Abstracts Workshop on Mathematical Biology and Nonlinear Analysis, December 19-21 2014

s Workshop on Mathematical Biology and Nonlinear Analysis, December 19-21 2014 Afrah Abdou [email protected] Title: Common Fixed Point for Infinite Mappings in Modular Metric Spaces Afrah A.N. Abdou King Abdulaziz University The notion of a modular metric on an arbitrary set and the corresponding modular spaces, generalizing classical modular over linear spaces like Orlicz spaces, were recently introduced. In this paper, we investigate the existence of common fixed points of a family of infinite multivalued mappings in more general setting in modular metric spaces. Folashade B. Agusto [email protected] Title: Mathematical Model of an Age-Structured Transmission Dynamics of Chikungunya Virus F.B. Agusto, Shamise Easley, Kenneth Freeman and Madison Thomas Austin Peay State University Abstract In this paper, we developed an age-structure deterministic model for the transmission dynamics of chikungunya virus. The model is analyzed to gain insights into the qualitative features of its associated equilibria. Some of the theoretical and epidemiological findings indicates that the model has locally asymptotically stable (LAS) disease free equilibrium when the associated reproduction number is less than unity. Furthermore, the model undergoes the phenomenon of backward bifurcation, where the stable diseasefree equilibrium of the model coexists with a stable endemic equilibrium when the associated reproduction number is less than unity. Further analyzes of the model indicates that the qualitative dynamics (with respect to the existence and asymptotic stability of the associated equilibria and the backward bifurcation property) of the model is not alter by the inclusion of age-structure to the chikungunya virus transmission model. A global uncertainty and sensitivity to determine the impact of the model parameters is implemented using Latin hypercube sampling and partial rank correlation coefficients methods. Following the results from the sensitivity analysis, three control strategies (mosquitoreduction, personal protection, and a universal strategy) are implemented. Numerical simulations indicates that the personal protection strategy is moreIn this paper, we developed an age-structure deterministic model for the transmission dynamics of chikungunya virus. The model is analyzed to gain insights into the qualitative features of its associated equilibria. Some of the theoretical and epidemiological findings indicates that the model has locally asymptotically stable (LAS) disease free equilibrium when the associated reproduction number is less than unity. Furthermore, the model undergoes the phenomenon of backward bifurcation, where the stable diseasefree equilibrium of the model coexists with a stable endemic equilibrium when the associated reproduction number is less than unity. Further analyzes of the model indicates that the qualitative dynamics (with respect to the existence and asymptotic stability of the associated equilibria and the backward bifurcation property) of the model is not alter by the inclusion of age-structure to the chikungunya virus transmission model. A global uncertainty and sensitivity to determine the impact of the model parameters is implemented using Latin hypercube sampling and partial rank correlation coefficients methods. Following the results from the sensitivity analysis, three control strategies (mosquitoreduction, personal protection, and a universal strategy) are implemented. Numerical simulations indicates that the personal protection strategy is more effective than the mosquito-reduction strategy and that the universal strategy is the most effective strategy in reducing chikungunya disease burden. Eric Avila-Vales [email protected] Title: Dynamics of an SIR epidemic model with nonlinear incidence rate, vertical transmission vaccination for the newborns and the capacity of treatment We study the dynamics of an SIR epidemic model with nonlinear incidence rate, vertical transmission vaccination for the newborns and the capacity of treatment. Treatment takes into account the limitedness of the medical resources and the efficiency of the supply of available medical resources. Under some conditions we prove the existence of backward bifurcation, the stability and the direction of Hopf bifurcation. We also explore how the mechanism of backward bifurcation affects the control of the infectious disease. Numerical simulations are presented to illustrate the theoretical findings. Alfonso Castro [email protected], [email protected] Title: Existence of non-degenerate continua of singular radial solutions for several classes of semilinear elliptic problems We establish the existence of countably many branches of uncountably many solutions to elliptic boundary value problems with subcritical, and sub-super critical growth. We also prove the existence of two branches of uncountably many solutions to a problem with jumping nonlinearities. This case is remarkable since, generically, this problem has only finitely many regular solutions. Jing Chen [email protected] Title: Modeling the geographic spread of rabies in China Abstract: Human rabies is one of the major public health problems in China. In the last 20 years or so, rural communities and areas in Mainland China