Vardayani Ratti




My research interest is in Mathematical Biology, in particular applications in ecology, infectious diseases, pollination biology and agriculture.
I am looking for undergraduate or graduate students who want to pursue research in this area. If you are interested, please email me.


Recent invited talks and conferences


  • Advancing Tick-borne Disease Modeling Workshop, Dartmouth College, Hanover USA, July 2021

  • California State University, Northridge, USA, April 2021

  • California State University, Chico, USA, 2019

  • Rochester Institute of Technology, Rochester, USA, 2018

  • Arizona State University, Arizona, USA, 2018

  • AMS Special Session at Spring Eastern Sectional Meeting, Northeastern University, Boston, 2018

  • Sonia Kovalevsky Day, Dartmouth College, USA, 2018

  • Bowdoin College, Maine, USA, 2018

  • Fields Institute's Workshop on Pollinators and Pollination Modeling, Toronto, 2018

  • Joint Mathematics Meeting, San Diego, USA, 2018

  • Minisymposium on Mathematical Biology, Dartmouth College, 2017

  • Joint Mathematics Meetings, Atlanta, USA, 2017

Contact

Assistant Professor
California State University, Chico
Department of Mathematics and Statistics
400 W 1st St
Chico, CA 95929
Email: vratti@csuchico.edu
Phone: (+1) 530-898-5508
Office: 136 Holt Hall
Zoom (by appointment only):Meeting ID: 530 898 5508, Passcode: RATTI



Teaching



MATH 260: Ordinary Differential Equations I

F 21

First order separable, linear, and exact equations; second order linear equations, Laplace transforms, series solutions at an ordinary point, systems of first order linear equations, and applications.


Textbook: Elementary Differential Equations and Boundary Value Problems by W.E. Boyce, R.C. DiPrima.


MATH 360: Ordinary Differential Equations II

Sp 21

Systems of first order linear equations, existence and uniqueness theorems, stability, Sturm separation theorems, power series methods


Textbook: Differential Equations & Linear Algebra by C.H. Edwards, D.E. Penney.
Final projects are based on the following research articles.
  • A mathematical model of Coronavirus Disease (COVID-19) containing asymptomatic and symptomatic classes
  • Dynamical Features of a Mathematical Model on Smoking
  • Stability of a Mathematical Model of Malaria Transmission with Relapse
  • A Mathematical Model of HIV dynamics Treated with a Population of Gene Edited Hematopoietic Progenitor Cells Exhibiting Threshold Phenomenon
  • Mathematical Model to Simulate Tuberculosis Disease Population Dynamics
  • An Avian Influenza Mathematical Model
  • Mathematical Model and Optimal Control of New-Castle Disease
  • Mathematical Analysis of an HIV/AIDS Epidemic Model
  • Bifurcation Analysis on the Mathematical Model of Measles Disease Dynamics

MATH 121: Analytic Geometry and Calculus II

Sp 21

The definite integral and applications to area, volume, work, differential equations, etc. Sequences and series, vectors and analytic geometry in 2 and 3-space, polar coordinates, and parametric equations.


Textbook: Active Calculus 2.1 , published by Grand Valley State University Libraries.


MATH 361: Boundary Value Problems and Partial Differential Equations

Fall 2020

Partial differential equations, separation of variables, orthogonal sets of functions, Sturm-Liouville problems, Fourier series, boundary value problems for the wave equation, heat equation, and Laplace equation; Bessel functions, Legendre polynomials


Textbook: Applied Partial Differential Equations. by Richard Haberman, 5th edition.

A rigorous introduction to discrete mathematical structures for computer science majors (fulfills a requirement for the minor in math). Topics include propositional and predicate calculus; basic proof methods; sets, functions, and operations with them; algorithms and their complexity; applications of number theory to computer science and computer security; matrices and matrix arithmetic; mathematical induction, recursive definitions and algorithms; combinatorics and counting techniques; relations and their representation by matrices and digraphs, applications to databases; equivalence relations and partitions of sets; partially ordered sets, lattices, and Boolean algebras; Boolean functions and circuits; graphs, trees, and their applications; formal languages and grammars; finite-state automata and language recognition, regular languages.


