Yusuke Asai and Eva Herrmann
Goethe-University Frankfurt
Objectives: Random ordinary differential equations (RODEs) are introduced to build mathematical models with noise processes. Numerical schemes for RODEs are developed via RODE-SODE transformation and they are applied to biological models. The accuracy of the schemes as well as their computational costs are compared.
Methods: Lotka-Volterra model and virus kinetic model are randomized by adding two different types of noise processes.
In the first case, a Wiener process is adopted to make the predatory rate fluctuated and noisy switching scenario is described in simple prey-predator system. The explicit numerical schemes, namely RODE-Taylor schemes, stochastic linear multi-step methods (SLMMs) and the averaged schemes, are applied to the model.
The second example is a three compartment virus kinetics model with spatial dependence. An Ornstein-Uhlenbeck process is taken as a stochastic process and the loss rate of virus is randomized. The random partial differential equation (RPDE) is discretized with respect to space and implicit schemes are tested to the corresponding system of RODEs.
Results: Two steady states, which have been reported in multiple preys models, could be observed in the simulations. The averaged schemes show the same convergence rate with 1-step and multi-step 1-order schemes. Their computational costs are quite small while the accuracy is relatively low comparing to higher order schemes. The dimension of first example is low and no big difference in computational cost is observed between Ito-Taylor schemes and SLMMs. On the other hand, the second example is of high dimension due to the spatial discretization and computational costs, especially between 1.5-order Ito-Taylor scheme and SLMM, is quite apparent.
Conclusions: The SLMMs have big advantage from the point of computational costs especially when they are applied to large systems. The implicit schemes are stable and can be applied to stiff systems or spatially discretized RODEs. Choosing appropriate type of noise is still an open question and further investigation will be necessary.
References: [1] Y. Asai, E. Herrmann, P.E. Kloeden, Stable integration of stiff random ordinary differential equations, Stoch. Anal. Applns, 31 (2013) 293-313.
[2] Y. Asai and P.E. Kloeden; Numerical schemes for random ODEs via stochastic differential equations, Communications in Applied Analisys 17 (2013), No.3 & 4, 511–528.
[3] M. Tansky; Switching effect in prey-predator system, J. theor. Biol. (1978) 70, 263–271
[4] K. Wang and W. Wang; Propagation of HBV with spatial dependence, Mathematical Biosciences 210 (2007), 78–95.
Reference: PAGE 24 (2015) Abstr 3396 [www.page-meeting.org/?abstract=3396]
Poster: Methodology - New Modelling Approaches