(MSIR). Infectious disease dynamics Mathematical models need to integrate the increasing volume of data being generated on host-pathogen interactions. Many theoretical studies of the population dynamics, structure and evolution of infectious diseases of plants and animals, including humans, are concerned with this problem. Research topics include: antigenic shift epidemiological networks evolution and spread of resistance immuno-epidemiology intra-host dynamics Pandemic pathogen population genetics persistence of pathogens within hosts phylodynamics role and identification of infection reservoirs role of host genetic factors spatial epidemiology statistical and mathematical tools and innovations Strain (biology) structure and interactions transmission, spread and control of infection virulence Mathematics of mass vaccination If the proportion of the population that is immune exceeds the herd immunity level for the disease, then the disease can no longer persist in the population and its transmission dies out. Thus, a disease can be eliminated from a population if enough individuals are immune due to either vaccination or recovery from prior exposure to disease. For example, smallpox eradication, with the last wild case in 1977, and certification of the eradication of indigenous transmission of 2 of the 3 types of wild poliovirus (type 2 in 2015, after the last reported case in 1999, and type 3 in 2019, after the last reported case in 2012). The herd immunity level will be denoted q. Recall that, for a stable state: In turn, which is approximately: S will be (1 − q), since q is the proportion of the population that is immune and q + S must equal one (since in this simplified model, everyone is either susceptible or immune). Then: Remember that this is the threshold level. Die out of transmission will only occur if the proportion of immune individuals exceeds this level due to a mass vaccination programme. We have just calculated the critical immunization threshold (denoted qc). It is the minimum proportion of the population that must be immunized at birth (or close to birth) in order for the infection to die out in the population. Because the fraction of the final size of the population p that is never infected can be defined as: Hence, Solving for , we obtain: When mass vaccination cannot exceed the herd immunity If the vaccine used is insufficiently effective or the required coverage cannot be reached, the program may fail to exceed qc. Such a program will protect vaccinated individuals from disease, but may change