# American Institute of Mathematical Sciences

2012, 9(2): 313-346. doi: 10.3934/mbe.2012.9.313

## The Malthusian parameter and $R_0$ for heterogeneous populations in periodic environments

 1 Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba Meguro-ku, Tokyo 153-8914, Japan

Received  June 2011 Revised  August 2011 Published  March 2012

Since the classical stable population theory in demography by Sharpe and Lotka, the sign relation ${\rm sign}(\lambda_0)={\rm sign}(R_0-1)$ between the basic reproduction number $R_0$ and the Malthusian parameter (the intrinsic rate of natural increase) $\lambda_0$ has played a central role in population theory and its applications, because it connects individual's average reproductivity described by life cycle parameters to growth character of the whole population. Since $R_0$ is originally defined for linear population evolution process in a constant environment, it is an important extension if we could formulate the same kind of threshold principle for population growth in time-heterogeneous environments.
Since the mid-1990s, several authors proposed some ideas to extend the definition of $R_0$ so that it can be applied to population dynamics in periodic environments. In particular, the definition of $R_0$ in a periodic environment by Bacaër and Guernaoui (J. Math. Biol. 53, 2006) is most important, because their definition of $R_0$ in a periodic environment can be interpreted as the asymptotic per generation growth rate, so from the generational point of view, it can be seen as a direct extension of the most successful definition of $R_0$ in a constant environment by Diekmann, Heesterbeek and Metz ( J. Math. Biol. 28, 1990).
In this paper, we propose a new approach to establish the sign relation between $R_0$ and the Malthusian parameter $\lambda_0$ for linear structured population dynamics in a periodic environment. Our arguments depend on the uniform primitivity of positive evolutionary system, which leads the weak ergodicity and the existence of exponential solution in periodic environments. For typical finite and infinite dimensional linear population models, we prove that a positive exponential solution exists and the sign relation holds between the Malthusian parameter, which is defined as the exponent of the exponential solution, and $R_0$ given by the spectral radius of the next generation operator by Bacaër and Guernaoui's definition.
Citation: Hisashi Inaba. The Malthusian parameter and $R_0$ for heterogeneous populations in periodic environments. Mathematical Biosciences & Engineering, 2012, 9 (2) : 313-346. doi: 10.3934/mbe.2012.9.313
##### References:
 [1] S. Anita, M. Iannelli, M.-Y. Kim and E.-J. Park, Optimal harvesting for periodic age-dependent population dynamics,, SIAM J. Appl. Math., 58 (1998), 1648.  doi: 10.1137/S0036139996301180.  Google Scholar [2] N. Bacaër and S. Guernaoui, The epidemic threshold of vector-borne diseases with seasonality. The case of cutaneous leishmaniasis in Chichaoua, Morocco,, J. Math. Biol., 53 (2006), 421.   Google Scholar [3] N. Bacaër and R. Ouifki, Growth rate and basic reproduction number for population models with a simple periodic factor,, Math. Biosci., 210 (2007), 647.  doi: 10.1016/j.mbs.2007.07.005.  Google Scholar [4] N. Bacaër, Approximation of the basic reproduction number $R_0$ for vector-borne diseases with a periodic vector population,, Bull. Math. Biol., 69 (2007), 1067.  doi: 10.1007/s11538-006-9166-9.  Google Scholar [5] N. Bacaër and X. Abdurahman, Resonance of the epidemic threshold in a periodic environment,, J. Math. Biol., 57 (2008), 649.  doi: 10.1007/s00285-008-0183-1.  Google Scholar [6] N. Bacaër and E. H. Ait Dads, Genealogy with seasonality, the basic reproduction number, and the influenza pandemic,, J. Math. Biol., 62 (2011), 741.   Google Scholar [7] N. Bacaër and E. H. Ait Dads, On the biological interpretation of a definition for the parameter $R_0$ in periodic population models,, J. Math. Biol., (2011).  doi: 10.1007/s00285-011-0479-4.  Google Scholar [8] G. Birkhoff, Extensions of Jentzsch's theorem,, Trans. Amer. Math. Soc., 85 (1957), 219.  