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Agent-based simulators (ABS) are a popular epidemiological modelling tool to study the impact of various non-pharmaceutical interventions in managing an epidemic in a city (or a region). They provide the flexibility to accurately model a heterogeneous population with time and location varying, person-specific interactions as well as detailed governmental mobility restrictions. Typically, for accuracy, each person is modelled separately. This however may make computational time prohibitive when the city population and the simulated time is large. In this paper, we dig deeper into the underlying probabilistic structure of a generic, locally detailed ABS for epidemiology to arrive at modifications that allow smaller models (models with less number of agents) to give accurate statistics for larger ones, thus substantially speeding up the simulation. We observe that simply considering a smaller aggregate model and scaling up the output leads to inaccuracies. We exploit the observation that in the initial disease spread phase, the starting infections create a family tree of infected individuals more-or-less independent of the other trees and are modelled well as a multi-type super-critical branching process. Further, although this branching process grows exponentially, the relative proportions amongst the population types stabilise quickly. Once enough people have been infected, the future evolution of the epidemic is closely approximated by its mean field limit with a random starting state. We build upon these insights to develop a shifted, scaled and restart-based algorithm that accurately evaluates the ABS's performance using a much smaller model while carefully reducing the bias that may otherwise arise. We apply our algorithm for Covid-19 epidemic in a city and theoretically support the proposed algorithm through an asymptotic analysis where the population size increases to infinity.