Large Language Models (LLMs) offer significant potential for delivering health information. However, their reliability in low-resource contexts remains uncertain. This study evaluates GPT-4, Gemini Pro, Llama~3, and Mistral-7B on health crisis-related enquiries concerning COVID-19, dengue, the Nipah virus, and Chikungunya in the low-resource context of Bangladesh. We constructed a question--answer dataset from authoritative sources and assessed model outputs through semantic similarity, expert-model cross-evaluation, and Natural Language Inference (NLI). Findings highlight both the strengths and limitations of LLMs in representing epidemiological history and health crisis knowledge, underscoring their promise and risks for informing policy in resource-constrained environments.