This app allows you to explore a model with 2 different modes of direct transmission. Read about the model in the “Model” tab. Then do the tasks described in the “What to do” tab.
This is a simple SIR model with births and deaths and 2 types of direct transmission, density-dependent and frequency-dependent (based on the terminology in (Begon et al. 2002) - see Further Information tab for references).
This model has the following compartments:
The processes being modeled are:
The force of infection, denoted by f here, depends on the transmission model. For density-dependent transmission, it is given by f = bd I / A, where A is the area of the scenario under consideration (e.g. a city). For frequency-dependent transmission, we have f = bf I / N, where N is the size of the population, N=S+I+R.
The flow diagram and the set of ordinary differential equations (ODE) which are used to implement this model are as follows:
Flow diagram for this model.
\[\dot S =m - f S + wR - nS\] \[\dot I = f S - g I -nI\] \[\dot R = g I - wR - nR\]
with f = bd I / A for density-dependent transmission and f = bf I / N for frequency-dependent transmission.
Note that quite often, we assume that the area is constant and do not include it in the model or even talk about it. Often, the population size is also assumed to be constant. In that case, the 2 types of transmission models behave the same. But as soon as area or population size change, the model results differ. You can explore the effect of this relationship using this simulation.
The tasks below are described in a way that assumes everything is in units of MONTHS (rate parameters, therefore, have units of inverse months). If any quantity is not given in those units, you need to convert it first (e.g. if it says a year, you need to convert it to 12 months).
simulate_directtransmission.R
. You can call this function directly, without going through the shiny app. Check the help file for the function for more information. If you go that route, you need to use the results returned from this function and produce useful output (such as a plot) yourself.Begon, M., M. Bennett, R. G. Bowers, N. P. French, S. M. Hazel, and J. Turner. 2002. “A Clarification of Transmission Terms in Host-Microparasite Models: Numbers, Densities and Areas.” Epidemiol Infect 129 (1): 147–53.