Textbook: Applied Discrete Structures by Al Doerr & Ken Levasseur, 2019


MATH 120: Analytic Geometry and Calculus

(F19, Sp20, F20, Blackboard, student access only)

Limits and continuity. The derivative and applications to related rates, maxma and minima, and curve sketching. Transcendental functions. An introduction to the definite integral and area.


Textbook: Stewart’s Calculus 8E Early Transcendentals, published by Cengage.


MATH 3: Calculus

(Winter and Fall 2018)

This course involves collaborative learning. It is an introduction to single variable calculus aimed at students who have seen some calculus before, either before matriculation or in introductory calculus course (Math 1). Math 3 begins by revisiting the core topics in Math 1 - convergence, limits, and derivatives - in greater depth before moving to applications of differentiation such as related rates, finding extreme values, and optimization. The course then turns to integration theory, introducing the integral via Riemann sums, the fundamental theorem of calculus, and basic techniques of integration.


Textbook: ``Calculus'' by Herman and Strang's, Volume 1. (open access PDF)


Math 8: Calculus of Functions of one and Several Variables

(Spring 2018)

This course is a sequel to Math 3 and provides an introduction to Taylor series and functions of several variables. The first third of the course is devoted to approximation of functions by Taylor polynomials and representing functions by Taylor series. The second third of the course introduces vector-valued functions. It begins with the study of vector geometry, equations of lines and planes, and space curves. The last third of the course is devoted to studying differential calculus of functions of several variables.


Textbook: ``Calculus", by James Stewart, 8th Edition, ISBN: 978-1-285-74062-1


MATH 76: Topics in Applied Mathematics

(Winter 2018)

This course is designed to provide students with the basic tools for building and analyzing mathematical models in Biology primarily using ordinary differential equations. In addition, you will learn how to analyze and simulate the models. You will also learn to interpret and communicate the results in the context of biology.


Textbooks:


MATH 22: Linear Algebra with Applications

(Fall 2017, Spring 2019)

This course presents the fundamental concepts and applications of linear algebra with emphasis on Euclidean space. Significant goals of the course are that the student develop the ability to perform meaningful computations and to write accurate proofs. Topics include bases, subspaces, dimension, determinants, characteristic polynomials, eigenvalues, eigenvectors, and especially matrix representations of linear transformations and change of basis. Applications may be drawn from areas such as optimization, statistics, biology, physics, and signal processing.


Textbook: Linear Algebra and Its Applications (5th Edition) by David C. Lay, Steven R. Lay, Judi J. McDonald, Pearson 2015


Math 13: Multivariable Calculus

(Winter 17)

This course is a sequel to Math 8 and provides an introduction to calculus of vector-valued functions. The course starts with iterated, double, triple, and surface integrals including change of coordinates. The remainder of the course is devoted to vector fields, line integrals, Green’s theorem, curl and divergence, and Stokes’ theorem.


Textbook: ``Calculus Early Transcendentals Multivariable", by Rogawski & Adams, 3rd Edition, ISBN: 978-1464171758


MATH 23: Differential Equations

(Fall 2016)

This course is a survey of important types of differential equations, both linear and nonlinear. Topics include the study of systems of ordinary differential equations using eigenvectors and eigenvalues, numerical solutions of first and second order equations and of systems, and the solution of elementary partial differential equations using Fourier series.


Textbook: Elementary Differential Equations and Boundary Value Problems (10th Edition) by Boyce & DiPrima, Wiley 2012

Graduate Teaching Assistant (2009 - 2015)



Research


Honeybee Disease Modeling



Recent years have seen a dramatic decline in the number of honeybee colonies in North America and Europe. There is no consensus that explains the colony losses. Possible factors include, but are not limited to, varroa mites and the deadly viruses they carry, pesticides, nosema, and extreme winter conditions. I develop mathematical models of honeybee colony infestation by Acute Bee Paralysis Virus (ABPV) that is carried and transmitted by a parasitic varroa mite. The models are formulated in terms of non-linear ordinary differential equations. Our first model uses constant coefficients, which are, strictly speaking, valid for a single season at most, under the assumption that seasonal averages are representative of environmental dynamics and under the assumption that the dynamics are fast enough, such that the system equilibrates quickly, in less than a single season.




