There has been an unprecedented increase in the frequency of serious global epidemic risks in the last decade. The COVID-19 pandemic has overwhelmed the world since 2020, causing over 3 million estimated deaths globally as May 2021. Influenza remains a perennial challenge, and the re-emergence of smallpox as bioterror attack is also of increasing concern. Modelling of interventions can provide critical support to public health decision making during an epidemic, comparing the effects of different interventions, and helping prioritize resources. This research uses modelling and risk analysis to inform policy and practice in relation to epidemic control for re-emergence of smallpox following a bioterror attack, the annual threat of seasonal influenza, and the response to the ongoing COVID-19 pandemic in Australia. 1. A SEIR mathematical model was constructed, simulating a smallpox re-emergence and targeted, ring and mass vaccination in response to the outbreak. Age specific distribution of immunosuppression and contact rates were used to estimate the number of doses needed of second and third generation vaccines and their most effective use, while estimating the number of cases and deaths in different scenarios. This study informs preparedness and response planning for smallpox vaccination distribution. 2. To estimate the level of residual immunity from previous vaccination in the population of NSW, serological data on vaccinia antibodies levels collected in 2003, were used. The data were analysed and compared to another data set showing levels following vaccination, and the decline in geometric mean titres (GMT) over time was modelled. This study informs current pre-existing immunity to smallpox in Australia. 3. Seasonal influenza vaccination is recommended around March-April. There is evidence of vaccine effectiveness waning over a season. This study quantified how changes in timing of vaccination uptake, waning immunity and vaccine coverage could impact prevention of influenza in Australia. Data on vaccine effectiveness, waning immunity over time, influenza notification and coverage estimate, were used to model the effect on vaccine effectiveness of shifting the time of vaccination by month from March to August. 4. Finally, for COVID-19 pandemic control, Australia implemented a travel ban on China from February 1st, 2020. Three scenarios were modelled to test the impact of this travel ban on epidemic control: no ban, and ban followed by full or partial lifting (were only students). Incidence data from China and air travel passenger movements between China and Australia during and after the epidemic peak in China, were used to inform an SEIR model reproducing the epidemic curve in Australia for each scenario. This research informs decisions on placing or lifting travel bans, applied to countries with high disease incidence, as a control measure for the COVID-19 epidemic. All chapters have been published in peer reviewed journals. This thesis informs outbreak response policy and practices against diseases of global interest, while highlighting the importance of modelling as a tool for public health. Finally, there are several recommendations proposed to enhance the accuracy, transparency, and quality of mathematical modelling results for informing public health responses.