Superinfection and the hypnozoite reservoir for Plasmodium vivax: a multitype branching process approximation

Plasmodium vivax malaria is a mosquito-borne disease of significant public health importance. A defining feature of the within-host biology of P. vivax is the accrual of a hypnozoite reservoir, comprising a bank of quiescent parasites in the liver that are capable of causing relapsing blood-stage infections upon activation. Superinfection, characterised by composite blood-stage infections with parasites derived from multiple mosquito inoculation or hypnozoite activation events, is another important attribute. We have previously developed a stochastic epidemic model of P. vivax malaria, formulated as a Markov population process with countably infinitely-many types, that is adjusted for both hypnozoite accrual and blood-stage superinfection. Here, we construct a Markovian branching process with countably infinitely-many types to approximate the early stages of this epidemic model. With $P_M$ denoting the mosquito population size, we consider the limit $P_M\to \infty$ with the ratio of the mosquito and human populations held fixed. With $\kappa <1/2$ an arbitrary constant, we use a classical coupling argument to obtain a total variation bound of order $O\left(P_M^{2\kappa -1}\right)$ that is valid until $o\left(P_M^{\kappa }\right)$ human-to-mosquito and mosquito-to-human transmission events have occurred. We characterise the probability of global disease extinction under the branching process to approximate the probability of elimination, as opposed to sustained endemic transmission, when the epidemic model is initialised with low-level human and/or mosquito infection. We apply our model to two scenarios of epidemiological interest, namely the re-introduction of P. vivax malaria in a region where elimination has previously been achieved; and a mass drug administration campaign with population-wide depletion of the hypnozoite reservoir.