# 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. [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. [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. [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. [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. [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. [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. [8] G. Birkhoff, Extensions of Jentzsch's theorem,, Trans. Amer. Math. Soc., 85 (1957), 219. doi: 10.2307/1992971. [9] G. Birkhoff and R. S. Varga, Reactor criticality and nonnegative matrices,, J. Soc. Indust. Appl. Math., 6 (1958), 354. doi: 10.1137/0106025. [10] G. Birkhoff, Lattices in applied mathematics,, in, (1961), 155. [11] G. Birkhoff, Positivity and criticality,, in, (1961), 116. [12] G. Birkhoff, Uniformly semi-primitive multiplicative process,, Trans. Am. Math. Soc., 104 (1962), 37. doi: 10.1090/S0002-9947-1962-0146100-6. [13] G. Birkhoff, Uniformly semi-primitive multiplicative processes. II,, J. Math. Mech., 14 (1965), 507. [14] G. Birkhoff, "Lattice Theory," 3rd ed.,, American Mathematical Society Colloquium Publications, (1967). [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. [16] C. Chicone and Y. Latushkin, "Evolution Semigroups in Dynamical Systems and Differential Equations,", Mathematical Surveys and Monographs, 70 (1999). [17] K. Deimling, "Nonlinear Functional Analysis,", Springer-Verlag, (1985). [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. [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. [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. [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). [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. [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). [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. [25] F. R. Gantmacher, "The Theory of Matrices," Vol. 2,, Chelsea Publishing Company, (1959). [26] J. K. Hale, "Ordinary Differential Equations,", Robert E. Krieger Pub. Co., (1980). [27] J. A. P. Heesterbeek and M. G. Roberts, Threshold quantities for helminth infections,, J. Math. Biol., 33 (1995), 415. doi: 10.1007/BF00176380. [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. [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. [30] M. Iannelli, "Mathematical Theory of Age-Structured Population Dynamics,", Giardini Editori e Stampatori in Pisa, (1995). [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. [32] H. Inaba, Weak ergodicity of population evolution processes,, Math. Biosci., 96 (1989), 195. doi: 10.1016/0025-5564(89)90059-X. [33] H. Inaba, Threshold and stability results for an age-structured epidemic model,, J. Math. Biol., 28 (1990), 411. doi: 10.1007/BF00178326. [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. [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. [36] H. Inaba, The net reproduction rate and the type-reproduction number in multiregional demography,, Vienna Yearbook of Population Research, (2009), 197. [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. [38] P. Jagers and O. Nerman, Branching processes in periodically varying environment,, The Annals of Probability, 13 (1985), 254. doi: 10.1214/aop/1176993079. [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. [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. [41] I. Marek, Frobenius theory of positive operators: Comparison theorems and applications,, SIAM J. Appl. Math., 19 (1970), 607. doi: 10.1137/0119060. [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. [43] A. M. Ostrowski, Positive matrices and functional analysis,, in, (1964), 81. [44] I. Sawashima, On spectral properties of some positive operators,, Nat. Sci. Report Ochanomizu Univ., 15 (1964), 53. [45] H. H. Schaefer and M. P. Wolff, "Topological Vector Spaces," 2nd edition,, Graduate Texts in Mathematics, 3 (1999). [46] H. R. Thieme, Renewal theorems for linear periodic Volterra integral equations,, J. Inte. Equ., 7 (1984), 253. [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. [48] H. R. Thieme, Semiflows generated by Lipschitz perturbations of non-densely defined operators,, Differential and Integral Equations, 3 (1990), 1035. [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. [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. [51] R. S. Varga, "Matrix Iterative Analysis," 2nd Edition,, Springer Series in Computational Mathematics, 27 (2000). [52] W. Wang and X.-Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments,, J. Dyn. Diff. Equat., 20 (2008), 699. [53] A. D. Ziebur, New directions in linear differential equations,, SIAM Review, 21 (1979), 57. doi: 10.1137/1021004.

