# American Institute of Mathematical Sciences

May  2017, 22(3): 809-829. doi: 10.3934/dcdsb.2017040

## Effects of superinfection and cost of immunity on host-parasite co-evolution

 Department of Applied Mathematics, University of Western Ontario, London, Ontario, N6A 3K7, Canada

Received  November 2015 Revised  May 2016 Published  January 2017

Fund Project: Research partially supported by NSERC of Canada.

In this paper, we investigate the cost of immunological up- regulation caused by infection in a between-host transmission dynamical model with superinfection. After introducing a mutant host to an existing model, we explore this problem in (A) monomorphic case and (B) dimorphic case. For (A), we assume that only strain 1 parasite can infect the mutant host. We identify an appropriate fitness for the invasion of the mutant host by analyzing the local stability of the mutant free equilibrium. After specifying a trade-off between the production and recovery rates of infected hosts, we employ the adaptive dynamical approach to analyze the evolutionary and convergence stabilities of the corresponding singular strategy, leading to some conditions for continuously stable strategy, evolutionary branching point and repeller. For (B), a new fitness is introduced to measure the invasion of mutant host under the assumption that both parasite strains can infect the mutant host. By considering two trade-off functions, we can study the conditions for evolutionary stability, isoclinic stability and absolute convergence stability of the singular strategy. Our results show that the host evolution would not favour high degree of immunological up-regulation; moreover, superinfection would help the parasite with weaker virulence persist in hosts.

