Pamela M. Burrage

Mitchell T. Griggs, Kevin Burrage, Pamela M. Burrage

This paper develops methods for numerically solving stochastic delay-differential equations (SDDEs) with multiple fixed delays that do not align with a uniform time mesh. We focus on numerical schemes of strong convergence orders $1/2$ and $1$, such as the Euler--Maruyama and Milstein schemes, respectively. Although numerical schemes for SDDEs with delays $τ_1,\ldots,τ_K$ are theoretically established, their implementations require evaluations at both present times such as $t_n$, and also at delayed times such as $t_n-τ_k$ and $t_n-τ_l-τ_k$. As a result, previous simulations of these schemes have been largely restricted to the case of divisible delays. We develop simulation techniques for the general case of indivisible delays where delayed times such as $t_n-τ_k$ are not restricted to a uniform time mesh. To achieve order of convergence (OoC) $1/2$, we implement the schemes with a fixed step size while using linear interpolation to approximate delayed scheme values. To achieve OoC $1$, we construct an augmented time mesh that includes all time points required to evaluate the schemes, which necessitates using a varying step size. We also introduce a technique to simulate delayed iterated stochastic integrals on the augmented time mesh, by extending an established method from the divisible-delays setting. We then confirm that the numerical schemes achieve their theoretical convergence orders with computational examples.

Kevin Burrage, Pamela M. Burrage, Justin N. Kreikemeyer et al.

In this paper, we adapt a two-species agent-based cancer model that describes the interaction between cancer cells and healthy cells on a uniform grid to include the interaction with a third species -- namely immune cells. We run six different scenarios to explore the competition between cancer and immune cells and the initial concentration of the immune cells on cancer dynamics. We then use coupled equation learning to construct a population-based reaction model for each scenario. We show how they can be unified into a single surrogate population-based reaction model, whose underlying three coupled ordinary differential equations are much easier to analyse than the original agent-based model. As an example, by finding the single steady state of the cancer concentration, we are able to find a linear relationship between this concentration and the initial concentration of the immune cells. This then enables us to estimate suitable values for the competition and initial concentration to reduce the cancer substantially without performing additional complex and expensive simulations from an agent-based stochastic model.

Kevin Burrage, Pamela M. Burrage, Ian W. Turner et al.

In this paper we study the class of mixed-index time fractional differential equations in which different components of the problem have different time fractional derivatives on the left hand side. We prove a theorem on the solution of the linear system of equations, which collapses to the well-known Mittag-Leffler solution in the case the indices are the same, and also generalises the solution of the so-called linear sequential class of time fractional problems. We also investigate the asymptotic stability properties of this class of problems using Laplace transforms and show how Laplace transforms can be used to write solutions as linear combinations of generalised Mittag-Leffler functions in some cases. Finally we illustrate our results with some numerical simulations.