invaded by rabies are gradually and significantly enlarged. Dogs are the main infection source, which contribute 85%-95% of human cases in China. Some provinces such as Shaanxi and Shanxi, used to be rabies free, have increasing numbers of human infections cases now. Recent phylogeographical analyses of rabies virus clades indicate that the human rabies cases in different and geographically unconnected provinces in China are epidemiologically related. In order to investigate how the movement of dogs changes the geographically interprovincial spread of rabies in Mainland China, we propose a multi-patch model for the transmission dynamics of rabies between dogs and humans, in which each province is regarded as a patch. In each patch the submodel consists of susceptible, exposed, infectious, and recovered subpopulations of both dogs and humans and describes the spread of rabies among dogs and from infectious Human rabies is one of the major public health problems in China. In the last 20 years or so, rural communities and areas in Mainland China invaded by rabies are gradually and significantly enlarged. Dogs are the main infection source, which contribute 85%-95% of human cases in China. Some provinces such as Shaanxi and Shanxi, used to be rabies free, have increasing numbers of human infections cases now. Recent phylogeographical analyses of rabies virus clades indicate that the human rabies cases in different and geographically unconnected provinces in China are epidemiologically related. In order to investigate how the movement of dogs changes the geographically interprovincial spread of rabies in Mainland China, we propose a multi-patch model for the transmission dynamics of rabies between dogs and humans, in which each province is regarded as a patch. In each patch the submodel consists of susceptible, exposed, infectious, and recovered subpopulations of both dogs and humans and describes the spread of rabies among dogs and from infectious dogs to humans. The existence of the disease-free equilibrium and the basic reproduction number will be discussed and calculated, and how the moving rates of dogs between patches affect the basic reproduction number will be studied. To investigate the rabies virus clades lineages observed in the phylogeographical analyses, the two-patch model will be used to simulate the human rabies data to study the inter-provincial spread of rabies between Guangxi and Guizhou, Fujian and Hebei and Sichuan and Guizhou, respectively. In order to reduce and prevent geographical spread of rabies in China, our results suggest that the management of dog market and trade need to be regulated and transportation of dogs need to be better monitored and under constant surveillance. Jerome Coville [email protected] Title: Persistence criteria in some nonlocal model in unbounded domain and applications I will report on a recent study made in collaboration with H.Berestycki and H. Vo concerning persistence criteria in some nonlocal models in € R . I will first present the persistence criteria that we have obtained and then, I will discuss the behaviour of this criteria with respect to the dispersal operator when it is conditioned by a cost function. Donald L. DeAngelis [email protected] Title: Effects of dispersal on total biomass in a patchy, heterogeneous system: Analysis and experiment D. L. DeAngelis, U. S. Geological Survey and University of Miami, Bo Zhang, University of Miami, Wei-Ming Ni, University of Minnesota and Center for Partial Differential Equations, East China Normal University, Abstract A recent result for a reaction-diffusion partial differential equation is that a population diffusing at an intermediate rate in an environment in which resources vary spatially will reach a higher total equilibrium biomass than the population in an environment in which the same total resources are distributed homogeneously. This has so far been proven only for the case in which the reaction term has only one parameter, m(x), varying with spatial distance x, which serves as both the growth rate coefficient and carrying capacity of the population. This result is too limited to apply to real populations. In order to make the model more relevant for ecologists, we have extended it to a logistic reaction term, with independent parameters, r(x) for intrinsic growth rate, and K(x) for carrying capacity. When r(x) and K(x) are proportional, the logistic equation takes a particularly simple form, and the earlier results still hold. We have proven the results for the more general case of a non-negative correlation between r(x) and K(x). We review natural and laboratory systems to which theseA recent result for a reaction-diffusion partial differential equation is that a population diffusing at an intermediate rate in an environment in which resources vary spatially will reach a higher total equilibrium biomass than the population in an environment in which the same total resources are distributed homogeneously. This has so far been proven only for the case in which the reaction term has only one parameter, m(x), varying with spatial distance x, which serves as both the growth rate coefficient and carrying capacity of the population. This result is too limited to apply to real populations. In order to make the model more relevant for ecologists, we have extended it