Citation: Liman Dai, Xingfu Zou. Effects of superinfection and cost of immunity on host-parasite co-evolution. Discrete and Continuous Dynamical Systems - B, 2017, 22 (3) : 809-829. doi: 10.3934/dcdsb.2017040
##### References:
 [1] J. Apaloo, J. S. Brown and T. L. Vincent, Evolutionary game theory: ESS, convergence stability, and NIS, Evolutionary Ecology Research, 11 (2009), 489-515. [2] R. M. Anderson and R. M. May, Population biology of infectious diseases: Part Ⅰ, Nature, 280 (1979), 361-367.  doi: 10.1038/280361a0. [3] G. Beck and G. S. Habicht, Immunity and the invertebrates, Scientific American, 275 (1996), 60-66.  doi: 10.1038/scientificamerican1196-60. [4] B. Boldin and O. Diekmann, Superinfections can induce evolutionarily stable coexistence of pathogens, Journal of Mathematical Biology, 56 (2008), 635-672.  doi: 10.1007/s00285-007-0135-1. [5] B. Boldin, S. A. H. Geritz and E. Kisdi, Superinfections and adaptive dynamics of pathogen virulence revisited: A critical function analysis, Evolutionary Ecology Research, 11 (2009), 153-175. [6] R. G. Bowers, The basic depression ratio of the host: The evolution of host resistance to microparasites, Proc. Roy. Soc. Lond. B, 268 (2001), 243-250.  doi: 10.1098/rspb.2000.1360. [7] A. Bugliese, The role of host population heterogeneity in the evolution of virulence, J. Biol. Dyn., 5 (2011), 104-119.  doi: 10.1080/17513758.2010.519404. [8] C. Combes, The Art of Being a Parasite University of Chicago Press, Chicago, 2005. [9] C. Darwin, On the Origin of Species John Murray, London, 1859. [10] T. Day, On the evolution of virulence and the relationship between various measures of mortality, Proceedings of the Royal Society of London. Series B: Biological Sciences, 269 (2002), 1317-1323.  doi: 10.1098/rspb.2002.2021. [11] T. Day, Virulence evolution and the timing of disease life-history events, Trends in Ecology & Evolution, 18 (2003), 113-118.  doi: 10.1016/S0169-5347(02)00049-6. [12] T. Day and J. G. Burns, A consideration of patterns of virulence arising from host-parasite coevolution, Evolution, 57 (2003), 671-676. [13] U. Dieckmann and R. Law, The dynamical theory of coevolution: A derivation from stochastic ecological processes, J. Math. Biol., 34 (1996), 579-612.  doi: 10.1007/BF02409751. [14] S. Gandon, M. van Baalen and V. A. A. Janseny, The evolution of parasite virulence, superinfection, and host resistance, The American Naturalist, 159 (2002), 658-669. [15] S. A. H. Geritz, G. Mesz and J. A. J. Metz, Evolutionarily singular strategies and the adaptive growth and branching of the evolutionary tree, Evolutionary Ecology, 12 (1998), 35-57.  doi: 10.1023/A:1006554906681. [16] S. A. H. Geritz, J. A. J. Metz, É. Kisdi and G. Meszéna, Dynamics of adaptation and evolutionary branching, Physical Review Letters, 78 (1997), 2024-2027.  doi: 10.1103/PhysRevLett.78.2024. [17] B. Hall and B. Hallgrímsson, Strickberger's Evolution Jones & Bartlett Learning, Burlington, 2008. [18] E. Kisdi, Trade-off geometries and the adaptive dynamics of two co-evolving species, Evolutionary Ecology Research, 8 (2006), 956-973. [19] O. Leimar, Evolutionary change and Darwinian demons, Selection, 2 (2002), 65-72.  doi: 10.1556/Select.2.2001.1-2.5. [20] O. Leimar, Multidimensional convergence stability, Evolutionary Ecology Research, 11 (2009), 191-208. [21] R. L. Lochmiller and C. Deerenberg, Trade-offs in evolutionary immunology: Just what is the cost of immunity, Oikos, 88 (2000), 87-98.  doi: 10.1034/j.1600-0706.2000.880110.x. [22] J. Ma and S. A. Levin, The evolution of resource adaptation: How generalist and specialist consumers evolve, Bull. Math. Bio., 68 (2006), 1111-1123.  doi: 10.1007/s11538-006-9096-6. [23] P. Marrow, U. Dieckmann and R. Law, Evolutionary dynamics of predator-prey systems: An ecological perspective, Journal of Mathematical Biology, 34 (1996), 556-578. [24] C. Matessi and C. Di Pasquale, Long-term evolution of multilocus traits, Journal of Mathematical Biology, 34 (1996), 613-653.  doi: 10.1007/BF02409752. [25] R. M. May and M. A. Nowak, Superinfection, metapopulation dynamics, and the evolution of diversity, Journal of Theoretical Biology, 170 (1994), 95-114.  doi: 10.1006/jtbi.1994.1171. [26] J. Mosquera and F. R. Adler, Evolution of virulence: a unified framework for coinfection and superinfection, Journal of Theoretical Biology, 195 (1998), 293-313.  doi: 10.1006/jtbi.1998.0793. [27] M. Nuño, Z. Feng, M. Martcheva and C. Castillo-Chavez, Dynamics of Two-Strain Influenza with Isolation and Partial Cross-Immunity, SIAM J. Appl. Math., 65 (2005), 964-982.  doi: 10.1137/S003613990343882X. [28] Y. Pei, Closed-form conditions of bifurcation points for general differential equations, International Journal of Bifurcation and Chaos, 15 (2005), 1467-1483.  doi: 10.1142/S0218127405012582. [29] T. O. Svennungsen and É. Kisdi, Evolutionary branching of virulence in a single-infection model, Journal of Theoretical Biology, 257 (2009), 408-418.  doi: 10.1016/j.jtbi.2008.11.014. [30] H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity, SIAM, J. Appl. Math., 70 (2009), 188-211.  doi: 10.1137/080732870. [31] M. E. J. Woolhouse, J. P. Webster, E. Domingo, B. Charlesworth and B. R. Levin, Biological and biomedical implications of the co-evolution of pathogens and their hosts, Nature Genetics, 32 (2002), 569-577.  doi: 10.1038/ng1202-569.

