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but the parasites should be killed off just as quickly no matter how long a person's had them.
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If we assume that the drug is efficient enough to kill all stages of the kind of parasite one is infected with (according to the reports on pubmed, an anti parasites drug is able to reduce the load, not to get rid of all of them), and if the drug molecules are able to reach all parts of the body unaltered where the parasites are, and if the dose is high enough, and if the drug is taken frequently enough so that all eggs and all stages are killed, and if all pets and others humans in contact with this person are also given these efficient drugs, and if the environment and foods and drinks do not have eggs on/in it, then maybe yes.
Do you realize how unrealistic what you write is ?