Executive Summary : | Age structure is an essential demographic characteristic that significantly impacts the dynamics and spread of infectious diseases. Two main motivations to assimilate "age-structure" models are: (i) "Age-structure" is inevitable in situations where infectivity and susceptibility are firmly age-related, (ii) "Age structure" is also inevitable to model chronic infections diseases, such as TB & HIV for which population demography, and especially survivorship needs more sensible distributions, instead of exponential types modelled using ODEs. Though the mathematical tools to obtain analytical solution and computational techniques for simulations of ODE based models in epidemic are well established & extensively used in modelling even with exceptionally complexity in epidemic systems, but tools for "age-structured" models are yet to develop. Hence "age-structured" epidemic modelling, development of analytical and simulation techniques will keep at demanding mathematicians for years to come. In an age-structured model, rates of control measures are typically age-dependent. Studying diseases like hepatitis C, HIV/AIDS, and TB having a long-term chronic stage needs to include the time passed after infection into the model. Besides the age of infection, some more class ages, like vaccination age, immunity age, latent age etc., may affect the spreading of communicable diseases. Numerous people do not remain unsusceptible to the infections forever, and as time elapses after vaccination, the effect of the vaccine may be shrunk. Thus, a model with the age of vaccination is essential. The primary aim of this project is to work out and investigate class 'age-structured' type models in epidemic systems like (i) Model having age of infection, (ii) Model with the age of recovery and (iii) Model having the age of vaccination etc. For the age-structured model, the optimal control problems are distinct compared to both the PDE and ODE models. The underlying space for the 'age-structured' model is L1, not Hilbert; we aim to develop 'optimal control' problems. In 'age-structured' models, an analytical solution is not easy to determine in most cases, we aim to develop computational techniques for "age-structured" epidemiological models. Incorporating "age-structure" improves the dynamical behavior of the epidemiological system. The mathematical set-up for this type of first-order PDEs model is different in comparison to traditional second-order PDEs, due to its requirement of working in L1 spaces that are not Hilbert. I expect to develop and demonstrate varying techniques for analysing and simulating 'age structured' infection models on multiple examples. |