Written by: Basil Hassan
Post Date
May 25 2023

The Power of Numbers: Unraveling Disease Dynamics through Mathematical Modeling

The use of mathematical modeling in the fight against diseases and pandemics is not new. Scientists and mathematicians have used mathematical modeling to identify the cause of diseases and estimate their transmission rates. Since it helps calculate the transmission rate and the proportion of the population at risk of infection, this methodology becomes crucial for pandemic control. For the recent novel coronavirus (COVID-19) too, such models were created, which helped gauge its spread and hence aided authorities in making suitable decisions to prevent its spread.  
Mohsin Ali, a Ph.D. student at the department of Mathematics at LUMS, formulated deterministic models for his Ph.D. thesis, which was successfully defended on Friday, May 5th ,2023. The thesis was titled “Mathematical Modelling of Some Infectious Diseases” and was examined by a committee including members from both LUMS and otter institutions such as Ajman University in UAE and the University of the Punjab in Lahore. The defense was supervised by Dr. Adnan Khan, an associate professor of Mathematics at LUMS.  


Figure 2 Mohsin presenting his thesis.


Dr. Mohsin’s primary data source was the NCBI database, and the initial model investigated the effects of asymptomatic persons in spreading the disease, which was the main issue with this disease i.e., the inability to observe symptoms early enough, making containing the disease a big challenge. Ali calculated the basic reproduction number based on incidence data from the initial outbreak in Wuhan. This estimate aligned with others and was considered low during the initial stages of the pandemic due to fewer infected individuals. He then covered isolation and quarantine procedures, as well as the medication and treatment needed to control the illness, into his calculations. From this, he predicted the COVID-19 epidemic curve during the pandemic's initial stages in Pakistan in 2020.

By analyzing incidence statistics for the pre-and post-relaxation period of social distancing measures, he calculated the fundamental reproduction number, which helped predict the epidemic curve. In the absence of significant attempts to manage the epidemic, the curve would have peaked around mid-September 2020. It was also found that the disease severity and illness-related factors are both linked to age and co-morbidities such as heart diseases and hypertension.  

Interestingly, in his thesis, potential patients were divided into two main groups; a low-risk group of teens with high immunities and a high-risk group comprising the elderly. It was evident that changes in the relative number of these low- and high-risk susceptible groups significantly influenced the effect of disease and mortality. Further, to understand how co-infection can affect pneumonia transmission, Dr. Mohsin examined the relationship between influenza and pneumonia respiratory illnesses. His model also exhibited a backwardly diverging curve in the presence of co-infection, indicating that bringing the disease’s reproduction rate to zero might not be sufficient. However, it can depend on the first susceptible population.

Numerical simulations and models were also presented to corroborate the theoretical conclusions of the thesis. An essential factor, 'Ro', was defined as the basic reproduction number that determined the decay or increase in the curves of the disease. 
The importance of basic Mathematics is often overlooked and imagined to be restricted to examinations and classrooms. However, with the right arithmetic and modeling, it is even possible to understand disease dynamics and take steps to counter it! 

Figure 1 Contours of Ro