In the following:
This is an account of a modelling scenario that uses the sir epidemic model. It was used in a third year applied mathematics subject. All students were enrolled in a mathematics degree of some type. Students are presented with the results of a test carried out on 100 individuals in a community containing 3000 people. From this they determined the number of infectious and recovered individuals in the population. Given the per capita recovery rate and making a suitable assumption about the number of infectious individuals at the start of the epidemic, they then estimate the infectious contact rate and from this the basic reproduction number. The mayor has asked the students to determine what will happen if no action is taken and to evaluate four policy options. They are asked to recommend the best course of action.
This scenario provides students with a problem where parameter values must be inferred from the information provided (one cannot be determined). They use the SIR model to provide public health recommendations, reinforcing their appreciation for the usefulness of mathematical modelling.
Our paper gives details of student presentations, and errors on the final exam, along with feedback to and from the instructor and the two student coauthors.
Carrin Goosenu, Mark Ian Nelson and Mahime Waranabeu. Applying the SIR model: can students advise the mayor of a small community? CODEE Journal, 17, Article 2, 2024. https://scholarship.claremont.edu/codee/vol17/iss1/2.
We investigate a three-component system involving the Belousov–Zhabotinsky reaction in water-in-oil microemulsions. Our goal is to investigate the connection between homoclinic snaking and semi-strength interaction in a three-variable reaction–diffusion system. A two-parameter bifurcation diagram of homogeneous, periodic and localized patterns is obtained numerically, and a natural asymptotic scaling for semi-strong interaction theory is found where an activator source term a=O(δ1) and b=O(δ2), with δ1 ≪ 1 being the diffusion ratio. Under this regime, singular perturbation techniques are used to construct localized steady-state patterns, and new types of non-local eigenvalue problems (NLEP) are derived to determine the stability of these patterns to O(1) time-scale instabilities. We extend the scope of the NLEP by considering a general scenario where both time-scaling parameters are non-zero. All analytical results are found to agree with numerics. Further numerical results are presented on the location of various types of breathing Hopf instability for localized patterns.
Fahad Al Saadi, Chunyi Gai and Mark Ian Nelson. Localized pattern formation: semi-strong interaction asymptotic analysis for three components model. Proceedings of the Royal Society A, 480, 20230591, 2024. https://doi.org/10.1098/rspa.2023.0591.
Numerical continuation is used to compute solution branches in a two-component reaction-diffusion model of Leslie–Gower type. Two regimes are studied in detail. In the first, the homogeneous state loses stability to supercritical spatially uniform oscillations, followed by a subcritical steady state bifurcation of Turing type. The latter leads to spatially localized states embedded in an oscillating background that bifurcate from snaking branches of localized steady states. Using two-parameter continuation, we uncover a novel mechanism whereby disconnected segments of oscillatory states zip up into a continuous snaking branch of time-periodic localized states, some of which are stable. In the second, the homogeneous state loses stability to supercritical Turing patterns, but steady spatially localized states embedded either in the homogeneous state or in a small amplitude Turing state are nevertheless present. We show that such behavior is possible when sideband Turing states are strongly subcritical and explain why this is so in the present model. In both cases, the observed behavior differs significantly from that expected on the basis of a supercritical primary bifurcation.
Fahad Al Saadi, E. Knobloch, M. Nelson, and H. Uecker. Time-dependent localized patterns in a predator-prey model. Chaos: An Interdisciplinary Journal of Nonlinear Science A, 34, Number 4, 043143, 2024. https://doi.org/10.1063/5.0197808.
Aviation fuels are manufactured using petroleum derived chemicals. An alternative is to use microorganisms as biocatalysts to produce biofuels. Such biofuels can either be blended with or replace existing fuels. This offers a route to replace petroleum-derived fuels by fuels obtained from renewable carbon sources.
One of the main bottlenecks in the microbial synthesis of sustain- able aviation fuels is that the chemicals produced are frequently toxic to the biocatalysts. One approach to overcome this problem is to redesign the bioreactors that are used in their production. We extend a stan- dard bioreactor model to include extraction of the biofuel from the reac- tor through a membrane. This reduces the concentration of the product within the bioreactor. We investigate how this technology mitigates the adverse effects of end-product toxicity.
Fahad Al Saadi and M.I. Nelson. Combatting Biofuel Toxicity Through Membrane Separation: Improving the Production of Sustainable Aviation Fuels. In Abid~Ali Khan, Mohammad~Sayeed Hossain, Mohammad Fotouhi, Axel Steuwer, Anwar Khan, and Dilek~Funda Kurtulus Proceedings of the First International Conference on Aeronautical Sciences, Engineering and Technology, pages 296-301, Singapore, 2024. Springer Nature Singapore. ISBN 978-981-99-7775-8. https://doi.org/10.1007/978-981-99-7775-8_31.
We adapt the Lanchester combat model to represent conflict between vampires and humans. It is assumed that vampires attack humans during the hours of darkness whilst humans attack vampires during the hours of light. The right-hand side of the differential equation model therefore depends upon the hour of the day. A key insight is that to answer the question `who wins' it is not required to `stitch together' the solutions to the differential equation over many days. Rather, the number of combatants surviving at the end of each day can be cast as a difference equation in terms of the numbers surviving at the end of the previous day. Two models are investigated and from the solutions the boundary delimiting the parameter regions of victory and defeat is found. Where appropriate we introduce discussion points, both mathematical and modelling. The mathematical techniques used include finding the eigenvalues and eigenvectors of a matrix, solving linear differential and discrete systems, and considering the physical meaning of the model and questioning its assumptions. Thus a variety of problem-solving skills are developed within the context of a model where students can question the underlying assumptions and propose, and investigate, models of their own.
S. Cuthbertson and M.I. Nelson. Modelling warfare between vampires and humans: who wins? International Journal of Mathematical Education in Science and Technology, 1–24, 2024. https://doi.org/10.1080/0020739X.2024.2329346.