Dynamic analysis of a stochastic vector-borne model with direct transmission and media coverage

: This paper presents a stochastic vector-borne epidemic model with direct transmission and media coverage. It proves the existence and uniqueness of positive solutions by constructing a suitable Lyapunov function. Immediately after that, we study the transmission mechanism of vector-borne diseases and give threshold conditions for disease extinction and persistence, in addition to which we show that the model has a stationary distribution determined by the synergistic action of two parameters, i


Introduction
Vector-borne disease seriously threatens world health, usually caused by vector-borne parasites, viruses, and bacteria transmitting pathogens between humans or from animals to humans.According to the World Health Organization, the disease accounts for 17 % of all infectious and has caused 700,000 deaths annually [1].Despite scientific and technological advances and growing affluence in all regions, vector-borne diseases remain one of the leading causes of global disease.Mathematical modeling has become an essential method for studying epidemic.Since the first modeling of malaria transmission by Ross [2] and subsequent modifications by Mac-Donald [3], a series of vector-borne disease models have been proposed [4][5][6][7].Various diease models based on influencing factors (e.g., time delay, vaccination, age structure, etc.) have been extensively studied [8][9][10][11].
It's commonly known that direct and indirect transmissions are two significant ways of the spread of various diseases.Although indirect transmission is not negligible, vector-borne diseases are often transmitted directly through blood transfusions, organ transplantation, laboratory exposure, or mother-to-baby during pregnancy, childbirth, and breastfeeding.It is worth noting that zika can be transmitted through sexual contact [12].Thus, direct transmission plays a vital role in the dynamics of vector-borne diseases and has attracted widespread attention [13][14][15][16].In the deterministic model proposed by Wei et al. [16], the host population is assumed to be divided into three subpopulations, i.e., susceptible, vector-borne infected, and recovered individuals.The infected individuals will not relapse once recovered, i.e., the recovered individuals will not become susceptible or infected.Let S (t), I(t), and R(t) be the numbers of susceptible, infected, and recovered individuals at time t.The vector population is divided into two parts, i.e., susceptible and infected vectors, denoted by M(t) and V(t) as the corresponding numbers at time t.Once infected without recovering, the vectors will carry the virus for life.The newly recruited vectors are susceptible when vertical transmission is ignored.On the other hand, media coverage is a crucial factor in controlling the spread of epidemics [17].Through the media, it is helpful to understand the progress of the epidemic and provide beneficial guidance [18].Many scholars have studied the impact of media coverage on disease transmission through mathematical modeling [19,20].Based on the above discussion, we introduce media coverage into the epidemic model and investigate the dynamic of Vector-borne diseases with direct transmission.Let β 1 be the transmission rate without media intervention, and β 2 I/(m + I) be the effect of media coverage on transmission, where β 1 > β 2 , and m measures how quickly people react to media reports [21].During the spread of the vector-borne epidemic, the susceptible can be infected through two transmission rates: the rate denoted by β 3 from an infected vector to a susceptible person, and the one denoted by β 4 from an infected person to a susceptible vector.Then, we propose a vector-borne model with direct transmission and media coverage as follows where Λ i , d i (i = 1, 2), and µ are the recruitment, natural, and disease-related death rates of people and vector population, α i (i = 1, 2, 3) are the saturated constants during different transmission processes, and γ is the recovery rate of infected people.Here, β 1 S I, β 3 S V, and β 4 MI measure the contagiousness of the vector-borne disease, and 1/(1 1) has two equilibria: a disease-free equilibrium E 0 and an endemic equilibrium E * = (S * , I * , R * , M * , V * ).This means that some individuals of both populations have been infected.
In the real world, random fluctuations are essential to ecosystems [22][23][24].Random factors, such as temperature and humidity, inevitably affect the epidemic's spread.Many stochastic models have been studied in recent years [25][26][27].Considering the complex environmental changes, Liu and Jiang claimed that the random perturbation may depend on the square of the state variables S and I in the system [28,29].Recently, nonlinear perturbations have received much attention [30][31][32].In addition to this, Sometimes ecosystems are also affected by violent random perturbations such as typhoons and tsunamis.To reflect reality better, Levy jumps were introduced in the model [33,34].However, this noise differs in detail and often leads to different results.It is worth noting that in the model of vector-borne diseases, Jovanović and Krstić [35] proposed that the random perturbation is proportional to the distance.Ran et al. [36] studied the dynamics of a stochastic vector-borne model with age structure and saturation incidence, considering the environmental noise on mosquito bite rate and transmission rate between vector and host.Son et al. [37] provided another stochastic vector-borne model with direct transmission, in which environmental noise affects the mortality of hosts and vectors.We don't want to add complex perturbations to make the model unmanageable; simple perturbations are more likely to reveal the inherent nature of the model.In our work, suppose that the environmental white noise is proportional to the number of subpopulations [38,39].Next, we extend the deterministic model (1.1) to a stochastic model.The recovered class is decoupled from the others in the model and then neglected.Then, we propose the following stochastic model where B i (t) (i = 1, 2, 3, 4) are independent standard Brownian motions, σ i (i = 1, 2, 3, 4) represent the white noise intensity, and the remaining parameters are the same as model (1.1).
The rest of this paper is organized as follows.Section 2 reviews some basic concepts and valuable lemmas used later.The uniqueness and positivity of the solution are proved in Section 3. Section 4 provides sufficient conditions for determining whether a disease is extinct.In Section 5, we explore the persistence in mean.In Section 6, we prove the existence of a unique ergodic stationary distribution under certain conditions.In Section 7, we validate the analysis results through numerical simulations.A brief conclusion is given in the last section.

