
Gonorrhea (Gr) is a common sexually transmitted infection associated with pelvic inflammatory disease, increased susceptibility to HIV infection, and vertical transmission to neonates. The brief duration of untreated Gr, combined with low average rates of sex partner acquisition in the population, make Gr dependent on relatively small numbers of individuals with high rates of sex partner acquisition (socalled "core groups") in order to avoid extinction. Dependence on core groups represents an important vulnerability for Gr, and targeted antimicrobial treatment of these groups has proven an extremely effective means of reducing Gr incidence to very low levels throughout the population. However, rapid emergence of resistance to several antibiotic classes in recent years has been associated with a contemporaneous resurgence in Gr rates. I will use a simple, behaviorstructured transmission model to show that treatment of core groups represents a highly effective strategy in the absence of resistance, but in the presence of antibiotic resistance treatment of core groups enhances dissemination of resistant strains, and undermines control efforts. In models that include two different treatment classes, a more durable reduction in Gr incidence is seen with random allocation of treatment than with sequential use of drug classes, but the preferred strategy involves use of rapid tests and targeted therapy based on microbial drug susceptibility. This model provides insights into the rapid of emergence of resistant Gr, and highlights the importance of developing pointofcare tests for antimicrobial resistance in order to maintain the advances in Gr control achieved over the past 40 years.
We discuss different timeoptimal control strategies for the basic SIR model with mass action contact rate. Among other things, the solution to an optimal control problem will depend on the cost function that the control is designed to minimize. In the literature, optimal vaccinationonly policies and isolationonly policies that minimize cost functions that penalize for using control resources, have been given for the basic SIR model. We discuss the slightly different problem of finding optimal control strategies under the constraint of limited resources. Practically, this can be viewed as finding the best strategy, given that there are a limited amount of funds with which to implement vaccination and/or isolation. In addition to addressing this question for the vaccinationonly and isolationonly models, we also present a solution for two different versions of a combined vaccinationisolation model. First we find the optimal combined policy under the assumption that the total vaccination and isolation resources have been separately allocated. Secondly, we give the solution for the case when only the total amount of resources have been allocated and the policymaker is free to choose how to divide these resources between vaccination and isolation. For example, when planning for an epidemic, funds can be used to stockpile vaccine or prepare isolation facilities. A major advantage to using the basic SIR model to address these questions is that the basic forms of the solutions can be found without using numerical simulations and can often be understood using simple graphical explanations.
The study of pattern formation for chemotaxis PDEs (partial differential equations) started with the identification of blowup solutions. If, however, the model is adapted to allow for global existence of solutions, then another interesting pattern formation process arises. Local maxima form and they show an interaction of merging (two local maxima coagulate) or emerging (a new maximum is formed). This dynamics can lead to steady states, periodic solutions or to (what we think is) chaotic behavior. I will show that this pattern interaction is very typical for a wide variety of chemotaxis models and I will discuss possible ideas on how to understand this complicated pattern interaction.
Joint work with K. Painter and Z. Wang.
With the assumptions that the infectious disease has a fixed latent period and the individuals in the latent period may disperse, we formulate an SIR model for a population living in an npatch environment (cities, towns, or countries, etc.). The model is given by a system of delay differential equations with a fixed delay accounting for the latency and nonlocal terms caused by the mobility of the individuals during the latent period. An expression for the basic reproduction number R_{0} is derived, and the diseasefree equilibrium is shown to be globally asymptotically stable when R_{0} < 1. At least one endemic equilibrium exists and the disease will be uniformly persistent for R_{0} > 1. For a simpler case with 2 patches only, two examples are given to illustrate that the population dispersal rate plays an important role for spread of the disease with latency, from which we can see the joint effect of the disease latency and population mobility on the disease dynamics. In addition to the existences of the diseasefree equilibrium and interior endemic equilibrium, existence of boundary equilibria and their stability are also discussed in both examples.
I analyze and compare two models, a laissezfaire and a Leslietype predatorprey model, with Holling type I functional responses. I show, numerically, that the two models can possess two limit cycles surrounding a stable equilibrium and that these cycles arise in global cyclicfold bifurcations. The Leslietype model may also exhibit supercritical and discontinuous Hopf bifurcations. I then present and analyze a new functional response, built around the arctan, that smoothes the sharp corner in a type I functional response. For this new functional response, both models undergo Hopf, cyclicfold, and Bautin bifurcations.
Indoor residual sprayingspraying insecticide inside houses to kill mosquitosis an important method for controlling malaria vectors in subSaharan Africa. While it has been responsible for suppressing at least one of the malariacarrying mosquito species, in recent years it has received relatively little attention. We propose a mathematical model for both regular and nonfixed spraying, using impulsive differential equations to account for the reduction in the mosquito population. First, we determine the stability properties of the nonimpulsive system. Next, we derive minimal effective spraying intervals and the degree of spraying effectiveness required to control mosquitos when spraying occurs at regular intervals. If spraying is not fixed, then we determine the "next best" spraying times and show that this solution is always suboptimal. We also consider the effects of an increased mosquito birth rate on the prevalence of mosquitos. We show that, if the mosquito birth rate increased by 25%, then the minimal effective spraying period would be reduced by half, whereas if the mosquito birth rate were doubled, then the minimal effective spraying period would be reduced by three quarters. The results are illustrated with numerical solutions. It follows that, although regular spraying is superior to nonfixed spraying, either will result in a significant reduction in the overall number of mosquitos, as well as the number of malaria cases in humans. We thus recommend that the use of indoor spraying be reexamined for widespread application in malariaendemic areas.
How does a given aquatic organism's wiggling result in propulsion? This has been well investigated in fish and in microorganisms such as bacteria where the viscous or inertial terms of the fluid equations can be ignored. Less has been done at intermediate Reynolds' Number, and furthermore, the actual interaction between the organism's musculature and the surrounding fluid is not well understood. In this talk we focus on the swimming behaviour of the nematode, a roundworm.
The immersed boundary method lends itself very well to the study of organism locomotion in fluid. Movement of passive nematodelike structures has been successfully modeled in complex flows. Active swimming of small organisms has also been successfully modeled when the restlength of each muscle segment is prescribed, and an energy minimum for organism configuration obtained. We are interested in modelling the development of swimming motion from rest, when motion is generated by the contraction of innervated muscle segments.
We have developed a threedimensional model for the body structure of the nematode, which explicitly models the organism's musculature. The immersed boundary method is then used to communicate between the nematode body and the surrounding fluid. This model allows us to study how the nematode musculature and surrounding fluid interact to create propulsion of the nematode. It also gives us the ability to pursue fundamental questions about how organism structure affects the swimming motion obtained and the fluid/muscle forces generated.