Consider the following paper titled: Stochastic regime switching SIR model driven by Lévy noise, authored by Yingjia Guo.
Link: https://www.sciencedirect.com/science/article/pii/S0378437117302145
The system reads:
\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.