doi: 10.2307/1992971.  Google Scholar [9] G. Birkhoff and R. S. Varga, Reactor criticality and nonnegative matrices,, J. Soc. Indust. Appl. Math., 6 (1958), 354.  doi: 10.1137/0106025.  Google Scholar [10] G. Birkhoff, Lattices in applied mathematics,, in, (1961), 155.   Google Scholar [11] G. Birkhoff, Positivity and criticality,, in, (1961), 116.   Google Scholar [12] G. Birkhoff, Uniformly semi-primitive multiplicative process,, Trans. Am. Math. Soc., 104 (1962), 37.  doi: 10.1090/S0002-9947-1962-0146100-6.  Google Scholar [13] G. Birkhoff, Uniformly semi-primitive multiplicative processes. II,, J. Math. Mech., 14 (1965), 507.   Google Scholar [14] G. Birkhoff, "Lattice Theory," 3rd ed.,, American Mathematical Society Colloquium Publications, (1967).   Google Scholar [15] P. J. Bushell, On the projective contraction ratio for positive linear mappings,, J. London Math. Soc. (2), 6 (1973), 256.  doi: 10.1112/jlms/s2-6.2.256.  Google Scholar [16] C. Chicone and Y. Latushkin, "Evolution Semigroups in Dynamical Systems and Differential Equations,", Mathematical Surveys and Monographs, 70 (1999).   Google Scholar [17] K. Deimling, "Nonlinear Functional Analysis,", Springer-Verlag, (1985).   Google Scholar [18] Ph. Clément, O. Diekmann, M. Gyllenberg, H. J. A. M. Heijmans and H. R. Thieme, Perturbation theory for dual semigroups. II. Time-dependent perturbations in the sun-reflexive case,, Proc. Royal Soc. Edinburgh Sect. A, 109 (1988), 145.   Google Scholar [19] O. Diekmann, H. J. A. M. Heijmans and H. R. Thieme, On the stability of the cell-size distribution. II. Time-periodic developmental rates. Hyperbolic partial differential equations, III,, Comp. Math. Appl. Part A, 12 (1986), 491.   Google Scholar [20] O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations,, J. Math. Biol., 28 (1990), 365.  doi: 10.1007/BF00178324.  Google Scholar [21] O. Diekmann and J. A. P. Heesterbeek, "Mathematical Epidemiology of Infectious Diseases. Model Building, Analysis and Interpretation,", Wiley Series in Mathematical and Computational Biology, (2000).   Google Scholar [22] O. Diekmann, J. A. P. Heesterbeek and M. G. Roberts, , The construction of next-generation matrices for compartmental epidemic models,, J. Roy. Soc. Interface 6, 7 (2010), 873.  doi: 10.1098/rsif.2009.0386.  Google Scholar [23] N. Dunford and J. T. Schwartz, "Linear Operators. Part I. General Theory,", With the assistance of W. G. Bade and R. G. Bartle, (1958).   Google Scholar [24] D. M. Ediev, On the definition of the reproductive value: Response to the discussion by Bacaër and Abdurahman,, J. Math. Biol., 59 (2009), 651.  doi: 10.1007/s00285-008-0246-3.  Google Scholar [25] F. R. Gantmacher, "The Theory of Matrices," Vol. 2,, Chelsea Publishing Company, (1959).   Google Scholar [26] J. K. Hale, "Ordinary Differential Equations,", Robert E. Krieger Pub. Co., (1980).   Google Scholar [27] J. A. P. Heesterbeek and M. G. Roberts, Threshold quantities for helminth infections,, J. Math. Biol., 33 (1995), 415.  doi: 10.1007/BF00176380.  Google Scholar [28] J. A. P. Heesterbeek and M. G. Roberts, Threshold quantities for infectious diseases in periodic environments,, J. Biol. Sys., 3 (1995), 779.  doi: 10.1142/S021833909500071X.  Google Scholar [29] J. M. Heffernan, R. J. Smith and L. M. Wahl, Perspectives on the basic reproductive ratio,, J. Roy. Soc. Interface, 2 (2005), 281.  doi: 10.1098/rsif.2005.0042.  Google Scholar [30] M. Iannelli, "Mathematical Theory of Age-Structured Population Dynamics,", Giardini Editori e Stampatori in Pisa, (1995).   Google Scholar [31] H. Inaba, A semigroup approach to the strong ergodic theorem of the multistate stable population process,, Math. Popul. Studies, 1 (1988), 49.  doi: 10.1080/08898488809525260.  Google Scholar [32] H. Inaba, Weak ergodicity of population evolution processes,, Math. Biosci., 96 (1989), 195.  doi: 10.1016/0025-5564(89)90059-X.  Google Scholar [33] H. Inaba, Threshold and stability results for an age-structured epidemic model,, J. Math. Biol., 28 (1990), 411.  doi: 10.1007/BF00178326.  