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##### References:
 [1] J. Apaloo, J. S. Brown and T. L. Vincent, Evolutionary game theory: ESS, convergence stability, and NIS, Evolutionary Ecology Research, 11 (2009), 489-515. [2] R. M. Anderson and R. M. May, Population biology of infectious diseases: Part Ⅰ, Nature, 280 (1979), 361-367.  doi: 10.1038/280361a0. [3] G. Beck and G. S. Habicht, Immunity and the invertebrates, Scientific American, 275 (1996), 60-66.  doi: 10.1038/scientificamerican1196-60. [4] B. Boldin and O. Diekmann, Superinfections can induce evolutionarily stable coexistence of pathogens, Journal of Mathematical Biology, 56 (2008), 635-672.  doi: 10.1007/s00285-007-0135-1. [5] B. Boldin, S. A. H. Geritz and E. Kisdi, Superinfections and adaptive dynamics of pathogen virulence revisited: A critical function analysis, Evolutionary Ecology Research, 11 (2009), 153-175. [6] R. G. Bowers, The basic depression ratio of the host: The evolution of host resistance to microparasites, Proc. Roy. Soc. Lond. B, 268 (2001), 243-250.  doi: 10.1098/rspb.2000.1360. [7] A. Bugliese, The role of host population heterogeneity in the evolution of virulence, J. Biol. Dyn., 5 (2011), 104-119.  doi: 10.1080/17513758.2010.519404. [8] C. Combes, The Art of Being a Parasite University of Chicago Press, Chicago, 2005. [9] C. Darwin, On the Origin of Species John Murray, London, 1859. [10] T. Day, On the evolution of virulence and the relationship between various measures of mortality, Proceedings of the Royal Society of London. Series B: Biological Sciences, 269 (2002), 1317-1323.  doi: 10.1098/rspb.2002.2021. [11] T. Day, Virulence evolution and the timing of disease life-history events, Trends in Ecology & Evolution, 18 (2003), 113-118.  doi: 10.1016/S0169-5347(02)00049-6. [12] T. Day and J. G. Burns, A consideration of patterns of virulence arising from host-parasite coevolution, Evolution, 57 (2003), 671-676. [13] U. Dieckmann and R. Law, The dynamical theory of coevolution: A derivation from stochastic ecological processes, J. Math. Biol., 34 (1996), 579-612.  doi: 10.1007/BF02409751. [14] S. Gandon, M. van Baalen and V. A. A. Janseny, The evolution of parasite virulence, superinfection, and host resistance, The American Naturalist, 159 (2002), 658-669. [15] S. A. H. Geritz, G. Mesz and J. A. J. Metz, Evolutionarily singular strategies and the adaptive growth and branching of the evolutionary tree, Evolutionary Ecology, 12 (1998), 35-57.  doi: 10.1023/A:1006554906681. [16] S. A. H. Geritz, J. A. J. Metz, É. Kisdi and G. Meszéna, Dynamics of adaptation and evolutionary branching, Physical Review Letters, 78 (1997), 2024-2027.  doi: 10.1103/PhysRevLett.78.2024. [17] B. Hall and B. Hallgrímsson, Strickberger's Evolution Jones & Bartlett Learning, Burlington, 2008. [18] E. Kisdi, Trade-off geometries and the adaptive dynamics of two co-evolving species, Evolutionary Ecology Research, 8 (2006), 956-973. [19] O. Leimar, Evolutionary change and Darwinian demons, Selection, 2 (2002), 65-72.  doi: 10.1556/Select.2.2001.1-2.5. [20] O. Leimar, Multidimensional convergence stability, Evolutionary Ecology Research, 11 (2009), 191-208. [21] R. L. Lochmiller and C. Deerenberg, Trade-offs in evolutionary immunology: Just what is the cost of immunity, Oikos, 88 (2000), 87-98.  doi: 10.1034/j.1600-0706.2000.880110.x. [22] J. Ma and S. A. Levin, The evolution of resource adaptation: How generalist and specialist consumers evolve, Bull. Math. Bio., 68 (2006), 1111-1123.  doi: 10.1007/s11538-006-9096-6. [23] P. Marrow, U. Dieckmann and R. Law, Evolutionary dynamics of predator-prey systems: An ecological perspective, Journal of Mathematical Biology, 34 (1996), 556-578. [24] C. Matessi and C. Di Pasquale, Long-term evolution of multilocus traits, Journal of Mathematical Biology, 34 (1996), 613-653.  doi: 10.1007/BF02409752. [25] R. M. May and M. A. Nowak, Superinfection, metapopulation dynamics, and the evolution of diversity, Journal of Theoretical Biology, 170 (1994), 95-114.  doi: 10.1006/jtbi.1994.1171. [26] J. Mosquera and F. R. Adler, Evolution of virulence: a unified framework for coinfection and superinfection, Journal of Theoretical Biology, 195 (1998), 293-313.  