The construction of next-generation matrices for compartmental epidemic models

Journal of the Royal Society Interface - Tập 7 Số 47 - Trang 873-885 - 2010
Odo Diekmann1, Hans Heesterbeek2, Michael C. Roberts3
1Department of Mathematics, Utrecht University, Budapestlaan 6, 3584 CD, Utrecht, The Netherlands
2Faculty of Veterinary Medicine, Utrecht University, Yalelaan 7, 3584 CL Utrecht, The Netherlands
3Centre for Mathematical Biology, Institute of Information & Mathematical Sciences, Massey University, Private Bag 102 904, North Shore Mail Centre, Auckland, New Zealand

Tóm tắt

The basic reproduction number ℛ 0 is arguably the most important quantity in infectious disease epidemiology. The next-generation matrix (NGM) is the natural basis for the definition and calculation of ℛ 0 where finitely many different categories of individuals are recognized. We clear up confusion that has been around in the literature concerning the construction of this matrix, specifically for the most frequently used so-called compartmental models. We present a detailed easy recipe for the construction of the NGM from basic ingredients derived directly from the specifications of the model. We show that two related matrices exist which we define to be the NGM with large domain and the NGM with small domain . The three matrices together reflect the range of possibilities encountered in the literature for the characterization of ℛ 0 . We show how they are connected and how their construction follows from the basic model ingredients, and establish that they have the same non-zero eigenvalues, the largest of which is the basic reproduction number ℛ 0 . Although we present formal recipes based on linear algebra, we encourage the construction of the NGM by way of direct epidemiological reasoning, using the clear interpretation of the elements of the NGM and of the model ingredients. We present a selection of examples as a practical guide to our methods. In the appendix we present an elementary but complete proof that ℛ 0 defined as the dominant eigenvalue of the NGM for compartmental systems and the Malthusian parameter r , the real-time exponential growth rate in the early phase of an outbreak, are connected by the properties that ℛ 0 > 1 if and only if r > 0, and ℛ 0 = 1 if and only if r = 0.

Từ khóa


Tài liệu tham khảo

10.1016/S0167-5877(97)00050-0

Diekmann O., 2000, Mathematical epidemiology of infectious diseases: model building, analysis and interpretation.

10.1007/BF00178324

10.1016/j.mbs.2008.08.005

10.1007/s002850100132

10.1137/080732870

10.1016/S0025-5564(02)00108-6

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Thông tin xuất bản

Journal of the Royal Society Interface

Tập 7 Số 47

873-885

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