Post Date
Nov 29 2023

Mathematical Modelling of Some Infectious Diseases

Dr. Adnan Khan
Mohsin Ali
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Mathematical modeling has a long history of being used to combat diseases and pandemics. Scientists and mathematicians have relied on mathematical modeling to uncover illnesses' origin and determine the speed at which they spread. This methodology is critical in pandemic control as it estimates the transmission rate and the percentage of the population at risk of infection. In our study for thesis, we developed various deterministic epidemic models to investigate the transmission of novel coronavirus disease (COVID-19) and co-infection of pneumonia and influenza to obtain in-depth insights into public health concerns. We employed dynamical system theory techniques to identify equilibria and different qualitative properties. We calculated the basic reproduction number for each model using the next-generation operator method. Our analysis revealed that the disease-free equilibrium (endemic equilibrium) is locally and globally (for some models) asymptotically stable whenever the threshold value is less (greater) than unity and unstable when it is greater (less) than unity. We discovered the existence of the backward bifurcation phenomenon in the co-infection of pneumonia and influenza. We conducted sensitivity analysis and contour plots to evaluate the sensitivity of parameters to the basic reproduction number. We also conducted numerical simulations based on realistic parameter values available in the literature. Additionally, we applied optimal control theory to determine the best strategies for controlling the outbreak.

Our first model explored asymptomatic individuals' role in transmitting the novel coronavirus COVID-19. We proposed optimal quarantine and isolation strategies, emphasizing that high levels of these measures are necessary during the early stages of the outbreak. Based on incidence data from the Wuhan first outbreak, we estimated the basic reproduction number to be 1.87, which was in agreement with other estimates in the literature and relatively low during the early phase of the pandemic. The second model discussed the medication and treatment required to control the disease, in addition to quarantine and isolation measures. The third model estimated and projected the epidemic curve of COVID-19 during the early phase of the pandemic in Pakistan in 2020. We estimated the basic reproduction number from incidence data for the pre- and post-relaxation period of social distancing measures. Using this analysis, we projected the epidemic curve and noted that if no substantial efforts were made to contain the epidemic, it would peak in mid-September 2020.

In the fourth model, we examined the severity of the disease and disease-related mortality, which is strongly correlated with age and the presence of co-morbidities. We incorporated this into our model by considering two susceptible classes: a low-risk group and a high-risk group. We observed that varying the relative proportion of low and high-risk susceptible populations strongly affects disease burden and mortality. Finally, the fifth model explored the interaction between influenza and pneumonia respiratory diseases, which is vital for understanding how co-infection might affect pneumonia transmission. We determined the effective reproduction number and showed that the disease-free, boundary, and endemic equilibria are locally and globally stable in the absence of co-infection. Furthermore, the model undergoes a backward bifurcation in the presence of co-infection, which means that to eliminate the disease, it might not be sufficient to bring the reproduction number below one. However, this may depend on the initial susceptible population. We also conducted numerical simulations to demonstrate the theoretical findings in the thesis.

List of publications:

[i] M. Ali, S. T. H. Shah, M. Imran, and A. Khan. The role of asymptomatic class, quarantine and isolation in the transmission of COVID-19. Journal of biological dynamics, 14(1):389–408, 2020.

[ii] M. Ali, M. Imran, and A. Khan. Analysis and prediction of the COVID-19 outbreak in Pakistan. Journal of Biological Dynamics, 14(1):730–747, 2020.

[iii] M. Ali, M. Imran, and A. Khan. Can medication mitigate the need for a strict lock down?: A mathematical study of control strategies for COVID-19 infection. medRxiv, 2020. ( Pre-Print)

[iv] A. Khan, M. Ali, W. Iqbal, and M. Imran. Effect of high and low risk susceptibles in the transmission dynamics of COVID-19 and control strategies. Plos one, 16(9):e0257354, 2021.

[v] M. Ali, M. Imran,S. Mirza and A. Khan. A Model for the transmission dynamics of Pneumococcal Pneumonia and Influenza Co-Infection. International Journal of Biomathematics 2023. (Submitted)