Analysis and optimal control of stochastic biological systems

This thesis is concerned with the analysis and control of stochastic biological processes that arise in cells and in the spread of diseases. The modelling methods associated with such processes are typically focused on the average values and based upon ordinary differential equations that are exact only in the limit of large numbers. When a small number of molecules or a low number of individuals are accounted for discrete valued stochastic approaches are suitable. The tools to synthesize control laws and the associated analysis of biological systems in a stochastic setting are however still inadequate when compared to their deterministic counterpart. Despite their practical relevance probabilistic models have indeed received considerably less attention. We try to address this gap in the literature by developing two tools: a) an analytical method to quantify noise for a model of a cellular signalling network, b) highlight the limitations of feedback policies to tame the spread of a disease in a community of individuals. Our first result is associated with analytically quantifying a bound for the second moment of a biochemical species when non linear reaction rates are considered. The existing research in this area has so far focused on introducing approximations that lead to non quantifiable errors. The work here presented introduces a novel method to address this problem without the need of any approximation. In particular, by making use of a discrete expansion we obtain an analytical expression providing a hard bound for the variance. The bound is then shown to be exact when the rates are linear. Our second results considers the problem of taming an epidemic at its early stages by modulating the transmission rate. We formulate an appropriate stochastic optimal control problem in order to select the optimal policy and by exploiting the structure of this problem we show the underlying limitations of using negative feedback in this context. In particular for costs which are linear with the number of infected individuals we show that in an optimal policy there is no decrease in the controlled transmission rate of the disease for increasing infected. From this it follows that negative feedback is not effective in taming the spread of the disease in this setting. The method adopted to derive this limitation is for a simplified epidemic model but accounts for arbitrary feedback policies, a large class of cost functions and for arbitrary system and cost parameters. The results in this thesis thus seek to address the possible pitfalls of adopting models with continuous variables and deterministic dynamics by developing tool for systems that are more adequately described by a stochastic formulations.

Name dc:type.qualificationname

Doctor of Philosophy (PhD)

Level dc:type.qualificationlevel

Doctoral

Grantor dc:publisher.institution

University of Cambridge

Year dc:date.issued

2023

Author dc:creator

• Pugliese Carratelli, Giovanni

Advisor dc:contributor.advisor

• Lestas, Ioannis

• Biological systems
• Engineering
• Master Equations
• Stochastic Control

Licence

• https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/57f95daf-00d6-4919-90fa-77557ee281b2/download
• https://www.rioxx.net/licenses/all-rights-reserved/

Language dc:language

eng

DOI dc:identifier.doi

https://doi.org/10.17863/CAM.106420

OAI identifier oai:identifier

oai:www.repository.cam.ac.uk:1810/364952