Epidemiological Models with Seasonality - Mathematical Methods and Models in Biomedicine - page 382

Fig. 1 Comparison between R and [ R 0 ] for some range of a parameter in the model

Fig. 2 Example showing that R 0 coincides with [ R 0 ] when [ R 0 ]= 1

For a malaria model with periodic coefficients it was proved in [ 3 ]thatif

[ then the DFE is globally asymptotically stable (unstable); this is

equivalent to the assertions ( 13 )and( 14 ). In the present chapter we establish Eqs.
( 13 )and( 14 ) for a model of waterborne diseases with periodic coefficients. Our
approach is similar to [ 3 ], but with some technical differences. In the concluding
section of the chapter we summarize the general features of our methods, and briefly
give additional examples.

R 0 ] < 1 ([ R 0 ] > 1 )

4 Statement of Results for a Waterborne Disease Model

Waterborne diseases are diseases which are transmitted through water. They include,
for instance, diarrhea and cholera, typhoid fever, hepatitis A, enteriditis salmonella,
and Giardia. Infection typically occurs through pathogen ingestion. Transmission
may happen when drinking sewage-contaminated water, or eating food grown
in sewage-contaminated land. Historically some of these diseases reappeared
periodically, as, for example, in the epidemic of cholera in the nineteenth century
London [ 10 ].