# American Institute of Mathematical Sciences

March 2018, 23(2): 609-628. doi: 10.3934/dcdsb.2018036

## Stability of travelling waves in a Wolbachia invasion

 School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia

* Corresponding author: Matthew H. Chan

Received  June 2016 Revised  September 2017 Published  December 2017

Fund Project: PSK was supported by the Australian Research Council, Discovery Project (DP160101597)

Numerous studies have examined the growth dynamics of Wolbachia within populations and the resultant rate of spatial spread. This spread is typically characterised as a travelling wave with bistable local growth dynamics due to a strong Allee effect generated from cytoplasmic incompatibility. While this rate of spread has been calculated from numerical solutions of reaction-diffusion models, none have examined the spectral stability of such travelling wave solutions. In this study we analyse the stability of a travelling wave solution generated by the reaction-diffusion model of Chan & Kim [4] by computing the essential and point spectrum of the linearised operator arising in the model. The point spectrum is computed via an Evans function using the compound matrix method, whereby we find that it has no roots with positive real part. Moreover, the essential spectrum lies strictly in the left half plane. Thus, we find that the travelling wave solution found by Chan & Kim [4] corresponding to competition between Wolbachia-infected and -uninfected mosquitoes is linearly stable. We employ a dimension counting argument to suggest that, under realistic conditions, the wavespeed corresponding to such a solution is unique.

Citation: Matthew H. Chan, Peter S. Kim, Robert Marangell. Stability of travelling waves in a Wolbachia invasion. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 609-628. doi: 10.3934/dcdsb.2018036
##### References:
 [1] L. Allen and T. J. Bridges, Numerical exterior algebra and the compound matrix method, Numerische Mathematik, 92 (2002), 197-232. doi: 10.1007/s002110100365. [2] N. H. Barton and M. Turelli, Spatial waves of advance with bistable dynamics: Cytoplasmic and genetic analogues of allee effects, The American Naturalist, 178 (2011), E48-E75. [3] C. Brelsfoard and S. Dobson, Short note: an update on the utility of Wolbachia for controlling insect vectors and disease transmission, Asia-Pacific Journal of Molecular Biology and Biotechnology, 19 (2011), 85-92. [4] M. H. Chan and P. S. Kim, Modelling a Wolbachia invasion using a slow-fast dispersal reaction-diffusion approach, Bulletin of Mathematical Biology, 75 (2013), 1501-1523. doi: 10.1007/s11538-013-9857-y. [5] R. Dautray and J. Lions, Mathematical Analysis and Numerical Methods for Science and Technology. Volume 5. Evolution Problems I, Springer-Verlag, Berlin, 1992. [6] A. Davey, An automatic orthonormalization method for solving stiff boundary-value problems, Journal of Computational Physics, 51 (1983), 343-356. doi: 10.1016/0021-9991(83)90098-0. [7] L. O. Drury, Numerical solution of Orr-Sommerfeld-type equations, Journal of Computational Physics, 37 (1980), 133-139. doi: 10.1016/0021-9991(80)90008-X. [8] P. Hancock, S. Sinkins and H. Godfray, Population dynamic models of the spread of wolbachia, The American Naturalist, 177 (2011), 323-333. [9] P. A. Hancock and H. C. J. Godfray, Modelling the spread of wolbachia in spatially heterogeneous environments, Journal of The Royal Society Interface, 9 (2012), 3045-3054. [10] K. Hilgenboecker, P. Hammerstein, P. Schlattmann, A. Telschow and J. H. Werren, How many species are infected with wolbachia?-a statistical analysis of current data, FEMS Microbiology Letters, 281 (2008), 215-220. [11] C. K. Jones, Stability of the travelling wave solution of the fitzhugh-nagumo system, Transactions of the American Mathematical Society, 286 (1984), 431-469. doi: 10.1090/S0002-9947-1984-0760971-6. [12] T. Kapitula and K. Promislow, Spectral and Dynamical Stability of Nonlinear Waves, Applied Mathematical Sciences. Springer New York, 2013. [13] M. Keeling, F. Jiggins and J. Read, The invasion and coexistence of competing wolbachia strains, Heredity (Edinb), 91 (2003), 382-388. [14] V. Ledoux, S. Malham and V. Thümmler, Grassmannian spectral shooting, Mathematics of Computation, 79 (2010), 1585-1619. doi: 10.1090/S0025-5718-10-02323-9. [15] M. A. Lewis and P. Kareiva, Allee dynamics and the spread of invading organisms, Theoretical Population Biology, 43 (1993), 141-158. [16] C. Mcmeniman, R. Lane, B. Cass, A. Fong, M. Sidhu, Y. Wang and S. O'Neill, Stable introduction of a life-shortening Wolbachia infection into the mosquito Aedes aegypti, Science, 323 (2009), 141-144. [17] M. Z. Ndii, R. I. Hickson and G. N. Mercer, Modelling the introduction of wolbachia into aedes aegypti mosquitoes to reduce dengue transmission, ANZIAM Journal, 53 (2012), 213-227. [18] B. S. Ng and W. H. Reid, An initial value method for eigenvalue problems using compound matrices, Journal of Computational Physics, 30 (1979), 125-136. doi: 10.1016/0021-9991(79)90091-3. [19] B. S. Ng and W. H. Reid, A numerical method for linear two-point boundary-value problems using compound matrices, Journal of Computational Physics, 33 (1979), 70-85. doi: 10.1016/0021-9991(79)90028-7. [20] B. Sandstede and A. Scheel, Absolute and convective instabilities of waves on unbounded and large bounded domains, Physica D: Nonlinear Phenomena, 145 (2000), 233-277. doi: 10.1016/S0167-2789(00)00114-7. [21] M. Turelli, Evolution of incompatibility-inducing microbes and their hosts, Evolution, 48 (1994), 1500-1513. [22] M. Turelli, Cytoplasmic incompatibility in populations with overlapping generations, Evolution, 64 (2010), 232-241. [23] A. P. Turley, M. P. Zalucki, S. L. O'Neill and E. A. McGraw, Transinfected Wolbachia have minimal effects on male reproductive success in Aedes aegypti, Parasites & Vectors, 6 (2013), p36. [24] T. Walker, P. Johnson, L. Moreira, I. Iturbe-Ormaetxe, F. Frentiu, C. McMeniman, Y. Leong, Y. Dong, J. Axford, P. Kriesner, A. Lloyd, S. Ritchie, S. O'Neill and A. Hoffmann, The wmel wolbachia strain blocks dengue and invades caged aedes aegypti populations, Nature, 476 (2011), 450-453.

