Modeling Super-spreading Events for Infectious Diseases: Case Study SARS

Super-spreading events for infectious diseases occur when some infected individuals infect more than the average number of secondary cases. Several super-spreading individuals have been identified for the 2003 outbreak of severe acute respiratory syndrome (SARS). We develop a model for super-spreading events of infectious diseases, which is based on the outbreak of SARS. Using this model we describe two methods for estimating the parameters of the model, which we demonstrate with the small-scale SARS outbreak at the Amoy Gardens, Hong Kong, and the large-scale outbreak in the entire Hong Kong Special Administrative Region. One method is based on parameters calculated for the classical susceptible - infected - removed (SIR) disease model. The second is based on parameter estimates found in the literature. Using the parameters calculated for the SIR model, our model predicts an outcome similar to that for the SIR model. On the other hand, using parameter estimates from SARS literature our model predicts a much more serious epidemic.Comment: 8 pages, 1 figure, 7 table

Topological aspects of poset spaces

We study two classes of spaces whose points are filters on partially ordered sets. Points in MF spaces are maximal filters, while points in UF spaces are unbounded filters. We give a thorough account of the topological properties of these spaces. We obtain a complete characterization of the class of countably based MF spaces: they are precisely the second-countable T_1 spaces with the strong Choquet property. We apply this characterization to domain theory to characterize the class of second-countable spaces with a domain representation.Comment: 29 pages. To be published in the Michigan Mathematical Journa

Reverse mathematics and equivalents of the axiom of choice

We study the reverse mathematics of countable analogues of several maximality principles that are equivalent to the axiom of choice in set theory. Among these are the principle asserting that every family of sets has a $\subseteq$-maximal subfamily with the finite intersection property and the principle asserting that if $P$ is a property of finite character then every set has a $\subseteq$-maximal subset of which $P$ holds. We show that these principles and their variations have a wide range of strengths in the context of second-order arithmetic, from being equivalent to $\mathsf{Z}_2$ to being weaker than $\mathsf{ACA}_0$ and incomparable with $\mathsf{WKL}_0$. In particular, we identify a choice principle that, modulo $\Sigma^0_2$ induction, lies strictly below the atomic model theorem principle $\mathsf{AMT}$ and implies the omitting partial types principle $\mathsf{OPT}$

Parameter Identification for a Stochastic SEIRS Epidemic Model: Case Study Influenza

A recent parameter identification technique, the local lagged adapted generalized method of moments, is used to identify the time-dependent disease transmission rate and time-dependent noise for the stochastic susceptible, exposed, infectious, temporarily immune, susceptible disease model (SEIRS) with vital rates. The stochasticity appears in the model due to fluctuations in the time-dependent transmission rate of the disease. All other parameter values are assumed to be fixed, known constants. The method is demonstrated with US influenza data from the 2004–2005 through 2016–2017 influenza seasons. The transmission rate and noise intensity stochastically work together to generate the yearly peaks in infections. The local lagged adapted generalized method of moments is tested for forecasting ability. Forecasts are made for the 2016–2017 influenza season and for infection data in year 2017. The forecast method qualitatively matches a single influenza season. Confidence intervals are given for possible future infectious levels

Reverse mathematics and uniformity in proofs without excluded middle

We show that when certain statements are provable in subsystems of constructive analysis using intuitionistic predicate calculus, related sequential statements are provable in weak classical subsystems. In particular, if a $\Pi^1_2$ sentence of a certain form is provable using E-HA${}^\omega$ along with the axiom of choice and an independence of premise principle, the sequential form of the statement is provable in the classical system RCA. We obtain this and similar results using applications of modified realizability and the \textit{Dialectica} interpretation. These results allow us to use techniques of classical reverse mathematics to demonstrate the unprovability of several mathematical principles in subsystems of constructive analysis.Comment: Accepted, Notre Dame Journal of Formal Logi

Stationary and convergent strategies in Choquet games

If NONEMPTY has a winning strategy against EMPTY in the Choquet game on a space, the space is said to be a Choquet space. Such a winning strategy allows NONEMPTY to consider the entire finite history of previous moves before making each new move; a stationary strategy only permits NONEMPTY to consider the previous move by EMPTY. We show that NONEMPTY has a stationary winning strategy for every second countable T1 Choquet space. More generally, NONEMPTY has a stationary winning strategy for any T1 Choquet space with an open-finite basis. We also study convergent strategies for the Choquet game, proving the following results. (1) A T1 space X is the open image of a complete metric space if and only if NONEMPTY has a convergent winning strategy in the Choquet game on X. (2) A T1 space X is the compact open image of a metric space if and only if X is metacompact and NONEMPTY has a stationary convergent strategy in the Choquet game on X. (3) A T1 space X is the compact open image of a complete metric space if and only if X is metacompact and NONEMPTY has a stationary convergent winning strategy in the Choquet game on X.Comment: 24 page

Reverse mathematics and properties of finite character

We study the reverse mathematics of the principle stating that, for every property of finite character, every set has a maximal subset satisfying the property. In the context of set theory, this variant of Tukey's lemma is equivalent to the axiom of choice. We study its behavior in the context of second-order arithmetic, where it applies to sets of natural numbers only, and give a full characterization of its strength in terms of the quantifier structure of the formula defining the property. We then study the interaction between properties of finite character and finitary closure operators, and the interaction between these properties and a class of nondeterministic closure operators.Comment: This paper corresponds to section 4 of arXiv:1009.3242, "Reverse mathematics and equivalents of the axiom of choice", which has been abbreviated and divided into two pieces for publicatio

The modal logic of Reverse Mathematics

The implication relationship between subsystems in Reverse Mathematics has an underlying logic, which can be used to deduce certain new Reverse Mathematics results from existing ones in a routine way. We use techniques of modal logic to formalize the logic of Reverse Mathematics into a system that we name s-logic. We argue that s-logic captures precisely the "logical" content of the implication and nonimplication relations between subsystems in Reverse Mathematics. We present a sound, complete, decidable, and compact tableau-style deductive system for s-logic, and explore in detail two fragments that are particularly relevant to Reverse Mathematics practice and automated theorem proving of Reverse Mathematics results