Overview
Modeling projects appear in applied mathematics courses, engineering, biology, epidemiology, economics and environmental science. The mathematics is often manageable; the difficulty lies in formulation (which variables, which mechanisms, which scales), in numerical solution (stability, step size, stiffness) and in interpretation (is the behavior a feature of the system or an artifact of the model?).
We work across the main model families: deterministic models based on ordinary and partial differential equations and difference equations; stochastic models including Markov chains and stochastic differential equations; Monte Carlo methods for uncertainty and integration; and agent-based models in which individual behavior produces collective patterns. Tools include Python (NumPy, SciPy, Mesa), MATLAB, R and NetLogo.
For students, we tutor the methods and review models you build. For thesis researchers and professionals, we can also help design the model, implement or review the simulation code, and plan sensitivity and uncertainty analyses. You remain responsible for the scientific claims.
Who it’s for and the problems it solves
This is for you if you are…
- Undergraduates in mathematical modeling, differential equations or numerical methods courses
- Students preparing for modeling competitions such as MCM/ICM, as coaching on technique
- Master's and PhD researchers building population, epidemic, pharmacokinetic, climate, traffic or economic models
- Researchers who need a Monte Carlo uncertainty analysis or an agent-based model reviewed
- Professionals prototyping models before investing in specialized software
Common problems we help with
- You have a research question but do not know how to translate it into equations
- Your ODE solver produces oscillations or blows up, and you are not sure whether the model or the numerics are at fault
- Parameter values come from different studies with different units, and results swing widely
- An agent-based model gives different results every run and you do not know how many runs are enough
- Reviewers asked for a sensitivity analysis or model validation you have not done
What’s included — and what isn’t
Included
- Model formulation: purpose, variables, assumptions, conservation laws, compartmental structures and dimensional analysis
- Ordinary differential equations: analytic methods where possible, equilibrium and stability analysis, phase planes, bifurcations
- Numerical solution: Euler and Runge–Kutta methods, adaptive step size, stiff solvers, error and stability, and solver choice in SciPy (solve_ivp), MATLAB (ode45, ode15s) or R (deSolve)
- Partial differential equations: finite differences, method of lines, stability conditions such as CFL, and when to move to finite element tools
- Discrete and stochastic models: difference equations, Markov chains, birth–death processes, Gillespie algorithm, stochastic differential equations
- Monte Carlo simulation: random number generation and seeding, variance reduction, convergence and confidence intervals for estimates
- Agent-based modeling: design using the ODD protocol, implementation in NetLogo or Python (Mesa), replication and output analysis
- Parameter estimation and model fitting: least squares, likelihood, identifiability issues
- Sensitivity and uncertainty analysis: one-at-a-time, Latin hypercube sampling, Morris screening and Sobol indices
- Model verification and validation, and clear reporting of assumptions and limitations
Not included
- Completing graded modeling assignments or competition entries for students
- Producing forecasts for regulatory, clinical or financial decisions without the client's own expert review
- Tuning a model until it matches a desired conclusion
What you receive
- Model specification
- A written description of the model: purpose, variables, equations or rules, parameters with units and sources, and assumptions, in a form you can adapt for a methods section.
- Tutoring notes and worked examples
- For students, notes and parallel examples covering the techniques your course or project uses.
- Simulation code or code review
- For researchers, documented and reproducible code in Python, MATLAB, R or NetLogo, or a review of your existing code with suggested fixes.
- Sensitivity and uncertainty report
- Tables and figures showing which parameters drive outcomes and how uncertain the results are, with interpretation notes.
- Walkthrough session
- A session explaining how the model and code work so you can modify, defend and report them.
How it works
Define purpose
We start with the question the model must answer, because that decides the level of detail and the outputs needed.
Formulate
We agree variables, mechanisms and assumptions, check dimensions and write the model specification.
Implement and verify
The model is solved analytically or numerically and checked against special cases, conservation properties or known solutions.
Explore and test robustness
We run scenarios, sensitivity and uncertainty analyses and, where data exist, calibration and validation.
Report
You receive the specification, code or review, figures and notes to support your own write-up.
Model types and numerical methods
- Compartmental ODE models
- SIR/SEIR epidemic models, predator–prey and competition models, pharmacokinetic compartments and chemical kinetics.
- Stiff solvers
- Implicit methods such as BDF or Rosenbrock (ode15s, solve_ivp with method='BDF' or 'Radau') for systems with very different timescales.