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##### References:
 [1] L. Allen and T. J. Bridges, Numerical exterior algebra and the compound matrix method, Numerische Mathematik, 92 (2002), 197-232. doi: 10.1007/s002110100365. [2] N. H. Barton and M. Turelli, Spatial waves of advance with bistable dynamics: Cytoplasmic and genetic analogues of allee effects, The American Naturalist, 178 (2011), E48-E75. [3] C. Brelsfoard and S. Dobson, Short note: an update on the utility of Wolbachia for controlling insect vectors and disease transmission, Asia-Pacific Journal of Molecular Biology and Biotechnology, 19 (2011), 85-92. [4] M. H. Chan and P. S. Kim, Modelling a Wolbachia invasion using a slow-fast dispersal reaction-diffusion approach, Bulletin of Mathematical Biology, 75 (2013), 1501-1523. doi: 10.1007/s11538-013-9857-y. [5] R. Dautray and J. Lions, Mathematical Analysis and Numerical Methods for Science and Technology. Volume 5. Evolution Problems I, Springer-Verlag, Berlin, 1992. [6] A. Davey, An automatic orthonormalization method for solving stiff boundary-value problems, Journal of Computational Physics, 51 (1983), 343-356. doi: 10.1016/0021-9991(83)90098-0. [7] L. O. Drury, Numerical solution of Orr-Sommerfeld-type equations, Journal of Computational Physics, 37 (1980), 133-139. doi: 10.1016/0021-9991(80)90008-X. [8] P. Hancock, S. Sinkins and H. Godfray, Population dynamic models of the spread of wolbachia, The American Naturalist, 177 (2011), 323-333. [9] P. A. Hancock and H. C. J. Godfray, Modelling the spread of wolbachia in spatially heterogeneous environments, Journal of The Royal Society Interface, 9 (2012), 3045-3054. [10] K. Hilgenboecker, P. Hammerstein, P. Schlattmann, A. Telschow and J. H. Werren, How many species are infected with wolbachia?-a statistical analysis of current data, FEMS Microbiology Letters, 281 (2008), 215-220. [11] C. K. Jones, Stability of the travelling wave solution of the fitzhugh-nagumo system, Transactions of the American Mathematical Society, 286 (1984), 431-469. doi: 10.1090/S0002-9947-1984-0760971-6. [12] T. Kapitula and K. Promislow, Spectral and Dynamical Stability of Nonlinear Waves, Applied Mathematical Sciences. Springer New York, 2013. [13] M. Keeling, F. Jiggins and J. Read, The invasion and coexistence of competing wolbachia strains, Heredity (Edinb), 91 (2003), 382-388. [14] V. Ledoux, S. Malham and V. Thümmler, Grassmannian spectral shooting, Mathematics of Computation, 79 (2010), 1585-1619. doi: 10.1090/S0025-5718-10-02323-9. [15] M. A. Lewis and P. Kareiva, Allee dynamics and the spread of invading organisms, Theoretical Population Biology, 43 (1993), 141-158. [16] C. Mcmeniman, R. Lane, B. Cass, A. Fong, M. Sidhu, Y. Wang and S. O'Neill, Stable introduction of a life-shortening Wolbachia infection into the mosquito Aedes aegypti, Science, 323 (2009), 141-144. [17] M. Z. Ndii, R. I. Hickson and