Malaria Model with Periodic Mosquito Birth Rate

1
Malaria Model with Periodic Mosquito Birth
Rate Bassidy Dembele, Avner Friedman,
Abdul-Aziz Yakubu Department of Mathematics,
Howard University, Washington, D.C
Mathematical Biosciences Institute, The Ohio
State University, OHIO
Table 1 Parameters for the Malaria model
Abstract
2. Malaria Model with a Periodic Mosquito Birth
Rate
Sh the proportion of susceptible individuals
(humans) Ih the proportion of infected
individuals Rh the proportion of recovered
individuals Sm the proportion of susceptible
mosquitoes Im the proportion of infected
mosquitoes. By definition Sh Ih Rh 1 and Sm
Im 1. The susceptible individuals are those
who are not sick but they can become infected
through bites from infected mosquitoes The
infected individuals are those who were bitten by
infected mosquitoes and show symptoms of the
disease The recovered individuals are those who
have been treated by ACT The susceptible
mosquitoes are those who do not carry any
parasite in their salivary glands The infected
mosquitoes are those who carry a Multitude of
parasites in their salivary glands and are thus
able to infect humans. Our malaria model with
periodic mosquito birth rate is given by the
following system of equations dSh/dt ?h (Ih
Rh) ßhRh - ?ShIm, (2.1) dIh/dt ?ShIm (?h
ah)Ih, (2.2) dRh/dt ahIh
(?h ßh)Rh, (2.3) dIm/dt aIhSm
?(t)Im, (2.4) dSm/dt
?(t)Im aIhSm, (2.5) The
parameter ßh represents the rate of loss of
immunity. The parameter ?h represents birth rate
children of infected and of recovered individuals
are born as susceptible individuals, and thus
there is a loss of the infected and recovered
populations at rate ?h. On the other hand, for
susceptibles, it is assumed that birth rate does
not reduce or increase their relative
proportions. Newborn mosquitoes are assumed to be
susceptible and thus new births decreases Im at a
rate ?(t) and increases Sm at a rate ?(t). Let
In this study, we introduce a model of malaria, a
disease that involves a complex life cycle of
parasites, requiring both human and mosquito
hosts. The novelty of the model is the
introduction of a periodic coefficient into the
system of ODEs, which accounts for the seasonal
variations (wet and dry seasons) in the mosquito
birth rate. We define a basic reproduction number
R0 which depends on the periodic coefficient and
prove that if R0 lt 1 then the disease becomes
extinct, whereas if R0 gt 1 then the disease is
endemic and may even be periodic.
1. Introduction
Malaria is one the most devastating diseases and
one of the leading causes of death in the
tropical regions of the world. Worldwide, there
are more than 500 million clinical cases of
malaria a year and almost 2.7 million deaths, a
large percentage of them children, according to
the 2006 UNICEF report on malaria. The symptoms
of the disease are fever, chills, sweats,
headache, nauseas and vomiting, body aches, and
general malaise. If not treated by drugs, the
symptoms become more severe, as blood cells are
continuously being destroyed, and may lead to
death. Malaria is spread in three ways. The most
common is by the bite of an infected female
Anopheles mosquito. However, malaria can also be
spread through a transfusion of infected blood or
by sharing a needle with an infected person. In
this project, we focus on infection via
human-mosquito interaction. Malaria is caused by
a parasite called plasmodium sporozoite. The
parasite has a complex life cycle that requires
both a human host and an insect host namely,
female anopheline mosquito. Humans can only be
infected by bites from infected mosquitoes and
Mosquitoes can only be infected by biting
infected humans. Animals can also get the
malaria disease. However, animal malaria cannot
spread to humans, and human malaria cannot spread
to animals. The human-mosquito interaction can be
described by the following diagram
Theorem 3. If R0 gt 1 then there exists at least
one T-periodic solution of the system
(2.1)-(2.5) that is, R0 gt 1 implies the
persistence of the infective human and mosquito
populations on periodic solutions.
4. Examples
Figures 2. The parameter values are ?h 0.5, ßh
0.4, ah0.5, µ 0.6, a 0.4, d 0.3, ? 2p,
and ? µ(?h ah)/ a - 0.3 1.2. The disease
goes extinct ( R0 lt 1).
Note that the average infectious period of a
single person is 1/(?hah). Also the average
infectious period of a single mosquito is
Figures 3. The parameter values are ?h 0.5, ßh
0.4, ah0.5, µ 0.6, a 0.4, d 0.3, ? 2p,
and ? µ(?h ah)/ a 0.3 1.8. The disease
persists ( R0 gt 1).
5. Conclusions
In this paper, we studied a mathematical model of
malaria consisting of a system of ODEs. The
novelty of the model is in the introduction of a
time dependent periodic coefficient which
accounts for the seasonal variation (wet and dry
seasons) in the birth rate of the mosquito
population. We defined the basic reproduction
number R0 which naturally depends on this
periodic coefficient, and proved by a rigorous
mathematical analysis that if R0 lt 1 then the
disease goes extinct, whereas if R0 gt 1 then the
disease remains endemic.
Hence, R0 may be viewed as the average value of
the expected number of secondary cases produced
by a single infected individual entering the
population at the DFE. R0 is called the basic
reproduction number.
3. Results
Theorem 1. If R0 lt 1 then the disease free
equilibrium point, DFE (1, 0, 0), is globally
asymptotically stable. Theorem 2. If R0 gt 1 then
there exist numbers d0 gt 0 and d1 gt 0 such that
for any initial values Ih(0), Im(0) there is a
time T T(Ih(0), Im(0) ) such that Ih(t)gtd0 and
Im(t) gt d1 for all t ? T(Ih(0), Im(0) ).
6. Future Work
It will be interesting to test our results with
the real population data. On the other hand, it
will also be interesting to take account of
spatial variations in the mosquito population, as
well as their migration from one location to
another.
This material is based upon work supported by the
National Science Foundation under Agreement No.
0112050 and by The Ohio State University.