Epidemiological Models with Seasonality - Mathematical Methods and Models in Biomedicine - page 383

Fig. 3 Flow diagram of model ( 15a )-( 15d ); broken lines indicate death

Here we consider a simple SIR model with added compartment W that tracks
pathogen concentration in water. We follow a recent model studied by Tien et al. [ 9 ]
(see also the references in [ 9 ]). The model includes four variables: susceptible (s),
infected (i) and recovered (r) individuals, and pathogens (w). In the nondimensional
form where s

+ + =

i r 1, the four variables satisfy the following system of equations:

s = μ − β

1 si

− β

2 sw

− μ s , (15a)

i

= β

1 si

+ β

2 sw

− γ i − μ i , (15b)

w = ξ ( i − w ) , (15c)

r = γ i − μ r , (15d)

where β 1 is the water reservoir-person contact rate, β 2 is the person-person contact

rate, 1 / γ is the infectious period, 1 / ξ is the pathogen lifetime in the water reservoir,

and μ is the birth/death rate. The phase space for the system ( 15a )-( 15d )is

Ω = { ( s , i , w , r , ) ; s ≥ 0 , i ≥ 0 , 0 ≤ w ≤ i , r ≥ 0 , s + i + r = 1 }. (16)

Figure 3 illustrates the flow diagram represented by model ( 15a )-( 15d ).
Some waterborne diseases are seasonal. An example is the epidemic of cholera
in the nineteenth century London [ 10 ]. To model these cases we take β 1 , β 2 and ξ

to be periodic functions in t with period ω . Thus, we shall consider the system