Doctoral Research

Mathematical Epidemiology & Optimal Control

Why does the flu sometimes turn deadly? I use mathematics to model how diseases spread together, and to work out the smartest way to vaccinate a population when resources are limited.

The Question

Why does the flu sometimes turn deadly?

Imagine a child with a simple flu. A few days later, she is in the hospital with pneumonia. The flu didn’t cause it directly — it weakened her body just enough to let a second germ, pneumococcus, walk in through the open door.

This double attack is called co-infection, and it is my research topic.

So I ask: what if we could predict how these two diseases spread together? And then: with limited vaccines and a limited budget, who should get vaccinated, and when, to save the most lives?

I answer these questions with mathematics. I build a computer model of a whole population — sick and healthy, young and old, vaccinated and not — and test different vaccination plans on it, without risking a single real person.

Right now, I’m finishing the analysis and writing it up as a research paper. The goal is a smarter way to protect people every flu season.

🤒
Flu infection
→
🛡️
Weakened immune defenses
→
🦠
Pneumococcus moves in
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🏥
Severe illness / hospitalization
The Method

What is an “optimal control problem”?

In plain terms: it’s a mathematical way of finding the best possible strategyfor steering a system over time, when you can’t just fix everything at once — you have a limited budget, a limited supply, and every decision today affects what happens tomorrow. Here, the “system” is an entire population living through flu season, and the “strategy” is a vaccination plan.

1. Model the population
Split everyone into groups — healthy, infected with flu only, infected with both, vaccinated, recovered — and write down equations for how people move between these groups over time.
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2. Choose a vaccination strategy
This is the “control” — how many doses go out, to which age groups, and on which day — subject to how many doses actually exist and what they cost.
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3. Optimize the outcome
Use mathematics to find the exact strategy from step 2 that minimizes deaths and hospitalizations, given the constraints from step 1 and step 2.

The whole point is that this can be tested and re-tested entirely inside a computer — thousands of different vaccination plans can be tried instantly, and the mathematics guarantees you land on the genuinely best one, not just a plan that seems reasonable.

Along the Way

Class & Short Projects

Smaller projects and reports from coursework across three institutions and two countries — pure mathematics, before the shift toward epidemiology and applied control theory.

May 2023

Schemes as Varieties

MATH-523: Elements of Algebraic Geometry
LUMS, Pakistan
Supervisor: Dr. Shaheen Nazir
View report →
April 2023

Euler Characteristics

MATH-507: Advanced General Topology
LUMS, Pakistan
Supervisor: Dr. Haniya Azam
View report →
June 2022

Big Theorems of Functional Analysis

M1 TER Project · Master-1 in Mathematics
University of Lille, France
Supervisor: Catalin Badea
View report →
March 2022

Construction of Borel Sets

Anglais Mathématique
University of Lille, France
Supervisor: Gautami Bhowmik
View report →
May 2021

The Fundamental Group and Classification of Covering Spaces

MS Thesis · International Mathematics Master
COMSATS University Lahore
Supervisor: Pavel Putrov (ICTP Italy) & Hani Shaker (COMSATS Lahore)
No public link
Get in Touch

Interested in this work?

I’m always happy to discuss mathematical epidemiology, optimal control, or the courses I teach at LUMS.