Stochastic dynamics in a delayed epidemic system with Markovian switching and media coverage

A stochastic SIR system with Lévy jumps and distributed delay is developed and employed to study the combined effects of Markovian switching and media coverage on stochastic epidemiological dynamics and outcomes. Stochastic Lyapunov functions are used to prove the existence of a stationary distribution to the positive solution. Sufficient conditions for persistence in mean and the extinction of an infectious disease are also shown.


Introduction
The dynamic effects of time delays and stochastic noise on disease outcomes in populations are important research themes in mathematical epidemiology (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] and the references therein). Models incorporating systems of delay differential equations have been shown to exhibit more complex dynamics and capture more of the observed biology underlying disease transmission and persistence (see [11,[15][16][17] and the references therein). Studies of environmental noise in models have also been shown to capture a broader range of disease outcomes, i.e. large fluctuations in environmental noise have been shown to render a disease extinct in a model that otherwise would have shown disease progression to a unique endemic equilibrium [5,6,17].
Recently, a stochastic SIR epidemic system with distributed delay has been proposed [17]. Specifically, the model was used to study the effects of white noise (given by B(t), representing standard Brownian motion) and a distributed delay in the infection term (incorporated using kernel H : [0, ∞) → [0, ∞), representing L 1 -weak generic kernel function H(t) = ρe -ρt with ρ > 0 such that ∞ 0 H(τ ) dτ = 1) on the extinction and persistence of a disease, given the following model structure: where S(t), I(t) and R(t) represent the proportions of susceptible, infective and recovered individuals in a population, Λ and δ denote birth and recovery rates, d i (i = 1, 2, 3) and c denote the natural and disease induced death rates, m measures the average contact rate per day, and ω 2 > 0 represents the intensity of white noise. Here, we propose an extension of this model to include Markov switching and telephone noise, Lévy jumps and media impact.
Telephone noise [18][19][20][21][22] (also known as telegraph noise or burst noise) can be regarded as a switching state (that is memoryless, with exponentially distributed waiting times [21,22]) that allows for instantaneous transitions between two or more environmental regimes. By analysing observation data from the real world and performing mathematical modelling analysis, it should be noted that the birth rate of a susceptible individual is usually subject to various noises [18][19][20], i.e. telegraph noise. Hence, the telegraph noise is only included in the birth rate in this paper. Here, we propose some hypothesis: (H1) An irreducible and continuous Markov chain {β(t), t ≥ 0} with finite state space N = {1, 2, . . . , k} (k ∈ Z + ) is utilised to depict telephone noise. β(t) is assumed to be generated by a transition rate matrix (μ ij ) k×k , which is where the transition rate from state i to state j is denoted by μ ij ≥ 0, and μ ii = -k i =j,i=1 μ ij holds for i = j. It follows from the irreducibility property of β(t) that there exists a unique stationary probability distribution ξ = (ξ 1 , ξ 2 , . . . , ξ k ) ∈ R 1×k subject to k j=1 ξ j = 1, ξ j > 0 hold for any j ∈ N. Recent studies have shown that Lévy jumps can effectively portray an unexpected outbreak of infectious disease and other suddenly severe perturbations arising in the real world [23][24][25][26][27][28] that cannot be accurately depicted by Brownian motion. Consequently, we consider Lévy jumps using the following hypothesis: (H2) S(t-) denotes the left limit of S(t). M is a measurable subset of R + , Y denotes an independent Poisson counting measure with a Lévy measure ψ on M with ψ(M) < +∞ satisfying U(dt, du) = U(dt, du)ψ(du) dt, by assuming that λ(u) > -1 and υ > 0 satisfying Finally, it is well known that there is a profound relationship between public health issues and mass media coverage. Media reports can affect individual behaviour during an infectious disease outbreak, thus affecting the transmission of the infectious disease [29][30][31][32][33] and the effects of intervention strategies that are also affected by individual behaviour [34][35][36]. Therefore, it is necessary to consider crucial effects of media coverage on epidemiology dynamics. Based on the above analysis, some hypothesis is as follows: (H3) A nonlinear function m 1 -m 2 I(t) q+I(t) is introduced to depict effective contact rate between susceptible and infective individual [33], m 1 represents maximal average contact rate and m 2 I(t) q+I(t) denotes maximal reduced average contact rate due to public health risk warning disseminated by mass media, where m 1 > m 2 > 0 and q > 0.
Based on hypotheses (H1)-(H3), a stochastic delayed SIR system with telephone noise and media coverage is constructed as follows: Recently, some delayed stochastic SIR systems have been used to investigate the combined dynamic effects of stochastic fluctuation and time delay on epidemiological dynamics [37][38][39][40][41][42]. Additionally, complex dynamical behaviours caused by media coverage have been investigated in stochastic epidemic systems in [29][30][31][32][33][34][35]43]. To the authors' best knowledge, combined dynamics of Markovian switching and media coverage on stochastic SIR epidemic system have not been investigated before. In the second section, stochastically ultimate boundedness of the solution is studied. Existence and uniqueness of globally positive solution to the proposed system are investigated. Existence of a stationary distribution to the positive solution is discussed. In the third section, sufficient conditions for persistence in mean of each individual and extinction of infectious disease are discussed. Numerical simulations are supported to illustrate the main theoretical results. Finally, this paper ends with a conclusion.