Google Scholar [34] H. Inaba and H. Nishiura, The basic reproduction number of an infectious disease in a stable population: The impact of population growth rate on the eradication threshold,, Mathematical Modelling of Natural Phenomena, 3 (2008), 194.  doi: 10.1051/mmnp:2008050.  Google Scholar [35] H. Inaba and H. Nishiura, The state-reproduction number for a multistate class age structured epidemic system and its application to the asymptomatic transmission model,, Math. Biosci., 216 (2008), 77.  doi: 10.1016/j.mbs.2008.08.005.  Google Scholar [36] H. Inaba, The net reproduction rate and the type-reproduction number in multiregional demography,, Vienna Yearbook of Population Research, (2009), 197.   Google Scholar [37] H. Inaba, On a new perspective of the basic reproduction number in heterogeneous environments,, J. Math. Biol., (2011).  doi: 10.1007/s00285-011-0463-z.  Google Scholar [38] P. Jagers and O. Nerman, Branching processes in periodically varying environment,, The Annals of Probability, 13 (1985), 254.  doi: 10.1214/aop/1176993079.  Google Scholar [39] M. G. Kreĭn and M. A. Rutman, Linear operators leaving invariant a cone in a Banach space,, Uspehi. Mat. Nauk. (N.S.), 3 (1948), 3.   Google Scholar [40] C.-K. Li and H. Schneider, Applications of Perron-Frobenius theory to population dynamics,, J. Math. Biol., 44 (2002), 450.  doi: 10.1007/s002850100132.  Google Scholar [41] I. Marek, Frobenius theory of positive operators: Comparison theorems and applications,, SIAM J. Appl. Math., 19 (1970), 607.  doi: 10.1137/0119060.  Google Scholar [42] P. Michel, S. Mischler and B. Perthame, General relative entropy inequality: An illustration on growth models,, J. Math. Pures Appl. (9), 84 (2005), 1235.  doi: 10.1016/j.matpur.2005.04.001.  Google Scholar [43] A. M. Ostrowski, Positive matrices and functional analysis,, in, (1964), 81.   Google Scholar [44] I. Sawashima, On spectral properties of some positive operators,, Nat. Sci. Report Ochanomizu Univ., 15 (1964), 53.   Google Scholar [45] H. H. Schaefer and M. P. Wolff, "Topological Vector Spaces," 2nd edition,, Graduate Texts in Mathematics, 3 (1999).   Google Scholar [46] H. R. Thieme, Renewal theorems for linear periodic Volterra integral equations,, J. Inte. Equ., 7 (1984), 253.   Google Scholar [47] H. R. Thieme, Asymptotic proportionality (weak ergodicity) and conditional asymptotic equality of solutions to time-heterogeneous sublinear difference and differential equations,, J. Diff. Equ., 73 (1988), 237.  doi: 10.1016/0022-0396(88)90107-6.  Google Scholar [48] H. R. Thieme, Semiflows generated by Lipschitz perturbations of non-densely defined operators,, Differential and Integral Equations, 3 (1990), 1035.   Google Scholar [49] H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity,, SIAM J. Appl. Math., 70 (2009), 188.  doi: 10.1137/080732870.  Google Scholar [50] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,, Math. Biosci., 180 (2002), 29.  doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar [51] R. S. Varga, "Matrix Iterative Analysis," 2nd Edition,, Springer Series in Computational Mathematics, 27 (2000).   Google Scholar [52] W. Wang and X.-Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments,, J. Dyn. Diff. Equat., 20 (2008), 699.   Google Scholar [53] A. D. Ziebur, New directions in linear differential equations,, SIAM Review, 21 (1979), 57.  doi: 10.1137/1021004.  Google Scholar

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##### References:
 [1] S. Anita, M. Iannelli, M.-Y. Kim and E.-J. Park, Optimal harvesting for periodic age-dependent population dynamics,, SIAM J. Appl. Math., 58 (1998), 1648.  doi: 10.1137/S0036139996301180.  Google Scholar [2] N. Bacaër and S. Guernaoui, The epidemic threshold of vector-borne diseases with seasonality. The case of cutaneous leishmaniasis in Chichaoua, Morocco,, J. Math. Biol., 53 (2006), 421.   Google Scholar [3] N. Bacaër and R. Ouifki, Growth rate and basic reproduction number for population models with a simple periodic factor,, Math. Biosci., 210 (2007), 647.  doi: 10.1016/j.mbs.2007.07.005.  Google Scholar [4] N. Bacaër, Approximation of the basic reproduction number $R_0$ for vector-borne diseases with a periodic vector population,, Bull. Math. Biol., 69 (2007), 1067.  