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This PR enhances the probability_emergence() function by adding the num_init_infect argument. This follows the probability_epidemic(), probability_extinct() and probability_contain() functions that have the num_init_infect argument. This is an extension of the Antia et al. (2003) method which only included a single introduction.

An illustration of the influence of multiple initial pathogen introductions on the probability of emergence is added to the emergence.Rmd vignette.

Multiple introductions is added to the unit test that checks for the equivalence between probability_emergence() and probability_epidemic().

The default value for the mutation_rate argument in probability_emergence() is removed as this should always be specified by the user.

@joshwlambert joshwlambert added the enhancement New feature or request label Jul 15, 2025
@joshwlambert joshwlambert merged commit f7d96f5 into main Jul 15, 2025
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@joshwlambert joshwlambert deleted the enhance-emerge branch July 15, 2025 13:24
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