Preliminaries
Let (Ω, F , P) be a complete probability space with a filtration {F t } t≥0 satisfying the usual conditions (i.e., it is increasing and right continuous while F 0 contains all P-null sets).Denote R n + = {y ∈ R n : y i > 0, 1 ≤ i ≤ n}.Consider an n-dimensional stochastic differential equation of the following form [40] with initial value y(0) = y 0 ∈ R n , where B(t) denotes an n-dimensional standard Brownian motion defined on the complete probability space (Ω, F , {F t } t≥0 , P). Define a differential operator L of Eq. (2.1) as follows g T (y, t)g(y, t) i j ∂ 2 ∂y i ∂y j .
If L acts on a nonnegative function function where where the diffusion matrix Ā(Y) = (b i j (y)) and b i j (y) = k r=1 σ i r (y)σ j r (y).Lemma 2. ( [42]) The Markov process Y(t) has a unique stationary distribution π(•) if there is a bounded domain D ∈ R n with a regular boundary such that its closure D ∈ R n has the following properties (i) In the open domain D and some of its neighbors, the smallest eigenvalue of the diffusion matrix A(t) is set far from zero.
(ii) If y ∈ R n \ D, the mean time τ at which a path issuing from y reaches the set D is finite, and sup y∈D Eτ y < ∞ for every compact subset.Moreover, if f (•) is a function integrable concerning the measure π, then
Proof.For a given initial value (S (0), I(0), M(0), V(0)) ∈ R 4  + , the coefficient in the model (1.2) satisfies the local Lipschitz continuity condition.Hence, there is a unique local solution when t ∈ [0, τ e ), where τ e is the explosion time [43,44].To obtain the global property of the solution, we need to prove τ e = ∞ a.s.Suppose that k 0 ≥ 1 is enough large such that S (0), I(0), M(0) and V(0) all lie within the interval [1/k 0 , k 0 ].For each integer k ≥ k 0 , define a stopping time where ∅ is an empty set and inf ∅ = ∞.It can be seen that τ k increases as k → ∞, and τ ∞ = lim k→∞ τ k with 0 ≤ τ ∞ ≤ τ e a.s.In other words, if τ e = ∞ a.s.does not hold, there must have constants T, k 1 > 0 and ϵ ∈ (0, 1) Obviously, W is a non-negative function.Applying Itô's formula to (3.2) yields where L W : R 4 + → R + can be written in the following form Hence, we have Integrate both sides of Eq. ( 3.3) from 0 to τ k ∧ T .It is easy to get that Setting Ω = {τ k ≤ T } for k ≥ k 1 and by Eq. (3.1), we get P(Ω k ) ≥ ϵ.Further, every ω from Ω has at least one of S (τ k , ω), Combining Eq. (3.4) and Eq.(3.5), we have where 1 Ω(ω) denotes an indicator function of set Ω. Letting k → ∞ leads to the contradiction It implies that τ e = ∞ a.s.The proof is complete.□ It is clear that model (1.1) has a disease-free equilibrium E 0 = (Λ 1 /d 1 0, Λ 2 /d 2 , 0), in which the disease tends to become extinct in the time limit.However, there is no disease-free equilibrium in the stochastic version of the model, which requires other ways to consider its extinction.Define a threshold value and . Let (S (t), I(t), M(t), V(t)) be the solution of system (1.2) with any initial value (S (0), Proof.According to Ref. [45], we have We integrate both sides of the proposed model (1.2) and obtain It is obvious that From (4.1) and (4.2), the limit of Eq. (4.3) is Similarly, we integrate on both sides of the last two equation of the model (1.2).Hence, Combined (4.1) and (4.2), we can get the following equation On the other hand, through the Itô's formula, it follows that The last term here uses the inequality 2IV ≤ (I + V) 2 .Integrate on both sides of the equation and divide it by t.Thus, where }, and β = max{β 1 , β 3 }.From (4.4) and (4.5), we can get According to Lemma 1, it is obtained that The proof is complete.□

