We discuss the effect of introducing telegraph noise, which is an example of an environmental noise, into the susceptible-infectious-recovered-susceptible (SIRS) model by examining the model using a finite-state Markov Chain (MC). First we start with a two-state MC and show that there exists a unique nonnegative solution and establish the conditions for extinction and persistence. We then explain how the results can be generalised to a finite-state MC. The results for the SIR (Susceptible-Infectious-Removed) model with Markovian Switching (MS) are a special case. Numerical simulations are produced to confirm our theoretical results
This article develops a highly efficient batch‐centric model for the detailed scheduling of straight multiple‐source pipelines. The continuous‐time model permits multiple batches to be injected/delivered over a slot, thus enabling finding better schedules for a given number of event points in the grid. It can consider either global or line batch numbering. In the former, one predefines empty batches to allow injection of new products at intermediate sources. Empty batches are avoided with line numbering at the expense of specifying a number of batches for every line in the system, instead of using a global value. Three benchmark problems from the literature are solved to evaluate the performance of the proposed formulation. Compared to our recent work, new best solutions are reported in two cases, while for the third, the computational time has been reduced by two orders of magnitude. The results also show that global batch numbering is better.
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