1. Course Background & Purposes

It was known that mathematical modeling can play an important role in the understanding of mechanisms in ecological/epidemic systems. This summer course is intended to introduce ecological models with nutrient(s) storage, data-Fitting, parameter estimation, and model simulations, techniques of modeling. We will also focus on mathematical models for cholera and COVID-19. The target reproduction number will also be included since it is a crucial population threshold quantity that measures the level of control efforts required to achieve a particular prevention, intervention, or control objective. Finally, we will introduce the maximum principle and the positive invertibility of general linear elliptic BVP's. This theory is crucial in the analysis of reaction–diffusion–advection models, which are common for spatially variable systems.

2. Course Outline & Descriptions

Topic 1. Ecological modeling on stoichiometry and Algae–Bacteria Interactions

(1)   Title: Stoichiometric modelling and novel analysis with applications

Speaker: Hao Wang (University of Alberta)

Time: 10:00-12:00, June 29, 2023

Abstract: I will present the background of ecological stoichiometry and a series of mechanistic modelling with modern methods of analysis. Applications are broad and have practical importance for policy makers and stakeholders.

(2)   Title: Introduction of monotone dynamical system and its application to modeling of harmful algae.

Speaker: Feng-Bin Wang (Chang Gung University)

Time: 14:00-16:00, June 29, 2023

Abstract: In this talk, I shall first introduce theory of monotone dynamical system and comparison principle for systems coupled with ODEs/reaction-diffusion equations. Then I shall introduce a two-vessel gradostat models describing the dynamics of harmful algae, in which one vessel represents a small cove connected to a larger lake.

We show that the algae is washed out eventually if the basic reproduction ratio R0 is less than unity, while there

exists at least one positive state and algal blooms occur when R0 is greater than unity. Finally, I shall introduce and analyze a advection-dispersion-reaction models arising from the dynamics of harmful algae in flowing-water habitats where a main channel is coupled to a hydraulic storage zone, representing an ensemble of fringing coves on the shoreline.

References:

(1)H. L. Smith, Monotone Dynamical Systems: an Introduction to the Theory of Competitive and Cooperative Systems, Math. Surveys Monogr 41, American Mathematical Society Providence, RI, 1995.

(2)Sze-Bi Hsu, Feng-Bin Wang and Xiao-Qiang Zhao, Global Dynamics of Zooplankton and Harmful Algae in Flowing Habitats, Journal of Differential Equations, Vol. 255 (2013), pp. 265-297. 

(3) Feng-Bin Wang, Sze-Bi Hsu and Wendi Wang, Dynamics of harmful algae with seasonal temperature variations in the cove-main lake, Discrete and Continuous Dynamical System Series-B, Vol. 21, 2016, pp. 313-335.

(3)   Title: Comparing the Droop and Monod forms for resource explicit population modelling.

Speaker: Christopher Heggerud (University of Alberta)

Time: 10:00 -11:00, June 30, 2023  (online lecture)

Abstract: Almost all biological models use either the Droop or Monod framework to describe the resource-based growth of a living organism. The Monod form is mathematically simpler, but in certain environments misses important mechanistic behaviors, such as internal stoichiometry. Furthermore, empirical evidence suggests the Droop form describes data more accurately than the Monod form. We begin by summarizing ecological stoichiometric and the history of resource explicit modelling. Then, focusing on phytoplankton, we illustrate the underlying logics behind the Monod and Droop forms via conceptual comparison, experimental data validation, transient, and asymptotic dynamics. Finally, we provide a summary of the appropriate uses for each framework.

(4)   Title: Algae–Bacteria Interactions with Nutrients and Light: A Reaction–Diffusion–Advection Model

Speaker: Yawen Yan(University of Alberta)

Time:11:00 -12:00, June 30, 2023  (online lecture)

Abstract: There are many ecosystem functions performed by bacteria and algae, both beneficial and harmful. This makes it important to determine the conditions where the two can coexist, as well as where one can invade an ecosystem in which the other is dominant. This is a complex problem, as bacteria and algae have multifaceted interactions (both bottom-up control and competition for nutrients), which are influenced by the spatial and abiotic factors of their aquatic environment. In this talk, we will present a proposed reaction-diffusion-advection system for modeling algae-bacteria interactions in a poorly mixed water column containing nutrients and light. We rigorously derive basic ecological reproductive indices for the invasion of algae and bacteria into aquatic ecosystems. By analyzing nonnegative steady-state solutions, we obtain all possible outcomes for the survival or extinction of algae and bacteria. Additionally, we investigate the influence of both spatial and abiotic factors on the dynamics of algal and bacterial populations. Our results show that bacteria effectively reduce the biomass of algae and prevent them from moving to the water surface and ultimately reduce the probability of harmful algal blooms.