to a logistic reaction term, with independent parameters, r(x) for intrinsic growth rate, and K(x) for carrying capacity. When r(x) and K(x) are proportional, the logistic equation takes a particularly simple form, and the earlier results still hold. We have proven the results for the more general case of a non-negative correlation between r(x) and K(x). We review natural and laboratory systems to which these results are relevant and discuss the implications of the results to population theory and conservation ecology. We show some preliminary tests of this in an experimental laboratory system Keng Deng [email protected] Title: Asymptotic behavior for a reaction-diffusion population model with delay In this paper, we study a reaction-diffusion population model with time delay. We establish a comparison principle for coupled upper/lower solutions and prove the existence/uniqueness result for the model. We then show the global asymptotic behavior of the model. Yihong Du [email protected] Title: Long-time behavior of nonlinear free boundary problems In this talk I'll report some recent results obtained with collaborators on a one dimensional nonlinear free boundary problem of the form u_t-u_{xx}=f(u), where $x$ varies over the changing interval $(g(t), h(t))$, and $x=g(t)$, $x=h(t)$ are free boundaries whose evolution is governed by $g'(t)=-\mu u_x(t, g(t))$, $h'(t)=\mu u_x(t, h(t))$, $u(t,g(t))=u(t, h(t))=0$. For monotstable, bistable, and combustion types of $f(u)$, we obtain a rather complete description of the longtime behavior of the positive solutions of this problem, which may be viewed as a model for the spreading of a new or invasive species, with population density $u$ and spreading fronts $x=g(t)$ and $x=h(t)$. Arnaud Ducrot , [email protected] Title: Asymptotic behaviour of travelling waves for the delayed Fisher-KPP equation In this talk we discuss the behaviour of travelling wave solutions for the diffusive logistic equation with time delay. Using a phase plane analysis we prove the existence of travelling wave solution for each wave speed € c ≥ 2 . We show that for each given and admissible wave speed, such travelling wave solutions converge to a unique maximal wavetrain. As a consequence the existence of a nontrivial maximal wavetrain is equivalent to the existence of travelling wave solution non-converging to the stationary state € u =1. This is a joint work with Grégoire Nadin, Univ. Paris 6. William Fagan [email protected] Title: Animal Movement: Memory, Learning, and Autocorrelation Real landscapes are dynamic in space and time, and the scales over which such variation occurs can determine the success of different conservation strategies for resident species. Within such landscapes, real species rely on a variety of individual-level behaviors for movement and navigation. Movement behaviors such as long-distance searching and fine-scale foraging are often intermixed but operate on vastly different spatial and temporal scales. Individual experience, life-history traits, and resource dynamics combine to shape population-level patterns such as range residency, migration, and nomadism. I will discuss how a combination of empirical movement data and powerful statistical approaches (“animal models” of pedigree effects; semi-variance functions that leverage autocorrelations present in animal tracking data) can be used to inform our understanding of animal movement on large spatial scales. Animal models can be used to control for genetic variation among individuals while exploring alternative hypotheses about other factors, such as learning and experience, that influence animal movement. Semi-variance approaches can be used to identify multiple movement modes and solve the sampling rate problem for tracking data, allowing for the identification of critical scales for movement and the delineation of animal home ranges. Together these approaches can help reveal the relationships among individual movements, landscape dynamics, and population level patterns. Louis Fan [email protected] Title: Stochastic spatial models for chemotaxis This talk highlights the connections between stochastic particle systems and a variety of PDE models for chemotaxis. These PDE models include the classical Keller-Segel model and its variations studied in the past decade. I will also mention the biological motivations of these models and the implications of advances in probability theory on the study of the stochastic particle systems. Zhilan Feng [email protected] Modeling the synergy between HSV-2 and HIV and potential impact of HSV-2 therapy Zhilan Feng, Zhipeng Qiu, Zi Sang, Christina Lorenzo and John Glasser Mathematical models are formulated to study the joint disease dynamics of HIV and HSV-2. The model takes into account the fact that an HSV-2 infection may increase susceptibility to HIV infection and that co-infection of both diseases may increase infectiousness. Heterogeneous mixing between a male group and two female groups is also included. The models are used to investigate the role of antiviral treatment of people with HSV-2 in mitigating the incidence of HIV in populations where both pathogens occur, and to demonstrate how the disease dynamics can be influenced by the gender structure of the population. Daozhou Gao [email protected] Title: Tragedy of the commons in