Module 4 4 Modeling Malaria Angela B Shiflet

Background • Vector - animal that transmits pathogen • Anopheles mosquitoes • Host - infected human • Investigate progress of malaria • Plasmodium - malaria-causing

Simplifying assumptions • Ignore Plasmodium • Population of humans is closed • As soon as vector bites human, individual becomes host • No immunity • No incubation period • Ignore the chance of relapse

Simplifying assumptions • Death rates for infected and uninfected mosquitoes are identical • All mosquitoes reproduce at same rate • At birth, mosquito is uninfected • Infected mosquito immediately becomes host

Determine variables • uninfected_humans, who are susceptible to disease • human_hosts, who have malaria and can infect mosquitoes that bite them • immune, who cannot get disease again • uninfected_mosquitoes that do not carry Plasmodium • vectors that carry Plasmodium

Establish relationships

Equations and functions • flow_to_immune = immunity_rate * human_hosts • human_host_deaths = malaria_induced_death_rate * human_hosts • recovered = recovery_rate * human_hosts • mosquito_births = mosquito_birth_rate * mosquitoes, where mosquitoes = uninfected_mosquitoes + vectors • uninfected_mosquito_deaths = mosquito_death_rate * uninfected_mosquitoes • vector_deaths = mosquito_death_rate * vectors

Equations and functions d(uninfected_humans)/dt = (recovery_rate)(human_hosts) – (infection_rate)(uninfected_humans) We can break infection_rate into two factors: the probability a human is bitten by a mosquito and the probability that a mosquito is a vector. d(uninfected_humans)/dt = (recovery_rate)(human_hosts) – (prob_bit)(prob_vector)(uninfected_humans)

Equations and functions prob_vector = vectors / mosquitoes d(uninfected_humans)/dt = (recovery_rate)(human_hosts) – (prob_bit)(vectors)(uninfected_humans) / mosquitoes Assuming that an uninfected mosquito that bites a human host always becomes infected, the rate of change from uninfected to infected mosquito is d(vector_formation)/dt = flow_to_host = (prob_bite_human)(prob_host)(uninfected_mosquitoes) where prob_bite_human = probability that a mosquito bites a human and prob_host = the probability that a human is a host

Equations and functions prob_host = human_hosts / humans = human_hosts / (uninfected_humans + human_hosts + immune)

Human submodel

Mosquito submodel

Equation Set 4. 4. 1

Graphs

Graphs