(5)   Title: Applications of nonlinear eigenvalue problem to ecological models

Speaker: Feng-Bin Wang (Chang Gung University)

Time: 14:00-16:00, June 30, 2023

Abstract:

The basic reproduction ratio R0 is an important thredhold value in population biology, and it is defined as the spectral radius of the next-generation operator. The next-generation operator is usually defined essentially from the linearization of the model system around the trivial/semi-trivial solution, and hence, it is usually a linear operator. For some ecological models with nutrient(s) storage in a spatially variable habitat, however, the technique of linearization is not applicable due to the singularity at trivial/semi-trivial solution. To overcome this difficulty, I shall introduce a nonlinear eigenvalue problem arising from the associated homogeneous system (of degree one) around the trivial/semi-trivial solution. Then the associated principal eigenvalue, given by a recent Krein-Rutman type theorem involving two separate cones, is shown to characterize the threshold for persistence/extinction of species in the ecosystem. Furthermore, R0 is defined as the cone spectral radius of a homogeneous/nonlinear next-generation operator, and R0-1 and the aforementioned principal eigenvalue have the same sign.

References:

(1)J. Mallet-Paret, and R.D. Nussbaum, Generalizing the Krein-Rutman theorem, measures of noncompactness and the fixed point index, Journal of Fixed Point Theory and Applications, Vol.7, 2010, pp.103-143

(2)Sze-Bi Hsu, King-Yeung Lam, and Feng-Bin Wang, Single species growth  consuming inorganic carbon with internal storage in a poorly mixed habitat, Journal of Mathematical Biology, Vol. 75, 2017, pp.1775-1825.

(3)Feng-Bin Wang, Lei Zhang, and Xiao-Qiang Zhao, Basic reproduction ratios for time-periodic homogeneous evolution systems, SIAM Journal on Applied Mathematics, Accepted.

Topic 2. Data-driven Indices and techniques of modeling Infectious Diseases

7/12  https://nationaltaiwanuniversity-ksz.my.webex.com/nationaltaiwanuniversity-ksz.my/j.php?MTID=m31892abf52ba3a82864c502e6bbf04da

7/13  https://nationaltaiwanuniversity-ksz.my.webex.com/nationaltaiwanuniversity-ksz.my/j.php?MTID=med6c3d112b1178868771897579c8c68b

(1)   Title: Data-driven Indices of Infectious Diseases

Speaker: Naveen K. Vaidya(San Diego State University)

Time: 10:00 -12:00, July 12, 2023

Abstract: In this lecture, I will present how critical indices of infectious disease transmission, such as Reproduction Numbers, can be computed using available data. I will also discuss how confidence intervals of these estimates of indices can be calculated.

(2)   Title: Techniques of Modeling Infectious Diseases

Speaker: Naveen K. Vaidya(San Diego State University)

Time: 14:00-16:00, July 12, 2023

Abstract:In this lecture, I will discuss the ways of developing various mathematical models, including difference and differential equation systems, to describe the transmission dynamics of infectious diseases. I will mainly focus on COVID-19, HIV, and dengue diseases. I will also show the method of formulating reproduction from the developed models.

(3)   Title: Data-Fitting, Parameter Estimation, and Model Simulations

Speaker: Naveen K. Vaidya(San Diego State University)

Time: 10:00 -12:00, July 13, 2023

Abstract: In this lecture, I will present the computational methods formodel simulations. I will discuss how the data fitting of the developed models can be used to estimate critical model parameters. For estimates of these parameters, I will also compute confidence intervals. 

(4)   Title: Uncertainly and Sensitivity Analysis

Speaker: Naveen K. Vaidya(San Diego State University)

Time: 14:00-16:00, July 13, 2023

Abstract: In this lecture, I will discuss the uncertainty and sensitivity analysis methods of model output on parameters. I will explore both the local and global sensitivity of the models.

