Parameter estimation of influenza epidemic model

SEIRS and SVEIRS epidemic models are considered here to capture the main characteristic of transmission of influenza epidemic governed by a system of differential equations. All parameters estimations involved in these models are based on the influenza epidemic which occurred in Australia in 1919, often called Spanish flu data Sydney. Least squares method, which involves minimization of the sum of squared differences between the measurements and the model predictions is used to estimate the unknown parameters for both models. Graphical as well as numerical methods are used to validate these models. It is shown that our models reflect considerably the dynamical behavior of the influenza epidemic field data used. An important view of the disease dynamics including vaccine efficacy and level of vaccination is also drawn. Mathematical modeling is an important tool to study the mechanisms of spread of diseases in order to predict the future outbreak and thus to make strategies to control an epidemic. There is lot of research going on, theoretical as well as computational aspects of dynamical systems modeling based on infectious diseases. For example, Balcan et al. [1] have investigated recurrent outbreaks of influenza epidemics by a combination of disease relevant human interactions and mobility across multiple spatial scales. Colizza et al. [2] have studied a metapopulation model based on data-driven mobility schemes to evaluate the level of interventions needed to contain epidemics of varying severity. Perc et al. [3,4] have investigated the network-based models and these type of models have attracted a great deal of attention from diverse disciplines. Computational models and multi-scale numerical simulations represent essential tools in understanding of the spread of infections, in particular for assessing scenarios, predicting epidemic evolutions, and managing health emergencies that can benefit a broad audience of users including policy makers and health institutions. Broeck et al. [5] have developed computational (modeling and prediction) tools for realistic computer-based simulations on the spread of infectious disease. The models known as deterministic and stochastic epidemic models are very popular among the researchers. Among the deterministic models, compartmental epidemic models are the most used for investigation by the Mathematician. In a com-partmental epidemic model, it is assumed that the total population size in a compartment is differentiable with respect to time and the epidemic process is deterministic. Therefore, the transformation rates of population from one compartment to another compartment can be expressed as a derivative with respect to time. As …