12+12bσ22+12σ32.
Takinga=σ/β,b=μ/αβ, then(7)LV≤Λ+a+b+1μ+γ+σ+aε+12aσ12+12bσ22+12σ32≔K.
So,(8)dVS,E,I≤Kdt+σ1S−adB1t+σ2E−bdB2t+σ3I−1dB3t.
Integrating the above equation, with both sides from 0 toτk∧T and taking expectation, which can be obtained from Theorem 5.8(ii) in [27],(9)EVSτk∧T,Eτk∧T,Iτk∧T≤VS0,E0,I0+KT.
DenoteΩk=τk≤T, then PΩk≥δ. For ∀ω∈Ωk, one or more of Sτk,ω,Eτk,ω,Iτk,ω is equal to k or 1/k; thus,(10)VSτk,ω,Eτk,ω,Iτk,ω≥mink+1−lnk,1k+1+lnk,k+a−alnka,1k+a+alnakk+b−blnkb,1k+b+blnbk≔Gk.
Consequently,(11)V(S0,E0,I0)+KT≥E(1ΩkωV(S(τk,ω),E(τk,ω),I(τk,ω))≥P(Ωkω)Gk≥δGk,where 1Ωk is the indicator function of Ωk. Letting k⟶∞, then(12)∞>VS0,E0,I0+KT=∞,which contradicts the hypothesis, so we get that τ∞=∞. Therefore, the solution of the model (2) will not explode in a finite time with probability one.


## 3. Extinction and Persistence of the Disease

In this section, we discuss the problem of disease survival. First, we give the following lemma.Lemma 1.
Let(St,Et,It) be the solution of the model (2), then limsupt⟶+∞St+Et+It<+∞, limt⟶+∞St/t=0,limt⟶+∞Et/t=0,limt⟶+∞It/t=0,limt⟶+∞lnSt/t=0,limt⟶+∞lnEt/t=0,limt⟶+∞lnIt/t=0,a.s.. And limt⟶+∞∫0tSrdB1r/t=0,limt⟶+∞∫0tErdB2r/t=0,andlimt⟶+∞∫0tIr·dB3r/t=0 a.s..Proof.
The proof is similar to Lemma 2.2 in [28], so it is omitted here.
Then, we analyze the persistence and extinction of disease in model (2).Theorem 2.
IfR0=βΛ/μγ+σ+μ+0.5σ32<1, then the disease of model (2) will be extinct at an exponential rate with probability one.Proof.
From the stochastic model (2),(13)dS+E+I=Λ−μS+E+I−σIdt+σ1SdB1t+σ2EdB2t+σ3IdB3t.
Calculating the integral from 0 tot on both sides of (13) and dividing by t,(14)1t∫0tdSr+Er+Ir≤Λ−μt∫0t(Sr+Er+Irdr+σ1t∫0tSrdB1r+σ2t∫0tErdB2r+σ3t∫0tIrdB3r.
So,(15)St+Et+Itt≤Λ+S0+E0+I0t−μt∫0tSr+Er+Irdr+σ1t∫0tSrdB1r+σ2t∫0tErdB2r+σ3t∫0tIrdB3r.
From Lemma1,(16)limsupt⟶+∞1t∫0t(Sr+Er+Ir)dr≤Λμ.
From the model (2), by using It o^ formula, we have(17)dlnI=βS+αβE−γ+σ+μ+12σ32dt+σ3dB3t.
Integrating (17) with both sides from 0 to t, dividing by t, we can obtain(18)lnItt≤lnI0t+β1t∫0t(Sr+Er)dr−γ+σ+μ+12σ32+σ3t∫0tdB3t.
Combining with (16) and Lemma 1, we have(19)limsupt⟶+∞lnItt≤βΛμ−γ+σ+μ+12σ32≤γ+σ+μ+12σ32R0−1<0,which implies that(20)limt⟶+∞It=0,a.s.
Thus, the disease will die out exponentially with probability one.Theorem 3.
IfR1=Λεαβ/μ+0.5σ22ε+μ+0.5σ12γ+σ+μ+0.5σ32>1, then the disease It will be strong persistent in the mean, and(21)liminft⟶+∞1t∫0tIrdr≥μ+0.5σ22R1−1βa1+α>0,where a1 is a positive constant, and it is defined in the proof.Proof.
Assign(22)V=−a1lnS−lnE−a2lnI,where a1,a2>0 are the undetermined coefficients. We use It o^ formula(23)dV=−a1SdS−1EdE−a2IdI+12a1S2dS2+121E2dE2+12a2I2dI2=LVdt−a1σ1dB1t−σ2dB2t−a2σ3dB3t,where(24)LV=−a1SΛ−βSI+γI−ε+μS−1EεS−αβEI−μE−a2IβSI+αβEI−γ+σ+μI+12a1σ12+12σ22+12a2σ32=−a1ΛS+a1βI−a1γIS+a1ε+μ−εSE+αβI+μ−a2βS−a2αβE+a2γ+σ+μ+12a1σ12+12σ22+12a2σ32≤−a1ΛS+εSE+a2αβE+a1ε+μ+12σ12+a2γ+σ+μ+12σ32+μ+12σ22+βa1+αI.
Using inequalitya+b+c≥3abc3, and taking a1=Λεαβ/ε+μ+1/2σ122γ+σ+μ+1/2σ32,a2=Λεαβ/ε+μ+1/2σ12γ+σ+μ+1/2σ322, we have(25)LV≤−3a1a23Λεαβ+a1ε+μ+12σ12+a2γ+σ+μ+12σ32+μ+12σ22+βa1+αI=−Λεαβε+μ+1/2σ12γ+σ+μ+1/2σ32+μ+12σ22+βa1+αI=−μ+12σ22R1−1+βa1+αI.
Thus,(26)dV≤−μ+12σ22R1−1+βa1+αIdt−a1σ1dB1t−σ2dB2t−a2σ3dB3t.
Integrating (26), with both sides over the interval [0, t] and dividing by t lead to(27)Vtt≤V0t−μ+12σ22R1−1+βa1+αt∫0tIrdr−a1σ1t∫0tdB1r−σ2t∫0tdB2r−a2σ3t∫0tdB3r.
By taking the limit inferior of both sides in (27) and combining with Lemma 1, one has(28)liminft⟶+∞1t∫0tIrdr≥μ+1/2σ22R1−1βa1+α>0,which shows that the disease is strong persistent.