antibiotic use The emergence and spread of antibiotic resistance has become a major public health threat. Individual incentives lead to the overuse of antibiotics, whereas restrictions to limit use would benefit society as a whole. Under such circumstances, the goals of the individual conflict with the goals of the community, a “tragedy of the commons” may result. The decision to prescribe antibiotics can be analyzed as a mathematical game through the analysis of individual incentives and community outcomes. We developed several mathematical models of the transmission of antibiotic resistance, and found that a tragedy of the commons can occur in both single disease (treatment of mild or early infection) and multiple diseases (treatment of one disease can lead to drug resistance in another organism) settings. Juan Gutierrez [email protected] Title: Hemodynamic model of malaria infection with detailed immune response Half the world population is currently at risk of malaria infection, with 200 million clinical cases and 600,000 deaths in 2012. Even though this disease has attracted substantial research resources in the last century, the detailed characterization of the dynamics of malaria is still an open question. Existing mathematical models of malaria infection are rudimentary, and lack the immune data to expand the level of detail to useful predictive levels. The Malaria Host Pathogen Interaction Center (MaHPIC), a research consortium comprised by UGA, Emory, GT, and CDC is producing information about the disease at unprecedented levels of detail. In this talk I will present recent developments by our MaHPIC group in the mathematical modeling of the blood stage of malaria infection using a coupled system of differential equations comprised of two transport PDEs and a set of ODEs. I will also present the challenges in calibrating this type of model with 'omic technologies (transcriptomics, lipidomics, proteomics, metabolomics, and clinical data). Our preliminary model is able to reproduce the clinical presentation of malaria: severe anemia on first infection, and coexistence of host and parasites in subsequent infections. Xiaoqing He [email protected] Title: Global dynamics of the two-species Lotka-Volterra competition-diffusion system In this talk, we investigate the combined effects of diffusion, spatial variation and competition ability on the global dynamics of a classical Lotka-Volterra competition-diffusion system. We establish the main results which determine the global asymptotic stability of semi-trivial as well as co-existence steady states. Hence a complete understanding of the change in dynamics is obtained immediately. Carol Horvitz [email protected] Title: Transient elasticities and the expected effects of an insect bio-control on the short term dynamics of an invasive pest plant in Hawaii Carol C. Horvitz University of Miami Julie S. Denslow, Institute of Pacific Islands Forestry, USDA Forest Service Orou Gaoue, University of Hawaii at Manoa Tracy Johnson, Institute of Pacific Islands Forestry, USDA Forest Service The goal bio-control is to reduce populations of targeted pests. Success has most often been measured by whether or not the control agent achieves a selfsustaining population. However, a better criterion would be whether or not the agent reduces the targeted pest population in the short and long term, measured respectively by transient and asymptotic dynamics. Even in the absence of biocontrol agents, pest dynamics vary over space and time. Thus, it is important, as a first step, to obtain and analyze pre-release demographic data on the pest in the context of such variation. Here, to address this step for an invasive pest plant, we model pre-release dynamics based on detailed spatially replicated, demographic data collected over multiple pre-release years. We utilize an integral projection model of population dynamics, combining one continuous domain for larger individuals (where the state variable is stem diameter at breast height) with six discrete stages for small, seedling-sized individuals. We perform proportional sensitivity analyses to determine expected effects of an insect biocontrol agent (Tectococcus ovatus) on the transient dynamics of Psidium cattleianum (Strawberry guava) at each of 4 sites in Hawaii. We found that prerelease asymptotic annual per-capita rates of population growth (λ) vary across the sites from 0.99 to 1.20. We explore, through site-specific transient elasticity analyses, how the expected impact of consumers would be dynamic in the short term and how they would differ among sites. The changing stage distribution of a population during the transient phase coupled with the analytical transient elasticities drive these changes and differences. Jon Jacobsen [email protected] Title: Integrodifference Models for Persistence in Temporally Varying River Environments We consider integrodifference population models for growth and dispersal in the presence of advective flow and study population persistence in the context of both periodic and random kernel parameters. For the random setting we consider two persistence metrics and show they are mathematically equivalent. This is joint work with Yu Jin and Mark Lewis. Yun Kang [email protected] Title: A two-patch prey-predator model with dispersal in predators driven by the