Topic 3.Target reproduction numbers and mathematical modeling of cholera and COVID-19

7/19  https://nationaltaiwanuniversity-ksz.my.webex.com/nationaltaiwanuniversity-ksz.my/j.php?MTID=md4f66d0e097e3f4e109fc1b3553eba3a

7/20  https://nationaltaiwanuniversity-ksz.my.webex.com/nationaltaiwanuniversity-ksz.my/j.php?MTID=m374caf8b3020e0636456e890f8c5d225

(1)   Title: Mathematical modeling of cholera

Speaker: XueyingWang(Washington State University)

Time: 10:00 -12:00, July 19, 2023

Abstract: Cholera is a water-borne infectious disease that continues to be a major public health concern in many parts of the world. Mathematical modeling can be an effective tool for understanding the transmission dynamics of cholera and informing control and prevention strategies. In this lecture, we will discuss three different types of mathematical models commonly used in cholera modeling: ordinary differential equations (ODEs) for a single patch, multi-patch models, and partial differential equations (PDEs). We will talk about the advantages and limitations of each model type, as well as practical considerations for selecting the most appropriate model for a given scenario. In addition, we will investigate the key components of cholera transmission, including the role of environmental factors, human behavior and within-host and between-host dynamics.

(2)   Title: Target reproduction numbers

Speaker: Xueying Wang(Washington State University)

Time: 13:30-15:30, July 19, 2023

Abstract:

The target reproduction number is a crucial population threshold quantity that measures the level of control efforts required to achieve a particular prevention, intervention, or control objective. This concept, as a generalization of type reproduction number, was first introduced for nonnegative matrices with immediate applications to compartmental population models of ordinary differential equations. This lecture consists of two parts. In the first part, we will review the concepts type reproduction number and target reproduction number for nonnegative matrices.  In the second part, we will introduce a general theory of target reproduction numbers for positive operators on an ordered Banach space, which allows us to apply to reaction-diffusion population models. We will demonstrate that the target reproduction number can be interpreted as the expected number of offspring in a specific target set that a primary newborn individual of the same set would produce during its lifetime. Furthermore, we will present a characterization of the target reproduction number that enables its efficient numerical computation for reaction-diffusion models. Finally, we will illustrate our theoretical findings through examples.

(3)   Title: Mathematical modeling of COVID-19

Speaker: Xueying Wang(Washington State University)

Time: 10:00 -12:00, July 20, 2023

Abstract:

The COVID-19 pandemic has created unprecedented public health challenges worldwide. Despite significant progress in understanding the disease pathogenesis and progression, the epidemiological triad of pathogen, host, and environment remains unclear. In this lecture, we will start with the development of a multiscale modeling framework that couples within-host and between-host dynamics of COVID-19. The model includes multiple transmission routes, both human-to-human and environment-to-human, and connects multiple scales, both population and individual levels. Our detailed analysis of the dynamics shows rich dynamics, including both forward and backward bifurcations, emerging with the coupling of viral infection and epidemiological models. Model validation with virological and epidemiological data facilitates evaluation of the influence of several infection characteristics and potential antiviral treatments on disease spread. Our work highlights the critical role of the environment in affecting the transmission of COVID-19 and the necessity of implementing control measures that reduce environmental transmission, in addition to interventions such as social distancing and mask-wearing aimed at stopping human-to-human transmission. In the remaining of the lecture, we will talk about the cases studies of COVID-19 outbreaks in different locations. Attendees will gain a deeper understanding of the complexities of COVID-19 dynamics and the importance of environmental transmission in disease control.

(4)   Title: Stochastic models of infectious diseases with application to cholera epidemics

Speaker: Xueying Wang (Washington State University)

Time: 14:00-16:00, July 20, 2023

Abstract:

Seasonal variations have a significant impact on the dynamics of many infectious diseases, such as influenza, cholera, and malaria. Understanding the time when infectious individuals are first introduced into a population is crucial for predicting whether a major disease outbreak will occur. In this lecture, we will investigate the impact of seasonality on disease outbreaks using stochastic models. The lecture consists of two parts. In the first part, we will review basic theory on stochastic processes. In the second part, we will formulate a time-nonhomogeneous stochastic process and extend existing approaches to obtain an analytic estimate for the probability of disease extinction at the initiation of infection by using the backward Kolmogorov differential equation method. We will also derive approximations for the moments of the first extinction time. Using cholera as an example, we will illustrate our stochastic method and extend it to more general infectious disease models by applying theory from multitype branching processes.

