A Mathematical Model
Presented to the
Graduate Faculty of the
College of Sciences
Rochester Institute of Technology
In Partial Fulfillment
Of the Requirements for the Degree
MASTER OF SCIENCE IN INDUSTRIAL
AND APPILED MATHEMATICS
Leishmaniasis is a parasitic disease that is found in parts of the tropics, subtropics, and
southern Europe. It is transmitted by the bite of infected female phlebotomine sandflies. Sand
flies become infected by biting an infected animal or Human. Sandflies become infected by
ingesting infected cells during blood meals, they inject the infective stage (i.e., promastigotes)
from their proboscis during blood meals. Promastigotes that reach the puncture wound are
phagocytized by macrophages and other types of mononuclear phagocytic cells. Progmastigotes
transform in these cells into the tissue stage of the parasite (i.e., amastigotes), which multiply by
simple division and proceed to infect other mononuclear phagocytic cells. Parasite, host, and
other factors affect whether the infection becomes symptomatic and whether cutaneous or
visceral leishmaniasis results. In sandflies, amastigotes transform into promastigotes, develop in
the gut (in the hindgut for leishmanial organisms in the Viannia subgenus; in the midgut for
organisms in the Leishmania subgenus), and migrate to the proboscis.
In the New World (the Western Hemisphere), it is found in some parts of Mexico,
Central America, and South America (Latin America). It is not found in Chile or Uruguay. In the
Old World (the Eastern Hemisphere), it is found in some parts of Asia, the Middle East, Africa,
and southern Europe. It is not found in Australia or the Pacific Islands. Overall, it is found in
focal areas of about 88 countries. Over 90 percent of the cases of cutaneous leishmaniasis occur
in parts of Afghanistan, Algeria, Iran, Iraq, Saudi Arabia, and Syria (in the Old World) and in
Brazil and Peru (in the New World). The number of new cases of cutaneous leishmaniasis each
year in the world is thought to be about 1.5 million.
However, the cases of leishmaniasis evaluated in the United States reflect travel and
immigration patterns. For example, cases in U.S. civilian travelers typically are cases of
cutaneous leishmaniasis acquired in common tourist destinations in Latin America (rather than in
places in the Old World). Almost all of the people in the United States who have leishmaniasis
became infected while traveling or living in other countries. Occasional cases of cutaneous
leishmaniasis have been reported in residents of Texas and Oklahoma.
Hence it necessary to mathematically model the progress of Cutaneous Leishmaniasis
to discover the likely outcome of it, from which necessary control measures can be taken.
How Mathematical Modeling is used
It is possible to mathematically model the progress of most infectious diseases to
discover the likely outcome of an epidemic. This is done using some basic assumptions and some
simple mathematics to find parameters for various infectious diseases.
The SIR model  is the simplest for many infectious diseases. Standard convention labels
these three compartments S, I and R. Therefore, this model is called the SIR model. The model is
given by the following differential equations:
From the above differential equations we observe the following details:
S(t) + I(t) + R(t) = Constant = N
The Basic concepts used in S I R model are:
S : The proportion of the population who are susceptible to the disease (neither immune nor
I : The proportion of the population who are infectious to the disease.
R : The proportion of the population who are recovered to the disease.
β : The Contact Rate.
ν: The Recovery Rate.
R0 (The basic reproduction number): The average number of other individuals each infected
individual will infect in a population that has no immunity to the disease. If R0 >1, means the
disease yields epidemic. If R0 <1, means the disease is in control.
The above differential equations are solved in equilibrium condition to analyze the system.
Statement of the problem:
There are very few mathematical models of Cutaneous Leishmaniasis. In the those
mathematical models they have included just the Human population and Sand flies population,
but they have not included the Animal population, which is the main reservoir for Cutaneous
Leishmaniasis. This proposal is to develop a new mathematical model to demonstrate how
interactions between the Sand flies population (Carriers), Human population and Animal
population (The main Reservoir for the disease) occur for the transmission of Cutaneous
Fly population Animals
The mathematical model is constructed by considering the following main details in each
1. Human population
How are they susceptible to the disease?
How the infected are recovered.
How the spreading of disease occurs.
2. Sand Fly Population
It acts as a carrier so we consider whether it is infected or not and it chances of
3. Animal Population
It also has the same details as Human population, but since they are the reservoirs,
it has different rates infection, recovery etc.