Persistence in the mean of the disease
The most interesting aspect in the study of epidemic modeling is the extinction and persistence of epidemic; in the previous section we studied disease extinction and in this section we will show that diseases are persistent in the mean. and then for any given initial value (S (0), I(0), M(0), V(0)) ∈ R 4 + , the solution of system (1.2) has the following properties Proof.(i) From the first equation of system (1.2) integrating the above inequality and dividing both sides by t, we get In view of Theorem 1, for any initial value (S (0), + .Thus, Through the strong law of large numbers for local martingales, we have which together with (5.1) yields . This is the required assertion (i).
(ii) From the third equation of system (1.2) integrating the above inequality and dividing both sides by t, we get Then According to the strong law of large numbers of local martingales, we have which together with (5.2) yields

This is the required assertion (ii).
(iii) First, define a function W 2 (S , I, V) = − ln S − ln I − ln M − ln V.According to the Itô's formula: . We integrate the above inequality in the interval (0, t), divide it by t, and take the limit to t.Thus, According to the powerful number law of a martingale, It follows that (5.3) becomes The proof is complete.□

Stationary distribution
The ergodic property for an epidemic model means that the stochastic model has a unique stationary distribution that forecasts the permanence of the epidemic in the future.That means the disease persists for all time regardless of the initial condition.
In this section, we provide a sufficient condition for the existence of a stationary distribution in the model (1.2).Denote , where 2) has a unique stationary distribution π(•) with ergodicity.Proof.The diffusion matrix for model (1.2) is , k and k is a sufficiently large integer.Therefore, the condition (i) in Lemma 2 is satisfied.Next, we prove the condition (ii) in Lemma 2. Let where Θ i (i = 1, 2) are sufficiently large positive constants, satisfying−Θ and where Thus, it follows that A closed subset is defined as where ϵ > 0 is sufficiently small constants satisfying the following conditions where H is a constant and is determined later.Denote Y = (S , I, M, V).We divide R 4 + \ D into the following eight cases Now, we will prove that L V(S , I, M, V) < −1 on R 4 + \ D; this is equivalent to proving that it is valid on the above eight subsets.
In Cases 1-8, we choose enough small value ϵ such that L V (S , I, M, V) < −1 for any (S , I, M, V) ∈ D i (i = 1, 2, . . ., 8).Thus, L V (S , I, M, V) < −1 for all (S , I, M, V) ∈ R 4 + \D.On the other hand, Assume that (S (0), I(0), M(0), V(0)) = (y 1 , y 2 , y 3 , y 4 ) = y ∈ R 4 + \D, and τ y is that time at which a path starting from y reach to the set D, Integrating the above equation from 0 to τ (n) (t) and solving the expectation with the help of Dykins' formula yields Since V (y) is non-negative, it is obvious that Eτ (n) (t) ≤ V (y).We can get P {τ e = ∞} = 1 from the proof of Theorem 1.Thus, the system is regular; that is, for t → ∞, n → ∞, we have τ n (t) → τ y almost surely.According to Fatou Lemma, we have Eτ y ≤ V (y) < ∞.It is easy to find that for every compact set R 4 + \D of R 4 + , sup y∈R 4 + \D Eτ y < ∞.Then, the condition (ii) in Lemma 2 is satisfied.□

Numerical simulations
Numerical simulations are presented to support our theoretical findings of the model (1.2) and reveal the impact of media coverage on the spread of disease.Using the Milstein method mentioned in Higham [46], we consider the discretized equations as follows: where the time increment ∆t > 0, ζ 1i , ζ 2i , ζ 3i , ζ 4i , are mutually independent Gaussian random variables which follow the distribution N(0, 1) for i = 0, 1, 2, ..., n.
Vector-borne diseases with two transmission routes may be more likely to become endemic than diseases with one transmission route.Therefore, we tend to choose lower transmission rates and recruitment when numerically modeling disease extinction.According to Theorem 2, the solution of the stochastic model (1.2) will eventually approach zero; this means the disease will die out almost surely.And from Fig. 1, it is observed that the number of infected individuals tends to zero.Theorem 3 implies that the disease is persistent in the mean.Interestingly, In Fig. 2, it is clear that the number of infected individuals is higher than that of susceptible individuals.we can obtain that R S 2 = {R S 3 , R S 4 } ≈ 49.720 > 1 and the conditions of Theorem 4 is satisfied.The Fig. 3 show that the histograms of solutions of model (1.2) with white noise.Theoretical conclusions and numerical simulations indicate that the disease will eventually prevail and persist for a long time.

FrequencyFigure 3 .
Figure 3. Histogram with 100 bins generated from 50,000 simulations of the model (1.2),where the red curves are the probability density functions.