Case fatality models for epidemics in growing populations.
The asymptotically homogeneous SIR model of Thieme (1992) for growing populations, with incidence depending in a general way on total population size, is reconsidered with respect to other parameterizations that give clear insight into epidemiological relevant relations and thresholds. One important feature of the present approach is case fatality as opposed to differential mortality. Although case fatality models and differential mortality models are equivalent via a transformation in parameter space, the underlying ideas and the dynamic behaviors are different, e.g. the basic reproduction number depends on differential mortality but not on case fatality. The persistent distributions and exponents of growth of infected solutions are computed and discussed in terms of the parameters. The notion of asymptotically exponentially growing state (as opposed to stationary state or exponential solution) coined by Thieme is interpreted in terms of stability theory. Of some interest are limiting cases of models without recovery where two infected solutions exist.
Biomathematics, University of Tübingen, 72076 Tübingen, Germany. Electronic address: hadeler@uni-tuebingen.de.
Karl Peter
KP
Hadeler
Medical Biometry, University of Tübingen, 72076 Tübingen, Germany. Electronic address: klaus.dietz@uni-tuebingen.de.
Klaus
K
Dietz
Biomathematics, University of Tübingen, 72076 Tübingen, Germany; Medical Biometry, University of Tübingen, 72076 Tübingen, Germany; Department of Mathematical Sciences, Faculty of Applied Science, Umm Al-Qura University, Makkah 21955, Saudi Arabia; Mathematics Department, Faculty of Science, Mansoura University, Mansoura 35516, Egypt. Electronic address: muntaser.safan@yahoo.com.
Muntaser
M
Safan
D050936:Basic Reproduction Number; D058872:Epidemics; D006801:Humans; D015994:Incidence; D008962:Models, Theoretical; D011158:Population Growth
Asymptotically homogeneous system; Basic reproduction number; Case fatality; Epidemic model; Growing population; Stability