# Asymptotic behaviors of equilibrium points of a switching SDE with Levy jumps?

Consider the following paper titled: Stochastic regime switching SIR model driven by Lévy noise, authored by Yingjia Guo.

\begin{align} \mathbb{d}S &=[\Lambda(k)-\beta(k)S(t)I(t)-\mu(k) S(t)]\mathbb{d}t+\sigma_1(k) S(t)\mathbb{d}B_1(t)+\int_MD_1(k,y)S(t)\tilde{N}(dt,dy)\\[2ex] \mathbb{d}I &=[\beta(k)S(t)I(t)-(\mu(k)+\epsilon(k)+\gamma(k) )I(t)]\mathbb{d}t+\sigma_2(k) I(t)\mathbb{d}B_2(t)\\ &+\int_M D_2(k,y)I(t)\tilde{N}(dt,dy)\\[1ex] \mathbb{d}R &=[\gamma(k)I(t)-\mu(k)R(t)]\mathbb{d}t+\sigma_3(k) R(t)\mathbb{d}B_3(t)+\int_M D_3(k,y)R(t)\tilde{N}(dt,dy)\\[1ex] \end{align}

The reproduction number for the switching model being $$\mathcal{R}_{0_k}=\frac{\Lambda(k) \beta(k)}{\mu(k)(\mu(k)+\epsilon(k)+\gamma(k))}$$

The reproduction number for the deterministic model being $$\mathcal{R}_0=\frac{\Lambda \beta}{\mu(\mu+\epsilon+\gamma)}$$

Theorem 3.1 states the following:

If $$\mathcal{R}_{0_k}<1$$, $$\sigma_1^2(k)<\mu(k)-\frac{3}{2}\int_M D_1^2(k,y)\nu(dy)$$ and $$\sigma_2^2(k)<2(\mu(k)+\epsilon(k)+\gamma(k))-3\int_M D_2^2(k,y)\nu(dy)$$ then for any given initial value $$(S(0),I(0),R(0))\in\mathbb{R}_+^3$$ there is the property

$$\lim_{t\rightarrow \infty} \sup E \int_0^t \left[\left(S(s)-\frac{\Lambda(k)}{\mu(k)}\right)^2 +I^2(s)+R(s)\right] \leq \frac{\Lambda^2(k)}{\tilde{K}\mu^2(k)}\left[2\sigma_1^2(k)+3\int_M D_1^2(k,y)\nu(dy) \right]$$

which is satisfied for the solution for the stochastic system above, where

$$\tilde{K}=min \lbrace{2(\mu(k)-\sigma_1^2(k))-3\int_M D_1^2(k,y)\nu(dy), \frac{2 \Lambda(k)(1-\mathcal{R}_0)(2\mu(k)+\epsilon(k)+\gamma(k))}{\gamma(k)\mathcal{R}_0}, 2(\mu(k)+\epsilon(k)+\gamma(k))-\sigma_2^2(k)-3\int_M D_2^2(k,y)\nu(dy)\rbrace}$$

The equilibrium point associated with Theorem 3.1 is the following

$$E_{0_k}=\left(\frac{\Lambda(k)}{\mu(k)},0,0 \right)$$

How did the author know these were the conditions needed and how did he get the property for Theorem 3.1? Or better, as Nawaf suggests, why are these conditions sufficient? A similar question for Theorem 4.1.

EDIT:

I have figured why and how the conditions for Theorem 3.1 and Theorem 4.1 must be in place.

EDIT 2:

Regarding the proof for Theorem 4.1, why and/or how was $$p_2(k)$$ chosen in this particular way? I know it says $$\mu +\epsilon +\gamma - \cdots >0$$ but where did this condition arise from?

EDIT 2 has also been understood.

• Is the question: what meaning (if any) the sufficient conditions have? It would be helpful to underscore which sufficient conditions are being referred to. Oct 13, 2022 at 12:31
• @NawafBou-Rabee: If you can access the paper, in Theorem 3.1, how did the author know the conditions( $\sigma_1^2(k)<\mu(k)-\frac{3}{2}...$) and how did he know the property $\lim_{t\rightarrow \infty} \sup E \int_0^t [|\cdots]<\cdots$?
– Math
Oct 13, 2022 at 12:36
• (i) I think this more precise version of the question in the comment should probably be incorporated in the question. (ii) The question "how did the author know ... ?", is a question that probably should be directed to the author themselves. A better MO question is "why are these conditions sufficient?" or something like that. Oct 13, 2022 at 12:43
• @NawafBou-Rabee I will add it in the question. That is what i meant to ask, sorry.
– Math
Oct 13, 2022 at 12:45