Modelling \textit{Aedes albopictus} management, incorporating immigration and bi-directional \textit{Wolbachia} interactions

\textit{Aedes albopictus} mosquitoes are competent vectors for the spread of at least 24 different arboviruses, including dengue, Ross River, and Japanese encephalitis viruses. However, they remain less studied than their more urban cousins, \textit{Aedes aegypti}. We model an Incompatible Insect Technique (IIT) strategy for mosquito control, with bi-directional incompatibility between two strains of \textit{Wolbachia} (\walba/\walbb\, $\times$ \arwp) and age-based cytoplasmic incompatibility decay in a well-mixed population. An important consideration in bi-directional IIT control programs is reversibility when immigration is included. We explore the establishment probability after female contamination of an artificially-infected \textit{Wolbachia} mosquito strain, finding a conservative threshold of 40\% likely driven by mating inefficiencies -- this threshold needs validation in future field and lab experiments. We consider the suppression dynamics and probability of mosquito management success for different release strategies, showing differences in success between release cessation and six months later for different immigration rates. Importantly, our model suggests bi-directional IIT control programs are reversible with low amounts of immigration. We determine a corresponding cost proxy (numbers of mosquitoes released), showing similar short-term costs with differences in medium- and longer-term costs between release strategies. This work demonstrates opportunities to optimise the suppression of these medically important mosquitoes.