Modeling of Anopheles gambiae and its treatment in western Kenya highlands



Malaria epidemics in the highlands have continued to wreak havoc on public health of the inhab- itants, resulting in high morbidity and mortality, especially in children under age five. Since the early 1980s, there have been massive percentage increases in P. falciparum burden at African highland locations. One of the factors thought to be a key drivers of the resurgent epidemics is the ecosystem heterogeneity. For instance, Emutete and Wamondo, despite being neighboring towns in Kenya’s western highlands, with similar rainfall and temperature, exhibit extremely different population levels of Anopheles gambiae. This difference in mosquito population leads to malaria being epidemic in Wamondo and endemic in Emutete. The different mosquito population levels could be due to heterogeneity in the shape of the two towns: Emutete is a U-shaped valley and Wamondo is a V-shaped valley. The existing model does not integrate a phenomenon that occurs during periods of heavy daily rainfall. Studies show that heavy daily rainfalls displace (and kill) larval instars from smaller, temporary habitats. This process, known as larvae flushing, will be incorporated into the extended model. Insecticidal indoor residual wall treatment is a major tool for the control of malaria, with the goals of reducing indoor vector density and life span and reducing transmission rates of disease. With collaborators, I studied a mathematical model to predict reduction in the malaria vector, Anopheles gambiae, in the Emutete region in the western Kenya highlands due to four types of indoor wall treatments: two types of indoor residual DDT spray and two types of pyrethrin-based INESFLY insecticidal paint. The primary difference between these treatments is the duration of their persistence on walls.

Mathematical modeling of treatment of HIV with gene edited hematopoietic progenitor cells



The use of CRISPR/Cas gene editing technology has the potential to excise the CCR5 gene from hematopoeitic progenitor cells, rendering their differentiated CD4+ T cell descendants HIV resistant. With collaborators, I developed a mathematical model to mimic the therapeutic potential of CRISPR/Cas gene editing of hematopoietic progenitor cells to produce a class of HIV resistant CD4+ T cells. We defined the requirements for the permanent suppression of viral infection using gene editing as a novel therapeutic approach. We developed nonlinear ODE model to replicate HIV production in an infected host, incorporating the most appropriate aspects found in the many existing clinical models of HIV infection, and extended this model

A malaria model, combined with mosquito model, to study the effect of land use and topography on disease transmission



A natural extension of the previous project is to incorporate malaria transmission into the time- varying mosquito model. Individuals repeatedly exposed to the malaria parasite are known to acquire partial immunity to disease. In the absence of repeated exposure to the parasite, this immunity is lost. The level of exposure needed to induce or maintain partial immunity and the rate at which immunity is lost in the absence of exposure are questions that yield a wide range of answers in the literature. With collaborators, I incorporated partial immunity arising from re-inoculation in order to distinguish the malaria transmission patterns in Emutete and Wamondo. We did not take into account “passive immunity”, i.e., the immunity gained by a newborn from an immune mother. After an infant loses passive immunity, he/she is susceptible to malaria. In regions of high malaria transmission, the infant will soon get malaria and begin to develop immune defenses, or “active immunity.” We categorized active immunity into clinical immunity and anti-parasite immunity.



Publications


Publications (undergraduate co-authors are in bold)



  1. Ratti, V.; Winter, J. M.,; Wallace, D. I. (2021) Diluting Lyme disease: Modeling the effects of incompetent hosts on Borrelia burgdorferi transmission. Ticks and tick-borne diseases. 12(4): 101724.

  2. Ratti, V.; Wallace, D. I. (2020) A malaria transmission model predicts holoendemic, hyperendemic, and hypoendemic transmission patterns under varied seasonal vector dynamics. Journal of Medical Entomology. 2020(00): 1-17.

  3. Wallace, D. I.; Ratti, V., Kodali, A., Winter, J. M., Ayres, M. P., Chipman, J. W., ... & Webb, M. J. (2019) Effect of Rising Temperature on Lyme Disease: Ixodes scapularis Population Dynamics and Borrelia burgdorferi Transmission and Prevalence. Canadian Journal of Infectious Diseases and Medical Microbiology. 2019(00): 1-15.

