This presentation is made available through a Creative Commons Attribution- Noncommercial license. Details of the license and permitted uses are available at © 2010 Dr. Juliet Pulliam Title: Dynamics of Vector-Borne Pathogens Attribution: Dr. Juliet Pulliam, Topics in Biomedical Sciences Source URL: Coursehttp://lalashan.mcmaster.ca/theobio/mmed/index.php/Honours Course For further information please contact Dr. Juliet Pulliam

Dynamics of vector-borne pathogens Topics in Biomedical Sciences BSc Honours Course in Biomathematics African Institute for the Mathematical Sciences Muizenberg, South Africa 20 May 2010 Dr. Juliet Pulliam RAPIDD Program Division of International Epidemiology Fogarty International Center National Institutes of Health (USA)

Transmission Infectious diseases Mode of transmission Direct transmission Direct contact Droplet spread Indirect transmission Airborne Vehicle-borne (fomites) Vector-borne (mechanical or biological) Portal of entry Portal of exit

Transmission Infectious diseases Mode of transmission Direct transmission Direct contact Droplet spread Indirect transmission Airborne Vehicle-borne (fomites) Vector-borne (mechanical or biological) Portal of entry Portal of exit Mosquitoes Ticks Sandflies Tsetse flies Reduviid bugs

Vector-borne pathogens “Typical” natural history Onset of symptoms Onset of shedding IncubationClinical disease Infectious periodLatent period Infection

Vector-borne pathogens “Typical” natural history Onset of symptoms Onset of shedding IncubationClinical disease Infectious periodLatent period Infection Onset of shedding InfectiousLatent DeathInfection HOST VECTOR

Vector-borne pathogens “Typical” natural history Often acute: timecourse of infection << normal lifespan of host BUT timecourse of infection ~ normal lifespan of vector Sometimes immunizing: infection may stimulate antibody production, preventing future infection… or may not… or somewhere in between

Vector-borne pathogens Examples Mosquitoes Anopheles spp., malaria vectors Culex spp., West Nile vectors Other biting flies Phlebotomus papatasi, Leishmania vector Glossina spp., African trypanosomiasis vectors True bugs Triatoma infestans, Chagas vector Ticks Amblyomma spp., heartwater vectors

A simple view of the world Vector-borne pathogens ^ not so Exposed & Infected Diseased Infectivity < 1 Infectious Onset of symptoms Onset of shedding IncubationClinical disease Infectious periodLatent period Infection HOST

A simple view of the world Vector-borne pathogens Don’t worry about symptoms and disease! ^ not so Exposed & Infected Infectivity < 1 Infectious Onset of shedding Infectious periodLatent period Infection HOST

 H = infectivity to humans x per capita (vector) biting rate A simple view of the world Vector-borne pathogens ^ not so Exposed & Infected Infectivity < 1 Infectious Onset of shedding Infectious periodLatent period Infection HOST

A simple view of the world Vector-borne pathogens ^ not so Exposed & infected (not infectious) Infectious Recovered Susceptible HOST

 V = infectivity to vectors x per capita (vector) biting rate A simple view of the world Vector-borne pathogens ^ not so Exposed & Infected Infectivity < 1 Infectious Infectious period Onset of shedding InfectiousLatent DeathInfection VECTOR

A simple view of the world Vector-borne pathogens ^ not so EHEH IHIH RHRH SHSH EVEV IVIV SVSV VECTOR HOST

A simple view of the world Vector-borne pathogens ^ not so EHEH IHIH RHRH SHSH VECTOR EVEV IVIV SVSV HOST

birth rate per capita mortality rate A simple view of the world Vector-borne pathogens ^ not so per capita birth rate per capita mortality rate 1/latent period 1/infectious period

A simple view of the world Vector-borne pathogens ^ not so EHEH IHIH RHRH SHSH VECTOR EVEV IVIV SVSV HOST

infectivity = proportion of susceptible individuals that become infected, given exposure per capita (vector) biting rate = bites by one individual vector per time unit A simple view of the world Vector-borne pathogens ^ not so  = infectivity x per capita contact rate exposure = bite by I V HOST  = infectivity x per capita (vector) biting rate VECTOR exposure = bite on I H

infectivity = proportion of susceptible individuals that become infected, given exposure per capita (vector) biting rate = bites by one individual vector per unit time A simple view of the world Vector-borne pathogens ^ not so  = infectivity x per capita contact rate exposure = bite by I V infectivity to host = host infections produced per bite by I V on S H  H = bites (potentially infectious to host) by one individual vector per unit time  H I V = bites (potentially infectious to host) per unit time  H I V /N H = bites (potentially infectious to host) per host per unit time  H S H I V /N H = infectious bites per unit time HOST  = infectivity x per capita biting rate

infectivity = proportion of susceptible individuals that become infected, given exposure per capita (vector) biting rate = bites by one individual vector per unit time A simple view of the world Vector-borne pathogens ^ not so  = infectivity x per capita contact rate exposure = bites on I H infectivity to vector = vector infections produced per bite by S V on I H  V = bites (potentially infectious to vector) by one individual vector per unit time  V S V = bites (potentially infectious to vector) per unit time  V S V /N H = bites (potentially infectious to vector) per host per unit time  V S V I H /N H = infectious bites per unit time VECTOR  = infectivity x per capita biting rate

A simple view of the world Vector-borne pathogens ^ not so EHEH IHIH RHRH SHSH VECTOR EVEV IVIV SVSV HOST

A simple view of the world Vector-borne pathogens ^ not so HOST VECTOR

A simple view of the world Vector-borne pathogens ^ not so HOST VECTOR

A simple method for complex models Vector-borne pathogens FV -1 = is the “next generation matrix” For all compartments x i containing infected individuals (ie, E H, I H, E V, I V ), the time derivative can be rewritten as where = the rate of appearance of new infections in compartment x i = the rate of transfer out of compartment x i = the rate of transfer of individuals into compartment x i, other than new infections

A simple method for complex models Vector-borne pathogens FV -1 = is the “next generation matrix” F and V are then the square matrices defined by where and

A simple view of the world Vector-borne pathogens ^ not so For our system, we have

A simple view of the world Vector-borne pathogens ^ not so For our system, we have and we find

A simple view of the world Vector-borne pathogens ^ not so For our system, we have which gives

A simple view of the world Vector-borne pathogens ^ not so For our system, we have “next generation matrix”

A simple view of the world Vector-borne pathogens ^ not so For our system, we have and