Qualitative analysis of stationary distribution
Setting W (t) = t -∞ ρe -ρ(t-τ ) I(τ ) dτ , it follows from the linear chain technique [44] that system (3) can be written as For every finite state space k ∈ N defined in (H1), it follows from the Markov chain law that system (4) can be investigated as a hybrid system switching among the following subsystems: First, we discuss stochastically ultimate boundedness of the solution. Existence and uniqueness of globally positive solutions to the proposed system are also studied.
Proof The proof of Lemma 2.1 can be found in Appendix A. In the following part, we will consider the existence of a stationary distribution to the positive solution (which is a stationary Markov process) by constructing appropriately Lyapunov functions. where Proof If m 1 qm 2 Q(ε) > 0, then we define According to the biological interpretations of the second equation and fourth equation in system (5), it can be obtained that Using Itô's formula [45] on Z 1 (S, I, R, W ), it follows from (8) and Lemma 2.1 that where η is sufficiently small and chosen randomly from η ∈ (0, By using Itô's formula [45] on Z 2 (S, I, R, W ), it follows from simple computations and Lemma 2.1 that where Following the above analysis, we define functions f j (S, I, R, W ) (j = 1, 2, 3) and Z 31 (S, I, R, W ) as follows: where the constant ϕ > 0 satisfies -ϕA 2 + 3 j=1 sup t≥0 f j (S, I, R, W ) ≤ -2 and A 2 has been defined in (9).
Note that Z 31 (S, I, R, W ) is a continuous function and tends to the boundary of R 4 + infinity when (S, I, R, W ) → ∞. Consequently, it is easy to show that there exists an extreme point (S,Ĩ,R,W ) for Z 31 in the interior of R 4 + . By defining a nonnegative function Based on (9) and (10), it can be obtained that Additionally, it can be shown that if (m 1 , m 2 ) ∈ D 1 ∩ D 2 , then holds for either S → 0 + or I → 0 + or R → 0 + . Furthermore, It follows from (11), (12) and simple computations that there exists a sufficiently small positive constant > 0 such that LZ 3 (S, I, R, W ) ≤ -1 holds for any (S, Based on Lemma 2.1 [10], it is straightforward to show that there exists a solution of system (5), which is a stationary Markov process.