Topic 4. Mathematics for wave propagation in bones and the NMR imaging

7/26  https://nationaltaiwanuniversity-ksz.my.webex.com/nationaltaiwanuniversity-ksz.my/j.php?MTID=mf700847eeebdbd3663505c20fa52c723

7/27  https://nationaltaiwanuniversity-ksz.my.webex.com/nationaltaiwanuniversity-ksz.my/j.php?MTID=m7ad3c3a6fd67b6a4c2f865055034fea2

Speaker: Miao-Jung Yvonne Ou (University of Delaware)

Time: 10:00 -12:00, 14:00-16:00, July 26, 27, 2023

Abstract:

In this course, the mathematics arising from the ultrasound propagation in bones and the nuclear magnetic resonance (NMR) imaging of brains will be described and explained. The lectures will cover the mathematical modeling of these problems, the numerical challenges of these problems as well as the tools from analysis for tackling these challenges. Some open problems related to these applications will also be introduced.

Topic 5. The maximum principle and the positive invertibility of general linear elliptic BVP's

8/02  https://nationaltaiwanuniversity-ksz.my.webex.com/nationaltaiwanuniversity-ksz.my/j.php?MTID=m98d23ad7a6d00acf2863e9f764168ed7

8/03  https://nationaltaiwanuniversity-ksz.my.webex.com/nationaltaiwanuniversity-ksz.my/j.php?MTID=m6b01840485d354d4de1bf71c46d4392a

8/09  https://nationaltaiwanuniversity-ksz.my.webex.com/nationaltaiwanuniversity-ksz.my/j.php?MTID=me9c45e9ac425d9112efd33c88dbcc644

8/10  https://nationaltaiwanuniversity-ksz.my.webex.com/nationaltaiwanuniversity-ksz.my/j.php?MTID=mf72295efefe823abe44ed711f16cf906

Speaker: Julian Lopez-Gomez (Complutense University of Madrid)

Time: 10:00 -12:00, August 2, 3, 9, 10

Contents:

(1)The Hopf minimum principle and the boundary lemma of Hopf-Oleinik.

(2)Uniform decay property of Walter along the boundary.

(3)First theorem of classification of super solutions.

(4)Construction of a positive super solution in smooth domains.  

(5)Second theorem of classification of super solutions.

(6)Existence of principal eigenvalues.

(7)Theorem of positive invertibility.

(8)Some applications.

References:

(1)J. Lopez-Gomez. Linear Second Order Elliptic Operators, World Scientific Publishing, Singapore, 2013

(2)J. Lopez-Gomez. Meta solutions of Parabolic Equations in Population Dynamics, CRC Press, Boca Raton, 2015

3. Join Online

7/12  https://nationaltaiwanuniversity-ksz.my.webex.com/nationaltaiwanuniversity-ksz.my/j.php?MTID=m31892abf52ba3a82864c502e6bbf04da

7/13  https://nationaltaiwanuniversity-ksz.my.webex.com/nationaltaiwanuniversity-ksz.my/j.php?MTID=med6c3d112b1178868771897579c8c68b

7/19  https://nationaltaiwanuniversity-ksz.my.webex.com/nationaltaiwanuniversity-ksz.my/j.php?MTID=md4f66d0e097e3f4e109fc1b3553eba3a

7/20  https://nationaltaiwanuniversity-ksz.my.webex.com/nationaltaiwanuniversity-ksz.my/j.php?MTID=m374caf8b3020e0636456e890f8c5d225

7/26  https://nationaltaiwanuniversity-ksz.my.webex.com/nationaltaiwanuniversity-ksz.my/j.php?MTID=mf700847eeebdbd3663505c20fa52c723

7/27  https://nationaltaiwanuniversity-ksz.my.webex.com/nationaltaiwanuniversity-ksz.my/j.php?MTID=m7ad3c3a6fd67b6a4c2f865055034fea2

8/02  https://nationaltaiwanuniversity-ksz.my.webex.com/nationaltaiwanuniversity-ksz.my/j.php?MTID=m98d23ad7a6d00acf2863e9f764168ed7

8/03  https://nationaltaiwanuniversity-ksz.my.webex.com/nationaltaiwanuniversity-ksz.my/j.php?MTID=m6b01840485d354d4de1bf71c46d4392a

8/09  https://nationaltaiwanuniversity-ksz.my.webex.com/nationaltaiwanuniversity-ksz.my/j.php?MTID=me9c45e9ac425d9112efd33c88dbcc644

8/10  https://nationaltaiwanuniversity-ksz.my.webex.com/nationaltaiwanuniversity-ksz.my/j.php?MTID=mf72295efefe823abe44ed711f16cf906

4. Schedule

5. Registration

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6. Agenda Download

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