A multi-host deterministic-stochastic framework for giardiasis transmission: branching-process extinction analysis and continuous-time Markov chain dynamics

Giardiasis remains a widespread waterborne disease with substantial public health importance, particularly in settings with inadequate sanitation and high environmental contamination. In this study, we formulate and analyze a deterministic compartmental model that captures the transmission dynamics of Giardia duodenalis among immunocompetent and immunocompromised human populations, a lamb reservoir, and the environmental cyst pool. We derive the disease-free equilibrium and establish its local and global stability in terms of the basic reproduction number R 0 , obtained via the next-generation matrix method. To complement the deterministic analysis, we construct a multi-type Galton-Watson branching process approximation near the DFE and compute type-specific extinction probabilities under different introduction scenarios. Sensitivity analysis is performed by varying key shedding parameters to quantify their influence on extinction likelihood. Furthermore, a continuous-time Markov chain (CTMC) model is developed to estimate the distribution of extinction times, providing additional insight into stochastic fadeout dynamics. Numerical experiments reveal that infections introduced through lambs or the environmental cyst reservoir exhibit markedly lower extinction probabilities and longer mean extinction times compared with human-initiated introductions. Overall, the combined deterministic-stochastic framework highlights the importance of reducing environmental contamination and targeting high-shedding hosts to enhance the probability of disease elimination. The results underscore the significance of early detection, timely treatment, and interventions that curtail environmental cyst persistence as effective strategies for mitigating giardiasis transmission.