## 4. Existence of Ergodic Stationary Distribution

In this section, we investigate the conditions for the existence of a unique ergodic stationary distribution.Lemma 2 (see [16]).
Assume that there exists a bounded open domainU⊂Rd with regular boundary Γ, and(1)
There is a positive numberM satisfied that Σi,j=1daijxξiξj≥Mξ2,x∈U¯,ξ∈Rd(2)
There exists a nonnegativeC2 -function V such that LV is negative for any Rd\U
Then, the Markov processXt has a ergodic stationary distribution μ·, and it is unique.Theorem 4.
IfR1>1, for any initial value (S0,E0,I0)∈R+3, the solution of model (2) admits a unique stationary distribution μ·.Proof.
First, the diffusion matrix of the model (2) is as follows(29)A=σ12S2000σ22E2000σ32I2.
LetU be any bounded open domain in R+3, ∃M>0M=minσ12S2,σ22E2,σ32I2,S,E,I∈U¯ that satisfies(30)∑i,j=13aijS,E,Iξiξj=σ12S2ξ12+σ22E2ξ22+σ32I2ξ32≥Mξ2,for all S,E,I∈U¯,ξ=ξ1,ξ2,ξ3∈R+3. So, condition (1) in Lemma 2 is satisfied.
And then, we are going to verify condition (2). Consider a nonnegative C2-function VS,E,I:R+3⟶R+ with(31)V=HS,E,I−HS0,E0,I0,where S,E,I∈U=p,1/p×p,1/p×p,1/p and p>0 is a sufficiently small constant and HS0,E0,I0 is a unique minimum value point of the C2-function HS,E,I.
Denote(32)HS,E,I=NV1+V2+V3+V4,where V1=−lnS−b1lnE−b2lnI, V2=−S+E+I,V3=1/l+1S+E+Il+1,andV4=−lnS−lnE. b1andb2 are positive numbers, and we will show that later 0<l<1 is a sufficiently small constant satisfying(33)ζ≔μ−ld2>0,where d=maxσ12,σ22,σ32. N>0 is a sufficient large constant such that(34)f1u+f2u−hN≤−2,(35)f1u+f3u−ΛN≤−2.
h=ε+μ+0.5σ12R1−1>0, f1u=supS∈0,+∞f1S,f2u=supE∈0,+∞f2E,f3u=supI∈0,+∞f3I, f1S,f2E,andf3I will be given as follows.
Using Ito^ formula, one has(36)LV1=−1SΛ−βSI+γI−ε+μS−b1EεS−αβEI−μE−b2I[βSI+αβEI−γ+σ+μI]+12σ12+12b1σ22+12b2σ32≤−ΛS+b1εSE+b2αβE+b1μ+12σ22+b2γ+σ+μ+12σ32+ε+μ+12σ12+β1+b1αI.≤−3b1b23Λεαβ+b1μ+12σ22+b2γ+σ+μ+12σ32+ε+μ+12σ12+β(1+b1α)I=−Λεαβμ+1/2σ22γ+σ+μ+1/2σ32+ε+μ+12σ12+β1+b1α,=−ε+μ+12σ12R1−1+β1+b1αI=−h+β1+b1αI,where b1=Λεαβ/μ+0.5σ222γ+σ+μ+0.5σ32andb2=Λεαβ/μ+0.5σ22γ+σ+μ+0.5σ322. Similarly, we can get(37)LV2=−Λ+μS+μE+σ+μI.
Therefore,(38)LV3=S+E+IlΛ−μS+E+I−σI+l2S+E+Il−1σ12S2+σ22E2+σ32I2≤ΛS+E+Il−μ−ld2S+E+Il+1≤Q−12ζS+E+Il+1,where Q=supS,E,I∈R+3ΛS+E+Il−1/2ζS+E+Il+1. And,(39)LV4=−1SΛ−βSI+γI−ε+μS−1EεS−αβEI−μE+12σ12+12σ22≤−ΛS+βI+αβI+ε+2μ+12σ12+12σ22.
From equations (36)–(39), we have(40)LV≤N−h+β1+b1αI+N−Λ+μS+μE+σ+μI+Q−12ζS+E+Il+1−ΛS+βI+αβI+ε+2μ+12σ12+12σ22≤−hN+Nβ1+b1αI−ΛN+NμS+NμE+Nσ+μI+Q−12ζSl+1−12ζEl+1−12ζIl+1−ΛS+β1+αI+ε+2μ+12σ12+12σ22=f1S+f2E+f3I,where(41)f3I=−hN+βN1+b1α+β1+α+Nσ+μI−12ζIl+1,f2E=−12ζEl+1+N−Λ+μE,f1S=Q+ε+2μ+12σ12+12σ22+NμS−12ζSl+1−ΛS.
Since(42)f1S+f2E+f3I≤f1S+f2u+f3u⟶−∞,asS⟶+∞orS⟶0+,f1u+f2E+f3u⟶−∞,asE⟶+∞,f1u+f2E+f3u⟶f1u+f3u−ΛN≤−2,asE⟶0+,f1u+f2u+f3I⟶−∞,asI⟶+∞,f1u+f2u+f3I⟶f1u+f2u−hN≤−2,asI⟶0+.
From the above analysis, we can conclude thatLV<0, S,E,I∈R+3/U. So, this tells us that condition (2) in Lemma 2 also satisfies. Hence, model (2) has a unique stationary distribution μ·.


