Final epidemic size and optimal control of socio-economic multi-group influenza model

Flu, a common respiratory disease is caused mainly by the influenza virus. The Avian influenza (H5N1) outbreaks, as well as the 2009 H1N1 pandemic, have heightened global concerns about the emergence of a lethal influenza virus capable of causing a catastrophic pandemic. During the early stages of an epidemic a favourable change in the behaviour of people can be of utmost importance. An economic status-based (higher and lower economic class) structured model is formulated to examine the behavioural effect in controlling influenza. Following that, we have introduced controls into the model to analyse the efficacy of antiviral treatment in restraining infections in both economic classes and examined an optimal control problem. We have obtained the reproduction number R-0 along with the final epidemic size for both the strata and the relation between reproduction number and epidemic size. Through numerical simulation and global sensitivity analysis, we have shown the importance of the parameters phi(i), phi(s),eta(2), beta and theta on reproduction number. Our result shows that by increasing phi(1), eta(2) and by decreasing beta, theta and phi(s), we can reduce the infection in both the economic group. As a result of our analysis, we have found that the reduction of infections and their level of adversity is directly influenced by positive behavioural patterns or changes as without control susceptible population is increased by 23%, the infective population is decreased by 48.54% and the recovered population is increased by 23.23% in the higher economic group who opted changed behaviour as compared to the lower the economic group (people living with normal behaviour). Thus normal behaviour contributes to the spread and growth of viruses and adds to the hassle. We also examined how antiviral drug control impacts both economic strata and found that in the higher economic strata, the susceptible population increased by 53.84%, the infective population decreased by 33.6% and the recovered population improved by 62.29% as compared to the lower economic group, the susceptible population has increased by 19.04%, the infective population is decreased by 17.29% and the recovered population is improved by 47.82%. Our results enlighten the role that how different behaviour in separate socio-economic class plays an important role in changing the dynamics of the system and also affects the basic reproduction number. The results of our study suggest that it is important to adopt a modified behaviour like social distancing, wearing masks accompanying the time-dependent controls in the form of an antiviral drug's effectiveness to combat infections and increasing the proportion of the susceptible population.