Permanence in mean and extinction of disease
For the deterministic version of system (5), i.e., system (5) without Brownian motion and Lévy jumps, the endemic equilibrium (S * , I * , R * , W * ) is as follows: where I * satisfies Based on the formulation of endemic equilibrium, it follows from the Vieta theorem that there exists a unique endemic equilibrium provided (m 1 , m 2 ) ∈ D 3 , and D 3 is defined as follows: In the following, we discuss permanence in mean of each individual and disease extinction in system (5). Some corresponding practical interpretations can be found in [17] and the references therein.  (15), and C 5 and C 6 are defined in (20) and (21).
Construct the function U 2 (t) = I(t) -I * -I * ln I(t) I * . By using Itô's formula to system (5), it follows from Lemma 2.1 that where Q(ε) and P(ε) have been defined in Lemma 2.1.

Theorem 3.2 For any initial value
where C 11 is defined in (34). If C 11 < 0, then lim t→+∞ I(t) = 0 almost surely. Furthermore, the distribution of S(t) weakly converges to the measure with the density σ (t), which is defined in (26).
Proof Firstly, by applying Itô's formula into the first equation of system (5), we obtain that By integrating both sides of (36) from 0 to t, it follows from Lemma 2.1 that Let F(t) = t 0 M ln(1 + λ(u)) U(dτ , du) dτ . It can be shown that based on the exponential martingales inequality.
According to the Borel-Cantelli lemma [45], it can be concluded that a random integer T k0 = T k0 (ω) exists for almost all ω ∈ Ω, yielding that holds for T k ≥ T k0 almost surely. It follows from (38) that holds for all 0 ≤ t ≤ T k almost surely. Substituting (39) into (37), it can be obtained that holds for all 0 ≤ t ≤ T k almost surely. Furthermore, it can be shown that holds for 0 ≤ T k -1 ≤ t ≤ T k almost surely. It is easy to show that lim t→∞ B(t) t = 0 almost surely. If (m 1 , m 2 ) ∈ D 1 ∩D 6 , then, following (40), Secondly, using similar proofs to Theorem 3.2 of this paper, if (m 1 , m 2 ) ∈ D 1 ∩ D 6 , then it can be shown that lim t→∞ I(t) = 0 a.s.
Hence, it completes the proof.
Remark 3.4 Following similar arguments given in [17,40], we can show that the basic reproduction numbers for the deterministic and stochastic versions of system (5) are obtained as follows: and respectively. Note that R s 0 < R d 0 and that R s 0 decreases when the intensity of the Lévy jump increases.
Remark 3.5 Based on the mathematical formulation of system (5), it can be concluded that the state variable R(t) does not impose dynamic effects on infectious disease transmission. Hence, we have discussed some sufficient conditions for disease extinction omitting R(t) in Theorems 3.2 and 3.3 of this paper.

Numerical simulation
Simulation studies are used to explore the combined dynamic effects of Markovian switching and media coverage on the stochastic epidemiological dynamics of system (5). Calculations are based on Milstein's higher order method [46]. Suppose state space N = {1, 2}. Using the Markovian chain law, system (5) can be investigated as a hybrid system switching between subsystems and

Conclusion
It is well known that the studies of stochastic perturbations and media coverage are two important and well-established disciplines in mathematical epidemiology [1,3,47,48]. Here, we have extended the model in [17] to include Markovian switching, telephone noise, Lévy jumps and media impact. These extensions have been motivated by the following facts: (I) Lévy jumps have been shown to effectively portray an unexpected outbreak of infectious disease and other sudden severe perturbations arising in the real world [23][24][25][26][27][28], which cannot be accurately depicted by Brownian motion; (II) Evidences from real-world observations point out that the birth rate of susceptible individuals is subject to both white noise and telephone noise [18][19][20][21][22] (which is generally memoryless and can be regarded as a switching state among some considerable environmental regimes [18][19][20]); (III) It is well known that there is a profound relationship between public health issues and mass media coverage, and that media reports can elicit changes in individual behaviour during an infectious disease outbreak, affecting the implementation of public health measures to mitigate infection [29,43]. and distributed delay are investigated in this paper, which has not been studied before.
Our analytical findings thus provide enhanced knowledge in the field of mathematical epidemiology.