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### Modified SIR for Vector-Borne Diseases

Gay Wei En Colin 4i310

Chua Zhi Ming 4i307

Jacob SavosAOS

Katherine KamisAOS

Aims and Objectives

- To create a universal modified SIR model for vector-borne diseases to make predictions of the spread of diseases

Rationale

- The SIR Model currently used is extremely simplistic
- Only considers three compartments, namely Susceptible, Infected and Recovered
- Two directions of change, namely from Susceptible to Infected or from Infected to Recovered.

Rationale

- Since most vector-borne diseases do not work in such a way, this project aims to modify this SIR model so that it can encompass much more factors that the original SIR model
- Death rates
- Movement from Recovered to Susceptible
- Make it more applicable to real life, thus increasing its usability in accurately predicting the spread of such vector-borne diseases.

Introduction

- A vector-borne disease is transmitted by a pathogenic microorganism from an infected host to another organism
- HCI will be creating a model using Dengue Fever
- AOS will be creating a model using a tick-borne disease

Literature Review – Dengue Fever

- A very old disease that reemerged in the past 20 years
- Transmitted via mosquito bites
- In 2009, there were a total of 4452 cases of dengue fever in Singapore, of which there were 8 deaths

Literature Review – Aedes Mosquitoes

- Aedes mosquitoes refers to the entire genus of mosquito – over 700 different species
- Multiple species able to transmit dengue fever
- Have characteristic black and white stripe markings on body and legs

Aedes albopictus – the most invasive mosquito in the world

Retrieved from http://www.comune.torino.it/ucstampa/2005/aedes-albopictus.jpg

Aedes aegypti – Main vector of dengue fever in Singapore

Retrieved from http://www.telepinar.icrt.cu/ving/images/stories/aedes-aegypti__785698.jpg

Literature Review - Ticks

- Ticks have a two-year life cycle
- Ticks acquire a vector-borne disease by feeding on an infected host
- Once infected, ticks transmit the disease by feeding on an uninfected host

Lone Star Tick

Deer Tick

Literature Review - SIR

- Susceptible
- Infected
- Recovered

Literature Review - SIR

- Neuwirth, E., & Arganbright, D. (2004). The active modeler: mathematical modeling with Microsoft Excel. Belmont, CA: Thomson/Brooks/Cole.
- Introduces basic modeling techniques such as dynamic modeling and graphing
- Rates of change are shown to have relations between the three compartments: S(t), I(t) and R(t) in the subtopic simple epidemics.
- Calculus can be used to help us solve the research questions mentioned.

SIR - Equations

- S’(t)=-k∙S(t)∙I(t)
- I’(t)=-S’(t)-R\'(t)
- R’(t)=c∙I(t)
- k – Transmittal constant
- c – Recovery rate

Research Questions

- Are we able to predict the spread of a disease using the SIR Model?
- What kind of situations are the basic SIR Model unable to take into account?
- How can the basic SIR Model be modified to handle real life situations effectively?
- Is there a pattern in the spread of vector-borne diseases?

Fields of Mathematics - Differentiation

- Used to determine the rate of change of a function
- Infection and recovery obtained via differentiation based on data acquired
- e.g. With the weekly number of cases of the disease, we are able to find the best fit graph, the function of which we can then differentiate to determine the infection rate in the form of a function.

Methodology

- Begin with a simple SIR model
- Develop variables needed to modify the model
- Attempt to modify the model to incorporate all vector-borne diseases

Bibliography

Academy of Science. Academy of Science Mathematics BC Calculus Text.

Breish, N., & Thorne, B. (n.d.). Lyme disease and the deer tick in maryland. Maryland: The University of Maryland.

Duane J. Gubler(1998, July). Clinical Microbiology Reviews, p. 480-496, Vol. 11, No. 3, 0893-8512/98/$00.00+0. Dengue and Dengue Hemorrhagic Fever. Retrieved November 3, 2010 from http://cmr.asm.org/cgi/content/full/11/3/480?view=long&pmid=9665979

Neuwirth, E., & Arganbright, D. (2004). The active modeler: mathematical modeling with Microsoft Excel. Belmont, CA: Thomson/Brooks/Cole.

Ministry of Health: FAQs. (n.d.). Dengue. Retrieved November 3, 2010, from http://www.pqms.moh.gov.sg/apps/fcd_faqmain.aspx?qst=2fN7e274RAp%2bbUzLdEL%2fmJu3ZDKARR3p5Nl92FNtJidBD5aoxNkn9rR%2fqal0IQplImz2J6bJxLTsOxaRS3Xl53fcQushF2hTzrn1PirzKnZhujU%2f343A5TwKDLTU0ml2TfH7cKB%2fJRT7PPvlAlopeq%2f%2be2n%2bmrW%2bZ%2fJts8OXGBjRP3hd0qhSL4

Bibliography

Ong, A., Sandar, M., Chen, M. l., & Sin, L. Y. (2007). Fatal dengue hemorrhagic fever in adults during a dengue epidemic in Singapore. International Journal of Infectious Diseases, 11, 263-267.

Stafford III, K. (2001). Ticks. New Haven: The Connecticut Agricultural Experiment Station.

Wei, H., Li, X., & Martcheva, M. (2008). An epidemic model of a vector-borne disease with direct transmission and time delay. Journal of Mathematical Analysis and Applications, 342, 895-908.

Dobson, A. (2004). Population Dynamics of Pathogens with Multiple Host Species. The American Naturalist, 164, 564-578.

Any questions?

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