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

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

Liman Dai and  Xingfu Zou

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.

Keywords: Dimorphic adaptive dynamics, mutant, co-evolution, superinfection, up-regulation, trade-off function.
Mathematics Subject Classification: Primary: 92D30, 92D25; Secondary: 34C25, 34C6.
Citation: Liman Dai, Xingfu Zou. Effects of superinfection and cost of immunity on host-parasite co-evolution. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 809-829. doi: 10.3934/dcdsb.2017040
References:
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J. Apaloo, J. S. Brown and T. L. Vincent, Evolutionary game theory: ESS, convergence stability, and NIS, Evolutionary Ecology Research, 11 (2009), 489-515.   Google Scholar

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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.   Google Scholar

[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.  Google Scholar

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T. Day and J. G. Burns, A consideration of patterns of virulence arising from host-parasite coevolution, Evolution, 57 (2003), 671-676.   Google Scholar

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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.  Google Scholar

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E. Kisdi, Trade-off geometries and the adaptive dynamics of two co-evolving species, Evolutionary Ecology Research, 8 (2006), 956-973.   Google Scholar

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O. Leimar, Evolutionary change and Darwinian demons, Selection, 2 (2002), 65-72.  doi: 10.1556/Select.2.2001.1-2.5.  Google Scholar

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O. Leimar, Multidimensional convergence stability, Evolutionary Ecology Research, 11 (2009), 191-208.   Google Scholar

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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.  Google Scholar

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P. Marrow, U. Dieckmann and R. Law, Evolutionary dynamics of predator-prey systems: An ecological perspective, Journal of Mathematical Biology, 34 (1996), 556-578.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.   Google Scholar

[2]

R. M. Anderson and R. M. May, Population biology of infectious diseases: Part Ⅰ, Nature, 280 (1979), 361-367.  doi: 10.1038/280361a0.  Google Scholar

[3]

G. Beck and G. S. Habicht, Immunity and the invertebrates, Scientific American, 275 (1996), 60-66.  doi: 10.1038/scientificamerican1196-60.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

C. Combes, The Art of Being a Parasite University of Chicago Press, Chicago, 2005. Google Scholar

[9]

C. Darwin, On the Origin of Species John Murray, London, 1859. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

T. Day and J. G. Burns, A consideration of patterns of virulence arising from host-parasite coevolution, Evolution, 57 (2003), 671-676.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

B. Hall and B. Hallgrímsson, Strickberger's Evolution Jones & Bartlett Learning, Burlington, 2008. Google Scholar

[18]

E. Kisdi, Trade-off geometries and the adaptive dynamics of two co-evolving species, Evolutionary Ecology Research, 8 (2006), 956-973.   Google Scholar

[19]

O. Leimar, Evolutionary change and Darwinian demons, Selection, 2 (2002), 65-72.  doi: 10.1556/Select.2.2001.1-2.5.  Google Scholar

[20]

O. Leimar, Multidimensional convergence stability, Evolutionary Ecology Research, 11 (2009), 191-208.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  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)
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Figure 2.  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
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Figure 3.  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
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Table 1.  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 $2$
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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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