strength of prey-predator interactions Foraging movements of predator play an important role in population dynamics of prey-predator interactions, which also can be considered as mechanisms that contributes to spatial self-organization. In nature, there are many examples of prey-predator interactions where prey is immobile while predator disperses between patches non-randomly through different factors such as stimuli following the encounter of a prey. In this work, we formulate a Rosenzweig-MacArthur prey-predator two patch model with mobility only in predator and the assumption that predators move towards patches with more concentrated prey-predator interactions. We provide local and global analysis of our model. Our analytical results combined with bifurcation diagrams suggest that: (1) dispersal may stabilize or destabilize the coupled system; (2) dispersal may general multiple interior equilibria that lead to rich bistable dynamics or may destroy interior equilibria that lead to the extinction of predator in one patch or both patches; (3) Under certain conditions, the large dispersal can promote the permanence of the system. In addition, we compare the dynamics of our model to the classic two patch model to obtain a better understanding how different dispersal strategies may have different impacts on the dynamics and spatial patterns. Michael R. (Mike)” Kelly [email protected] Title: Optimal fish harvesting for a population modeled by a nonlinear, parabolic partial differential equation As the human population continues to grow, there is a need for better management of our natural resources in order for our planet to be able to produce enough to sustain us. One important resource we must consider is marine fish populations. The tool of optimal control is used to investigate harvesting strategies for maximizing yield of a fish population in a heterogeneous, finite domain. We determine whether these solutions include notake marine reserves as part of the optimal solution. The fishery stock is modeled using a nonlinear, parabolic partial differential equation with logistic growth, movement by diffusion and advection, and with Robin boundary conditions. The objective for the problem is to find the harvest rate that maximizes the discounted yield. Optimal harvesting strategies are found numerically. Yang Kuang [email protected] Title: A data-validated density-dependent diffusion model of glioblastoma growth Glioblastoma multiforme is an aggressive brain cancer that is extremely fatal. It is characterized by both proliferation and large amounts of migration, which contributes to the difficulty of treatment. Previous models of this type of cancer growth often include two separate equations to model proliferation or migration. We propose a single equation which uses density dependent diffusion to capture the behavior of both proliferation and migration. We analyze the model to determine the existence of traveling wave solutions. To support the viability of the density-dependent diffusion function chosen, we compare our model with well-known in vitro experimental data. This is a joint work with Tracy Stepien and Erica Rutter King Yeung (Adrian) Lam [email protected] Title: Resident-invader dynamics in infinite-dimensional dynamical systems We study the resident-invader dynamics for a class of models of spatial population with a one-dimensional trait, or strategy. We prove various global dynamical results on coexistence and exclusion, based on local invasibility criterions including the notions of evolutionary stability and convergence stability in adaptive dynamics. Applications of our abstract results include reactiondiffusion-advection models and nonlocal dispersal models. This leads to the novel conclusion that a recently established evolutionarily stable dispersal strategy in [Lam-Lou, J. Math. Biol. (2013)] is a neighborhood invader strategy. This is joint work with R.S. Cantrell (Miami) and C. Cosner (Miami). Dung Le [email protected] Title: Global existence of classical solutions to cross diffusion systems of more than 2 equations given on a planar domain The global existence of classical solutions to cross diffusion systems of more than 2 equations given on a planar domain is established. The results can apply to generalized Shigesada-Kawasaki-Teramoto (SKT) and food pyramid models whose diffusion and reaction can have polynomial growth of any order. If time permits I will also talk about the existence of their attractors and how the results can be extended to arbitrary dimension domains. Suzanne Lenhart [email protected] Title: Modeling of Johne's disease in dairy cattle Johne's disease in dairy cattle is a chronic infectious disease in the intestines caused by the bacilli, Mycobacterium avium ssp. paratuberculosis. We have modeled this disease with several approaches to illustrate different features. A system of difference equations represented an epidemiological situation in dairy farm to compare the effects of two types of diagnostic tests. Then an agent based model at the farm level was developed to see the effects of stochasticity. Lastly, a PDE/ODE model illustrated a novel way to link a within-host model with an epidemiological model Simon Levin [email protected] Title: Critical transitions in space and time A fundamental characteristic of complex