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300 5. Nonlinear Systems
Table 5.1 Sample parameters
Parameter Name Sample Value
M T /H T population ratio 2
a biting rate 0.2–0.5 per day
b effective bites infecting humans 0.5
c effective bites infecting mosquitos 0.5
r recovery rate 0.01–0.05 per day
μ mortality rate 0.05–0.5 per day
by condition (5.33), and after considerable simplification. Thus, (h ∗ ,m ∗ )is
asymptotically stable.
To give some idea of parameter values used in computation, we list a sample
set in Table 5.1.
EXERCISES
1. (Malaria) In this exercise develop and analyze a simplified version of the
malaria model under the condition that r is much less that μ.
a) Beginning with (5.27)–(5.28), nondimensionalize these equations by
rescaling time by taking τ = μt. Obtain
where
ε = r μ ,
dh
= λm(1 − h) − εh,
dτ
dm
= ηh(1 − m) − m,
dτ
λ = ab M T
,
μ H T
η = ac
μ .
b) Assuming ε is very small, neglect the εh term in the host equation and
draw the phase portrait. Include the equilibria, nullclines, direction
field, and a local stability analysis for the equilibria.
c) For the simplified dimensionless model in part (b), with the values
given in Table 5.1, specifically, a =0.5, r =0.01, and μ =0.5, use
a numerical method to draw time series plots of h and m for various
initial initial conditions.
2. (Schistosomiasis ∗ ) Schistosomiasis is a macroparasitic disease of humans
caused by trematode worms, or blood flukes. Trematodes are a class of