doi: 10.1007/s11538-006-9166-9.  Google Scholar [5] N. Bacaër and X. Abdurahman, Resonance of the epidemic threshold in a periodic environment,, J. Math. Biol., 57 (2008), 649.  doi: 10.1007/s00285-008-0183-1.  Google Scholar [6] N. Bacaër and E. H. Ait Dads, Genealogy with seasonality, the basic reproduction number, and the influenza pandemic,, J. Math. Biol., 62 (2011), 741.   Google Scholar [7] N. Bacaër and E. H. Ait Dads, On the biological interpretation of a definition for the parameter $R_0$ in periodic population models,, J. Math. Biol., (2011).  doi: 10.1007/s00285-011-0479-4.  Google Scholar [8] G. Birkhoff, Extensions of Jentzsch's theorem,, Trans. Amer. Math. Soc., 85 (1957), 219.  doi: 10.2307/1992971.  Google Scholar [9] G. Birkhoff and R. S. Varga, Reactor criticality and nonnegative matrices,, J. Soc. Indust. Appl. Math., 6 (1958), 354.  doi: 10.1137/0106025.  Google Scholar [10] G. Birkhoff, Lattices in applied mathematics,, in, (1961), 155.   Google Scholar [11] G. Birkhoff, Positivity and criticality,, in, (1961), 116.   Google Scholar [12] G. Birkhoff, Uniformly semi-primitive multiplicative process,, Trans. Am. Math. Soc., 104 (1962), 37.  doi: 10.1090/S0002-9947-1962-0146100-6.  Google Scholar [13] G. Birkhoff, Uniformly semi-primitive multiplicative processes. II,, J. Math. Mech., 14 (1965), 507.   Google Scholar [14] G. Birkhoff, "Lattice Theory," 3rd ed.,, American Mathematical Society Colloquium Publications, (1967).   Google Scholar [15] P. J. Bushell, On the projective contraction ratio for positive linear mappings,, J. London Math. Soc. (2), 6 (1973), 256.  doi: 10.1112/jlms/s2-6.2.256.  Google Scholar [16] C. Chicone and Y. Latushkin, "Evolution Semigroups in Dynamical Systems and Differential Equations,", Mathematical Surveys and Monographs, 70 (1999).   Google Scholar [17] K. Deimling, "Nonlinear Functional Analysis,", Springer-Verlag, (1985).   Google Scholar [18] Ph. Clément, O. Diekmann, M. Gyllenberg, H. J. A. M. Heijmans and H. R. Thieme, Perturbation theory for dual semigroups. II. Time-dependent perturbations in the sun-reflexive case,, Proc. Royal Soc. Edinburgh Sect. A, 109 (1988), 145.   Google Scholar [19] O. Diekmann, H. J. A. M. Heijmans and H. R. Thieme, On the stability of the cell-size distribution. II. Time-periodic developmental rates. Hyperbolic partial differential equations, III,, Comp. Math. Appl. Part A, 12 (1986), 491.   Google Scholar [20] O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations,, J. Math. Biol., 28 (1990), 365.  doi: 10.1007/BF00178324.  Google Scholar [21] O. Diekmann and J. A. P. Heesterbeek, "Mathematical Epidemiology of Infectious Diseases. Model Building, Analysis and Interpretation,", Wiley Series in Mathematical and Computational Biology, (2000).   Google Scholar [22] O. Diekmann, J. A. P. Heesterbeek and M. G. Roberts, , The construction of next-generation matrices for compartmental epidemic models,, J. Roy. Soc. Interface 6, 7 (2010), 873.  doi: 10.1098/rsif.2009.0386.  Google Scholar [23] N. Dunford and J. T. Schwartz, "Linear Operators. Part I. General Theory,", With the assistance of W. G. Bade and R. G. Bartle, (1958).   Google Scholar [24] D. M. Ediev, On the definition of the reproductive value: Response to the discussion by Bacaër and Abdurahman,, J. Math. Biol., 59 (2009), 651.  doi: 10.1007/s00285-008-0246-3.  Google Scholar [25] F. R. Gantmacher, "The Theory of Matrices," Vol. 2,, Chelsea Publishing Company, (1959).   Google Scholar [26] J. K. Hale, "Ordinary Differential Equations,", Robert E. Krieger Pub. Co., (1980).   Google Scholar [27] J. A. P. Heesterbeek and M. G. Roberts, Threshold quantities for helminth infections,, J. Math. Biol., 33 (1995), 415.  doi: 10.1007/BF00176380.  Google Scholar [28] J. A. P. Heesterbeek and M. G. Roberts, Threshold quantities for infectious diseases in periodic environments,, J. Biol. Sys., 3 (1995), 779.  doi: 10.1142/S021833909500071X.  Google Scholar [29] J. M. Heffernan, R. J. Smith and L. M. Wahl, Perspectives on the basic reproductive ratio,, J. Roy. Soc. Interface, 2 (2005), 281.  