  4. Ratti, V.; Nanda, S.; Eszterhas, S. K.; Howell, A.; Wallace, D.I.(2019) A Mathematical Model of HIV dynamics Treated with a Population of Gene Edited Hematopoietic Progenitor Cells Exhibiting Threshold Phenomenon. Mathematical medicine and biology: a journal of the IMA, 2019 (00): 1-31.

  5. Ratti, V.; Rheingold, E.; Wallace, D.I. (2018) Reduction of Mosquito Abundance Via Indoor Wall Treatments: A Mathematical Model. Journal of Medical Entomology. 55(4): 833-845.

  6. Ratti, V.; Kevan, P.G.; Eberl, H.J. (2017) A Mathematical Model of Forager Loss in Honeybee Colonies Infested with Varroa destructor and the Acute Bee Paralysis Virus. Bulletin of Mathematical Biology. 79(6): 1218-1253.

  7. Ratti, V.; Kevan, P.G.; Eberl, H.J. (2016) A discrete-continuous modeling framework to study the role of swarming in a honeybee colony infested with Varroa destructor and Acute Bee Paralysis Virus. The 2015 AMMCS-CAIMS Congress. DOI: 10.1007/978-3-319-30379-6_28. In book: Mathematical and Computational Approaches in Advancing Modern Science and Engineering, pp.299-308.

  8. Ratti, V.; Kevan, P.G.; Eberl, H.J.(2015) A mathematical model of the honeybee-varroa destructor-acute bee paralysis virus complex with seasonal effects. Bulletin of Mathematical Biology. 77(8):1493-1520.

  9. Ratti, V.; Kevan, P.G.; Eberl, H.J. (2013) A mathematical model of the honeybee-varroa destructor-acute bee paralysis virus complex. Canadian Applied Math Quarterly, 21(1):63-93.



Under revision (undergraduate co-authors are in bold)


  1. Ratti, V.; Chipman, J.W.; Wallace, D. I. (2021). Flushing Alters Endemicity Patterns in Regions with Similar Habitat Abundance. Current Research in Parasitology & Vector-Borne Diseases.

  2. Chen, J.; DeGrandi-Hoffman, G.; Ratti, V.; Kang, Y. (2021). A review of the mathematical models in honeybee colonies, Mathematical Biosciences and Engineering.

  3. Sreekanth, O.G., Ratti, V., Wallace, D. I. (2020). A New Approach to Detecting Patterns of ENSO Teleconnections with Temperature and Rainfall Patterns in the Western Kenya Highlands Sep- arates Seasonal, Auto-correlated, and Random Effects. Theoretical and Applied Climatology.

In preparation (undergraduate co-authors are in bold)


  1. Wallace, D.I.; Kachalia, A.A.; Ratti, V. Effect of habitat diversity on the population dynamics of Anopheles gambiae.

  2. A mathematical model of beevectoring applied to the biocontrol of Botrytis cineria by clonostachys rosea f. rosea)

Book Chapter


  1. Eberl, H.J.; Kevan, P.G.; Ratti, V.(2014) Infectious disease modeling for honeybee colonies in J. Dellivers(ed). “In Silico Bees”, p.87-134, CRC, Press Boca Raton.

Theses


  1. Ratti, V. Predictive Modeling of the Disease Dynamics of Honeybee-Varroa destructor-Virus Systems. Ph.D. Thesis

  2. Ratti, V. Local Stability Analysis of the Honeybee-Varroa destructor-Acute Bee Paralysis Virus. M.Sc. Thesis


Advising and Outreach


Advising


  1. Abraham Serrano Lopez, Summer 2021, CSC2

  2. Pablo Saravia, Summer 2020, CSC2

  3. Phuc Ngo, University of Guelph

  4. Colin Gui, Presendial Scholar, Dartmouth College

  5. Omkar Sreekanth, Dartmouth College

  6. Aparna Kachalia, Dartmouth College
I have been serving as a mentor in the National Alliance for Doctoral Studies in the Mathematical Sciences.

Outreach


I gave a plenary talk at Sonia Kovalevsky day which was very well-received.

In addition to the classroom teaching, I encourage my students to participate in activities like Association of Women in Mathematics (AWM) national essay competition. One of my students won this competition.

I also participated in TechWomen Ambassadors program whose purpose is to promote women in STEM fields.

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