show all references

##### 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. [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. [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. [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. [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. [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. [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. [8] G. Birkhoff, Extensions of Jentzsch's theorem,, Trans. Amer. Math. Soc., 85 (1957), 219. doi: 10.2307/1992971. [9] G. Birkhoff and R. S. Varga, Reactor criticality and nonnegative matrices,, J. Soc. Indust. Appl. Math., 6 (1958), 354. doi: 10.1137/0106025. [10] G. Birkhoff, Lattices in applied mathematics,, in, (1961), 155. [11] G. Birkhoff, Positivity and criticality,, in, (1961), 116. [12] G. Birkhoff, Uniformly semi-primitive multiplicative process,, Trans. Am. Math. Soc., 104 (1962), 37. doi: 10.1090/S0002-9947-1962-0146100-6. [13] G. Birkhoff, Uniformly semi-primitive multiplicative processes. II,, J. Math. Mech., 14 (1965), 507. [14] G. Birkhoff, "Lattice Theory," 3rd ed.,, American Mathematical Society Colloquium Publications, (1967). [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. [16] C. Chicone and Y. Latushkin, "Evolution Semigroups in Dynamical Systems and Differential Equations,", Mathematical Surveys and Monographs, 70 (1999). [17] K. Deimling, "Nonlinear Functional Analysis,", Springer-Verlag, (1985). [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. [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. [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. [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). [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. [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). [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. [25] F. R. Gantmacher, "The Theory of Matrices," Vol. 2,, Chelsea Publishing Company, (1959). [26] J. K. Hale, "Ordinary Differential Equations,", Robert E. Krieger Pub. Co., (1980). [27] J. A. P. Heesterbeek and M. G. Roberts, Threshold quantities for helminth infections,, J. Math. Biol., 33 (1995), 415. doi: 10.1007/BF00176380. [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. [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. [30] M. Iannelli, "Mathematical Theory of Age-Structured Population Dynamics,", Giardini Editori e Stampatori in Pisa, (1995). [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. [32] H. Inaba, Weak ergodicity of population evolution processes,, Math. Biosci., 96 (1989), 195. doi: 10.1016/0025-5564(89)90059-X. [33] H. Inaba, Threshold and stability results for an age-structured epidemic model,, J. Math. Biol., 28 (1990), 411. doi: 10.1007/BF00178326. [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. [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. [36] H. Inaba, The net reproduction rate and the type-reproduction number in multiregional demography,, Vienna Yearbook of Population Research, (2009), 197. [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. [38] P. Jagers and O. Nerman, Branching processes in periodically varying environment,, The Annals of Probability, 13 (1985), 254. doi: 10.1214/aop/1176993079. [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. [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. [41] I. Marek, Frobenius theory of positive operators: Comparison theorems and applications,, SIAM J. Appl. Math., 19 (1970), 607. doi: 10.1137/0119060. [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. [43] A. M. Ostrowski, Positive matrices and functional analysis,, in, (1964), 81. [44] I. Sawashima, On spectral properties of some positive operators,, Nat. Sci. Report Ochanomizu Univ., 15 (1964), 53. [45] H. H. Schaefer and M. P. Wolff, "Topological Vector Spaces," 2nd edition,, Graduate Texts in Mathematics, 3 (1999). [46] H. R. Thieme, Renewal theorems for linear periodic Volterra integral equations,, J. Inte. Equ., 7 (1984), 253. [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. [48] H. R. Thieme, Semiflows generated by Lipschitz perturbations of non-densely defined operators,, Differential and Integral Equations, 3 (1990), 1035. [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. [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. [51] R. S. Varga, "Matrix Iterative Analysis," 2nd Edition,, Springer Series in Computational Mathematics, 27 (2000). [52] W. Wang and X.-Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments,, J. Dyn. Diff. Equat., 20 (2008), 699. [53] A. D. Ziebur, New directions in linear differential equations,, SIAM Review, 21 (1979), 57. doi: 10.1137/1021004.
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