## 5. Numerical Simulations

The numerical analysis supporting the theoretical results in model (2) can be obtained by Milstein method [29]. The discretized equations are listed as follows:(43)Si+1=Si+Λ−βSiIi+γIi−ε+μSiΔt+Siσ1ξiΔt+σ122ξi2−1Δt,Ei+1=Ei+εSi−αβEiIi−μEiΔt+Eiσ2ηiΔt+σ222ηi2−1Δt,Ii+1=Ii+βSiIi+αβEiIi−θ+μ+γIiΔt+Iiσ3ζiΔt+σ322ζi2−1Δt,where the time increment Δt>0 and ξi,ηi,ζi are independent N0,1− distributed Gaussian random variables.Next, some numerical simulations are given to verify our theorem.Example 1.
Consider model (2) with parameter values (some of the parameters are from [30, 31]): Λ=0.44,β=0.01498,γ=0.427,ε=0.1431,α=0.5,μ=0.016,σ=0.05, σ1=0.2,σ2=0.6,andσ3=0.3 and the initial point is (S0,E0,I0)=12,9,5. And, we can compute that R0≈0.7662<1, the conditions of Theorem 2 are satisfied. Theorem 2 tells us that diseases become extinct, see Figure 1. From Figure 1(c), we can see that compared with the deterministic model (1), random white noise will accelerate the rate of disease extinction.Figure 1
S1t,E1t,I1t is the solution to the deterministic model (1), and S1t,E1t,I1t is the solution to the stochastic model (2).
(a)(b)(c)Example 2.
Consider model (2) with parameter values (some of the parameters are from [30, 31].): Λ=0.44,β=0.062,γ=0.427,ε=0.1431,α=0.9,μ=0.016,σ=0.05, σ1=0.1,σ2=0.1,σ3=0.05 and the initial point is (S0,E0,I0)=12,9,5. Simple calculation leads to R1≈2.0628>1. Therefore, the conditions of Theorem 3 hold. According to Theorem 3, the disease in model (2) has a strong persistence in the mean, see Figure 2. Comparing Figure 2(c) with Figure 2(d), we can see that the disease of model (2) goes to strong persistence in the mean. Meanwhile, under this condition, model (2) has a unique ergodic stationary distribution, see Figure 3.Figure 2
S1t,E1t,I1t is the solution to the deterministic model (1), and S2t,E2t,I2t is the solution to the stochastic model (2). Here, c, and d is the case of It in t∈0,150 and t∈0,1000, respectively.
(a)(b)(c)(d)Figure 3
a is the solution of the stochastic model (2). b,c,d, respectively, represent the density function of S,E,I in model (2).
(a)(b)(c)(d)


## 6. Conclusion

In this paper, we investigate the dynamics of a stochastic tungiasis model in public health education. Firstly, the existence and uniqueness of the global positive solution of model (2) are proved by constructing appropriate Lyapunov functions. Then, the sufficient conditions for disease extinction and persistence are analyzed. At the same time, the existence and uniqueness of the ergodic stationary distribution of model (2) are obtained under the condition of disease persistence. Through the combination of theoretical research and numerical simulation, we found that the smaller white noise can make the model (2) has a unique ergodic stationary distribution when R1>1. In comparison, the larger white noise can cause the extinction of disease in the model (2) when R0<1. Compared with the deterministic model, in the epidemic model, the combination of random white noise interference and public health education has a greater impact on the persistence and extinction of the disease, enriching the dynamic behavior of the epidemic model.



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