systems, and especially of complex adaptive systems, is the potential for sudden shifts from one basin of attraction to another in relation to temporal, spatial and other gradients. Many, but not all, such transitions are anticipated by more modest, reversible changes that warn of impending irreversible shifts. This lecture will discuss some examples from ecological systems, with possible extensions to social dynamics and a range of other applications. Xing Liang [email protected] Title: Spreading speed of integro-difference models in periodic habitat The main aim of this work is to understand what kind of diffusion mechanism can guarantee the existence of the spreading speed for an evolution system in the periodic media. The following three parts are included in this work: First, the uniform irreducibility of Radon measures on the circle is defined, and it is proved that the generalized convolution operator generated by a uniformly irreducible and nonnegative measure has the principal eigenvalue. Next, an abstract framework of the spreading speeds for general spatially periodic noncompact systems is established, the variational formula of the spreading speeds is given under the hypothesis that the principal eigenvalues of the linearized systems exist. Finally, based on the above two preparations, it is shown that the uniform irreducibility of the diffusion can guarantee the existence of the spreading speed in the periodic media through investigating the integro-difference system. Rongsong Liu [email protected] Title: An advection and age-structured approach to modeling bird migration and indirect transmission of avian influenza We model indirect transmission, via contact with viruses, of avian influenza in migratory and non-migratory birds, taking into account age-structure. Migration is modeled via a reaction-advection equation on a closed loop parameterized by arc length (the migration flyway) that starts and ends at the location where birds breed in summer. Our modeling keeps the birds together as a flock, the position of which is implicitly determined and known for all future time. Births occur when the flock passes the breeding location and are modeled using ideas from impulsive differential equations. For a migratory species the model derivation starts from age structured reaction-advection equations with location-dependent parameters that describe local conditions. In the derivation of delay equations for the time-dependent variables representing numbers of juvenile and adult birds, these location-dependent parameters are evaluated at the flock's position, so that seasonal effects are captured indirectly but through rigorous modeling whereby we keep track of the flock's exact position and local conditions there. Sufficient conditions are obtained for the local stability of the disease-free equilibrium (for a non-migratory species) and for the disease-free periodic solution (for a migratory species). Julián López Gómez [email protected] A Title: The theorem of characterization of the maximum principle for periodic-parabolic systems of cooperative type and the existence of principal eigenvalues for a class of associated weighted boundary value problems. We are generalizing a classical result by I. Antón and the author published in 1996 in the Proceedings of the First World Congress of Nonlinear Analysts, Tampa, Florida. August 19-26, 1992, when Hurricane Andrew heated Florida. This result generalized the theorem of characterization of the maximum principle by M. Molina-Meyer and the author in DIE 1994, later generalized by the author and H. Amann in JDE 1998, who was the leitmotiv of the book of the author on Second Order Linear Elliptic Operators, WSP, Singapore 2013. The main result is used to derive a number of improvements of some very classical, extremely elegant, results pioneered by R. S. Cantrell together with K. Schmitt in SIAM JMA 1986. Oleg Makarenkov [email protected] Title: Topological degree in the generalized Gause prey-predator model We consider a generalized Gause prey-predator model with T-periodic continuous coefficients. In the case where the Poincare map P over time T is well defined, the result of the paper can be explained as follows: we locate a subset U of R such that the topological degree d(I-P,U) equals +1. The novelty of the paper is that the later is done under only continuity and (some) monotonicity assumptions for the coefficients of the model. The approach uses a perturbation technique to locate a trapping region for the perturbed system and discovers suitable a-priori estimates that make it possible to catch the periodic solutions when the perturbation gradually disappears. We then introduce an integral operator that corresponds to the periodic problem for the system under consideration and use a version of Krasnoselski's irreversibility theorem to compute the topological degree of this operator. The formula for the topological degree can be used by future researchers to investigate periodic solutions of Gause prey-predator models with delays and other functionals. The full paper is available at http://dx.doi.org/10.1016/j.jmaa.2013.08.052. The work is partially supported by RFBR Grant 13-01-00347 Raul Manasevich [email protected] Title: Solutions with a prescribed number of zeros for a nonlinear elliptic equation with weights on