- Monte Carlo methods
- Estimating expectations, probabilities and integrals by repeated random sampling, with seeds recorded and standard errors reported.
- Agent-based models and ODD
- The Overview, Design concepts and Details protocol for describing agent-based models so they can be understood and replicated.
- Global sensitivity analysis
- Latin hypercube sampling with partial rank correlation, Morris elementary effects and variance-based Sobol indices (for example with SALib in Python).
- Calibration
- Least squares, maximum likelihood or approximate Bayesian methods, with checks for parameter identifiability.
Is it the model or the numerics?
Strange simulation behavior has two possible sources: the model itself or the way it is solved. Oscillations that shrink when you reduce the step size are numerical; oscillations that persist are probably real. We test numerical results against analytical special cases, check conservation of quantities that should be conserved, and compare solvers, so you can trust that the behavior you report belongs to the model.
How many simulation runs are enough?
Stochastic and agent-based models need replicate runs. The right number depends on the variability of the outputs you care about, not on a fixed rule. We estimate it by watching how the mean and its confidence interval stabilize as runs increase, and we report the number of replicates and seeds so the analysis can be reproduced.
Sensitivity analysis as part of the argument
Reviewers increasingly expect a model paper to show which conclusions are robust. A one-at-a-time analysis is easy but misses interactions; global methods explore the whole parameter space.
- Screening with Morris to find influential parameters cheaply
- Sobol indices to apportion output variance, including interactions
- Scenario analysis for structural assumptions that are not simple parameters
What the work can look like
Illustrative example — fictional data
Sample model specification excerpt: vaccination in an SEIR model
A master's student in public health wanted to explore how vaccination timing affects an outbreak in a closed population of 50,000. All values below are invented for illustration and are not real epidemiological estimates.
Structure
Compartments S, E, I, R and V. Assumptions: homogeneous mixing, no births or deaths over the 180-day horizon, vaccination moves people from S to V at a constant daily rate starting on day t_v, and vaccine protection is complete.
Numerics
Solved with SciPy solve_ivp (RK45, relative tolerance 1e-8). Verification: total population stays at 50,000 to within rounding, and with vaccination switched off the final size matches the analytical final-size relation.
Sensitivity
Latin hypercube sampling over plausible ranges of transmission rate, latent period and vaccination rate (1,000 samples). Partial rank correlation shows peak size is most sensitive to transmission rate and t_v. The student interprets and writes up the results.
Illustrative example with invented parameters. It is not public health guidance and real models must use sourced parameters and expert review.
What you’ll need to provide
- The research or course question and any required modeling approach
- Existing equations, diagrams, code or published models you are building on
- Parameter values and data with sources and units
- Software preference and computing resources (laptop, cluster)
- Course or institutional rules if the work is assessed
You don’t need everything ready to ask for a quote — tell us what you have and we’ll say what’s missing.
What affects price and turnaround
Price depends on
- Model type and complexity: a small ODE system differs from a spatial agent-based model
- Tutoring sessions, code review or model development for research clients
- Computational cost, including replicate runs and global sensitivity analysis
- Availability and condition of data for calibration and validation
Turnaround depends on
- How clearly the question and assumptions are defined at the start
- Simulation run times and whether a computing cluster is needed
- Time for you to make scientific decisions about model structure
Every project is quoted individually. How we price our work.
Questions people ask before starting
Can you build the model for my course project?
For assessed coursework we teach the techniques, work through a different example and review the model you build. For thesis and research projects we can help design, implement and review the model, and you remain responsible for the scientific decisions and interpretation, acknowledging assistance as your institution requires.
Which software should I use?
Python with SciPy is flexible and free; MATLAB has mature ODE solvers and Simulink; R with deSolve suits statisticians and ecologists; NetLogo is accessible for agent-based models, while Python's Mesa suits larger projects. We recommend based on your field's conventions and what your supervisor can review.
Do you help with modeling competitions such as MCM/ICM?
We coach techniques before the contest: formulation, common model families, sensitivity analysis and report writing. We do not help during the contest, since competition rules require the team's own work.
Can you check whether my model is realistic?
We can check internal consistency, numerical correctness, dimensional consistency and whether behavior makes sense in limiting cases. Whether the assumptions are realistic for your system depends on domain knowledge; we raise questions and help you compare with data, but you and your supervisor judge scientific validity.
What if my parameters come from different studies?
That is common. We help you record sources and units, convert consistently, flag conflicts and treat uncertain parameters with ranges in the sensitivity analysis rather than single values.