A primer on the use of probability generating functions in infectious disease modeling

We explore the application of probability generating functions(PGFs)to invasive processes,focusing on infectious disease introduced into large populations.Our goal is to acquaint the reader with applications of PGFs,moreso than to derive new results.PGFs help predict a number of properties about early outbreak behavior while the population is still effectively infinite,including the probability of an epidemic,the size distribution after some number of generations,and the cumulative size distribution of non-epidemic outbreaks.We show how PGFs can be used in both discrete-time and continuous-time settings,and discuss how to use these results to infer disease parameters from observed outbreaks.In the large population limit for susceptible-infected-recovered(SIR)epidemics PGFs lead to survival-function based models that are equivalent to the usual mass-action SIR models but with fewer ODEs.We use these to explore properties such as the final size of epidemics or even the dynamics once stochastic effects are negligible.We target this primer at biologists and public health researchers with mathematical modeling experience who want to learn how to apply PGFs to invasive diseases,but it could also be used in an applications-based mathematics course on PGFs.We include many exercises to help demonstrate concepts and to give practice applying the results.We summarize our main results in a few tables.Additionally we provide a small python package which performs many of the relevant calculations.

Agent-based modeling of COVID-19 outbreaks for New York state and UK:Parameter identification algorithm

This paper uses Covasim,an agent-based model(ABM)of COVID-19,to evaluate and scenarios of epidemic spread in New York State(USA)and the UK.Epidemiological parameters such as contagiousness(virus transmission rate),initial number of infected people,and probability of being tested depend on the region's demographic and geographical features,the containment measures introduced;they are calibrated to data about COVID-19 spread in the region of interest.At the first stage of our study,epidemiological data(numbers of people tested,diagnoses,critical cases,hospitalizations,and deaths)for each of the mentioned regions were analyzed.The data were characterized in terms of seasonality,stationarity,and dependency spaces,and were extrapolated using machine learning techniques to specify unknown epidemiological parameters of the model.At the second stage,the Optuna optimizer based on the tree Parzen estimation method for objective function minimization was applied to determine the model's unknown parameters.The model was validated with the historical data of 2020.The modeled results of COVID-19 spread in New York State and the UK have demonstrated that if the level of testing and containment measures is preserved,the number of positive cases in New York State remain the same during March of 2021,while in the UK it will reduce.

Mathematical models of SIR disease spread with combined non-sexual and sexual transmission routes

The emergence of Zika and Ebola demonstrates the importance of understanding the role of sexual transmission in the spread of diseases with a primarily non-sexual transmission route.In this paper,we develop low-dimensional models for how an SIR disease will spread if it transmits through a sexual contact network and some other transmission mechanism,such as direct contact or vectors.We show that the models derived accurately predict the dynamics of simulations in the large population limit,and investigateℛ0 and final size relations.

The influence of mobility among high-risk populations on HIV transmission in Western Kenya

Western Kenya suffers a highly endemic and also very heterogeneous epidemic of human immunodeficiency virus(HIV).Although female sex workers(FSW)and their male clients are known to be at high risk for HIV,HIV prevalence across regions inWestern Kenya is not strongly correlated with the fraction of women engaged in commercial sex.An agentbased network model of HIV transmission,geographically stratified at the county level,was fit to the HIV epidemic,scale-up of interventions,and populations of FSW in Western Kenya under two assumptions about the potential mobility of FSW clients.In the first,all clients were assumed to be resident in the same geographies as their interactions with FSW.In the second,some clients were considered non-resident and engaged only in interactions with FSW,but not in longer-term non-FSW partnerships in these geographies.Under both assumptions,the model successfully reconciled disparate geographic patterns of FSW and HIV prevalence.Transmission patterns in the model suggest a greater role for FSW in local transmission when clients were resident to the counties,with 30.0%of local HIV transmissions attributable to current and former FSW and clients,compared to 21.9%when mobility of clients was included.Nonetheless,the overall epidemic drivers remained similar,with risky behavior in the general population dominating transmission in highprevalence counties.Our modeling suggests that co-location of high-risk populations and generalized epidemics can further amplify the spread of HIV,but that large numbers of formal FSW and clients are not required to observe or mechanistically explain high HIV prevalence in the general population.