doi: 10.1098/rsif.2005.0042.  Google Scholar [30] M. Iannelli, "Mathematical Theory of Age-Structured Population Dynamics,", Giardini Editori e Stampatori in Pisa, (1995).   Google Scholar [31] H. Inaba, A semigroup approach to the strong ergodic theorem of the multistate stable population process,, Math. Popul. Studies, 1 (1988), 49.  doi: 10.1080/08898488809525260.  Google Scholar [32] H. Inaba, Weak ergodicity of population evolution processes,, Math. Biosci., 96 (1989), 195.  doi: 10.1016/0025-5564(89)90059-X.  Google Scholar [33] H. Inaba, Threshold and stability results for an age-structured epidemic model,, J. Math. Biol., 28 (1990), 411.  doi: 10.1007/BF00178326.  Google Scholar [34] H. Inaba and H. Nishiura, The basic reproduction number of an infectious disease in a stable population: The impact of population growth rate on the eradication threshold,, Mathematical Modelling of Natural Phenomena, 3 (2008), 194.  doi: 10.1051/mmnp:2008050.  Google Scholar [35] H. Inaba and H. Nishiura, The state-reproduction number for a multistate class age structured epidemic system and its application to the asymptomatic transmission model,, Math. Biosci., 216 (2008), 77.  doi: 10.1016/j.mbs.2008.08.005.  Google Scholar [36] H. Inaba, The net reproduction rate and the type-reproduction number in multiregional demography,, Vienna Yearbook of Population Research, (2009), 197.   Google Scholar [37] H. Inaba, On a new perspective of the basic reproduction number in heterogeneous environments,, J. Math. Biol., (2011).  doi: 10.1007/s00285-011-0463-z.  Google Scholar [38] P. Jagers and O. Nerman, Branching processes in periodically varying environment,, The Annals of Probability, 13 (1985), 254.  doi: 10.1214/aop/1176993079.  Google Scholar [39] M. G. Kreĭn and M. A. Rutman, Linear operators leaving invariant a cone in a Banach space,, Uspehi. Mat. Nauk. (N.S.), 3 (1948), 3.   Google Scholar [40] C.-K. Li and H. Schneider, Applications of Perron-Frobenius theory to population dynamics,, J. Math. Biol., 44 (2002), 450.  doi: 10.1007/s002850100132.  Google Scholar [41] I. Marek, Frobenius theory of positive operators: Comparison theorems and applications,, SIAM J. Appl. Math., 19 (1970), 607.  doi: 10.1137/0119060.  Google Scholar [42] P. Michel, S. Mischler and B. Perthame, General relative entropy inequality: An illustration on growth models,, J. Math. Pures Appl. (9), 84 (2005), 1235.  doi: 10.1016/j.matpur.2005.04.001.  Google Scholar [43] A. M. Ostrowski, Positive matrices and functional analysis,, in, (1964), 81.   Google Scholar [44] I. Sawashima, On spectral properties of some positive operators,, Nat. Sci. Report Ochanomizu Univ., 15 (1964), 53.   Google Scholar [45] H. H. Schaefer and M. P. Wolff, "Topological Vector Spaces," 2nd edition,, Graduate Texts in Mathematics, 3 (1999).   Google Scholar [46] H. R. Thieme, Renewal theorems for linear periodic Volterra integral equations,, J. Inte. Equ., 7 (1984), 253.   Google Scholar [47] H. R. Thieme, Asymptotic proportionality (weak ergodicity) and conditional asymptotic equality of solutions to time-heterogeneous sublinear difference and differential equations,, J. Diff. Equ., 73 (1988), 237.  doi: 10.1016/0022-0396(88)90107-6.  Google Scholar [48] H. R. Thieme, Semiflows generated by Lipschitz perturbations of non-densely defined operators,, Differential and Integral Equations, 3 (1990), 1035.   Google Scholar [49] H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity,, SIAM J. Appl. Math., 70 (2009), 188.  doi: 10.1137/080732870.  Google Scholar [50] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,, Math. Biosci., 180 (2002), 29.  doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar [51] R. S. Varga, "Matrix Iterative Analysis," 2nd Edition,, Springer Series in Computational Mathematics, 27 (2000).   Google Scholar [52] W. Wang and X.-Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments,, J. Dyn. Diff. Equat., 20 (2008), 699.   Google Scholar [53] A. D. Ziebur, New directions in linear differential equations,, SIAM Review, 21 (1979), 57.  doi: 10.1137/1021004.  Google Scholar
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