doi: 10.1006/jtbi.1998.0793. [27] M. Nuño, Z. Feng, M. Martcheva and C. Castillo-Chavez, Dynamics of Two-Strain Influenza with Isolation and Partial Cross-Immunity, SIAM J. Appl. Math., 65 (2005), 964-982.  doi: 10.1137/S003613990343882X. [28] Y. Pei, Closed-form conditions of bifurcation points for general differential equations, International Journal of Bifurcation and Chaos, 15 (2005), 1467-1483.  doi: 10.1142/S0218127405012582. [29] T. O. Svennungsen and É. Kisdi, Evolutionary branching of virulence in a single-infection model, Journal of Theoretical Biology, 257 (2009), 408-418.  doi: 10.1016/j.jtbi.2008.11.014. [30] H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity, SIAM, J. Appl. Math., 70 (2009), 188-211.  doi: 10.1137/080732870. [31] M. E. J. Woolhouse, J. P. Webster, E. Domingo, B. Charlesworth and B. R. Levin, Biological and biomedical implications of the co-evolution of pathogens and their hosts, Nature Genetics, 32 (2002), 569-577.  doi: 10.1038/ng1202-569.
Dependence of the value of evolutionary singular point on the cost of immunological up-regulation $k_1$ and the superinfection rate $\varphi$, where $\delta= 0.095$, $b= 0.6$, $c_2=0.3$, and $\bar{g}=0.15$. From two figures, both $c_1^*(k_1)$ and $c_1^*(\varphi)$ are decreasing functions in first quadrant. In (a) and (b), the four curves are obtained by varying the value of $\mu$, respectively. In (a), the curves are moved up when $\mu$ increases. However, the movement in (b) are in two direction and more complicated than it in (a)
Singularity and Isoclinic stability: when $\delta= 0.95$, $b=10$, $\beta=0.4$, $\mu=0.2$, $k_1=0.5$, and $k_2=0.8$. We only observe the regions in first quadrant. In figure (a) and (b), we plot the solutions when n (26) and (27) are equal to zero. In figures (c) and (d), the red solid curves represents function (35) and the blue dash curves represent function (36). In shadows, both conditions (33) and (34) for isoclinic stability can be met. We adjust the value of superinfection rates $\varphi$ to observe its effects. When superinfection rate increase, the values of $\tilde{c}_1^*$ and $\tilde{c}_2^*$ also increase. The shadow area has significant change when superinfection rate changes
Absolute stability: when $\delta= 0.3$, $\varphi=10$, $b=2$, $\beta=0.4$, $\mu=0.2$, $k_1=0.1$, and $k_2=0.8$. The red dot curve represents function k1 = 0.1, and k2 = 0.8. The red dot curve represents function (35) and the blue dash curve represents function (36), too. The golden solid line stands for the formula in inequality (37). In two shadows, the conditions for absolute stability can be satisfied
Descriptions of the variables and parameters in section 3
 Notation Meaning $S_1$ Abundance of susceptible residents $S_2$ Abundance of susceptible mutants $I_{11}$ Abundance of residents infected by the parasite strain $1$ $I_{12}$ Abundance of residents infected by the parasite strain $2$ $I_{21}$ Abundance of mutants infected by the parasite strain $1$ $I_{22}$ Abundance of mutants infected by the parasite strain $2$ $b$ Birth rate of a host $\mu$ Background mortality rate of a host $\beta$ Infection rate of a host $\delta$ Disease induced death rate per host $\varphi$ Superinfection rate per host $c_1$ ($c_{1h}$) Recovery rate of a resident (mutant) host infected by parasite strain $1$ $c_2$ ($c_{2h}$) Recovery rate of a resident (or mutant) host infected by parasite strain $2$
 Notation Meaning $S_1$ Abundance of susceptible residents $S_2$ Abundance of susceptible mutants $I_{11}$ Abundance of residents infected by the parasite strain $1$ $I_{12}$ Abundance of residents infected by the parasite strain $2$ $I_{21}$ Abundance of mutants infected by the parasite strain $1$ $I_{22}$ Abundance of mutants infected by the parasite strain $2$ $b$ Birth rate of a host $\mu$ Background mortality rate of a host $\beta$ Infection rate of a host $\delta$ Disease induced death rate per host $\varphi$ Superinfection rate per host $c_1$ ($c_{1h}$) Recovery rate of a resident (mutant) host infected by parasite strain $1$ $c_2$ ($c_{2h}$) Recovery rate of a resident (or mutant) host infected by parasite strain $2$

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