Epidemiological Models with Seasonality - Mathematical Methods and Models in Biomedicine - page 376

difficult to compute R 0 in this case, and, in particular, to determine when R 0 is
less than 1 or larger than 1. In the present chapter we develop general methods to
determine when R 0 < 1. However, for the sake of clarity we shall

first apply the method to a special case of waterborne diseases and then, in the final
section of this chapter, we extract from this special case the general features of our
methods and give some other examples.
Consider a dynamical system in

1andwhen R 0 >

n

R

d x
d t =

f ( x , γ ) , (1)

where γ =( γ 1 ,..., γ k ) varies in a k -dimensional parameter space Ω , and suppose

that x 0

is a stationary point independent of γ , i.e.,

x 0 f ( , γ )= 0forall γ ∈ Ω . (2)

x 0 We denote the eigenvalues of the Jacobian matrix ( ∂ f / ∂ x )( , γ ) by λ i ( γ ) and

arrange them so that

Re { λ n ( γ ) }≤ Re { λ n − 1 ( γ ) }≤···≤ Re { λ 2 ( γ ) }≤ Re { λ 1 ( γ ) }.

0, then x 0
is unstable. In epidemiological models there is a special interest in the steady state x 0
which represents DFE. Associated with x 0

0, then x 0 is asymptotically stable, and if Re If Re { λ 1 ( γ ) } < { λ 1 ( γ ) } >

is the concept of the basic reproduction

1 then the DFE x 0 is asymptotically stable (so

number R 0 , and it is shown that if R 0 <

1 then the DFE x 0 that Re { λ 1 } < 0) and if R 0 > is not stable (so that Re { λ 1 } > 0 ) .

Consider next a nonautonomous system

d x
d t =

x 0

( , γ ( )) ,

f t (3)

where γ ( t ) is ω -periodic, and assume, as before, that Eq. ( 2 ) holds for all γ = γ ( t ) ∈

Ω . For an epidemiological model of the form Eq. ( 3 ) one can still define the concept

of the basic reproduction number R 0 [ 12 ] and again show that if R 0 < 1thenthe

DFE is asymptotically stable, whereas if R 0 > 1 then the DFE is not stable. In the

autonomous case Eq. ( 1 ) R 0 is the spectral radius of a matrix defined in terms of
some submatrices of x 0

. In the nonautonomous case Eq. ( 3 ), R 0 is the
spectral radius of a linear integral operator on

( ∂ f / ∂ x )( , γ )

ω -periodic functions with a kernel

x 0

which is defined in terms of some submatrices of .

Suppose we express the basic reproduction number for the DFE x 0

( ∂ f / ∂ x )( , γ ( t ))

of Eq. ( 1 )as

a function

R 0 = R 0 ( γ 1 ,..., γ k ) ,

and then define R 0 ( t )= R 0 ( γ 1 ( t ) ,..., γ k ( t ) , [ R 0 ]= R 0 ( γ 1 ,..., γ k ) where γ i =

ω

0 γ

1
ω

( )

t d t . i