G. N. Mercer, Modelling the introduction of wolbachia into aedes aegypti mosquitoes to reduce dengue transmission, ANZIAM Journal, 53 (2012), 213-227. [18] B. S. Ng and W. H. Reid, An initial value method for eigenvalue problems using compound matrices, Journal of Computational Physics, 30 (1979), 125-136. doi: 10.1016/0021-9991(79)90091-3. [19] B. S. Ng and W. H. Reid, A numerical method for linear two-point boundary-value problems using compound matrices, Journal of Computational Physics, 33 (1979), 70-85. doi: 10.1016/0021-9991(79)90028-7. [20] B. Sandstede and A. Scheel, Absolute and convective instabilities of waves on unbounded and large bounded domains, Physica D: Nonlinear Phenomena, 145 (2000), 233-277. doi: 10.1016/S0167-2789(00)00114-7. [21] M. Turelli, Evolution of incompatibility-inducing microbes and their hosts, Evolution, 48 (1994), 1500-1513. [22] M. Turelli, Cytoplasmic incompatibility in populations with overlapping generations, Evolution, 64 (2010), 232-241. [23] A. P. Turley, M. P. Zalucki, S. L. O'Neill and E. A. McGraw, Transinfected Wolbachia have minimal effects on male reproductive success in Aedes aegypti, Parasites & Vectors, 6 (2013), p36. [24] T. Walker, P. Johnson, L. Moreira, I. Iturbe-Ormaetxe, F. Frentiu, C. McMeniman, Y. Leong, Y. Dong, J. Axford, P. Kriesner, A. Lloyd, S. Ritchie, S. O'Neill and A. Hoffmann, The wmel wolbachia strain blocks dengue and invades caged aedes aegypti populations, Nature, 476 (2011), 450-453.
The essential spectrum of $\mathcal{L}$ is given by $\lambda$ in the shaded regions. The blue dashed and solid lines represent $\lambda_-^{1, 2}$ and $\lambda_+^{1, 2}$ respectively. The red line indicates the absolute spectrum given by Eq. (17) and the red dot at the origin represents an eigenvalue. Note that these are not drawn to scale for visualisation purposes
The contour $C$
The image of $C$ under $D(\lambda)$, where $r_s =0.1$ and $r_b =10$
The image of $C$ under $D(\lambda)$, where $r_s =0.001$ and $r_b =500$
Plots $(a)$ and $(b)$ show the change in argument for $D[C]$ corresponding to Figures 3 and 4 respectively
Plot of the Evans function given by Eq. (26). The blue and red solid lines in both plots show $D(\lambda)$ with $\alpha = 1.1$ and $\alpha = 1$ respectively. The dashed lines in plot $(a)$ mark the edge of the absolute spectrum corresponding to each $\alpha$. For $\alpha = 1.1$, the only roots are at $\lambda =0$ and at the edge of the absolute spectrum $\lambda =-0.002607$, whereas for $\alpha = 1$ we were unable to detect a zero at the edge of the absolute spectrum due to its proximity to the origin
Solution to the boundary value problem (6). Figure $(a)$ shows the wave profile of $\hat{u}(z)$ and $\hat{v}(z)$, represented by solid and dashed lines respectively. Figure $(b)$ shows the heteroclinic connection between equilibrium states $\mathbf{e_{-}} = (1-\alpha \mu, 0)$ and $\mathbf{e_{+}} = (0, 1-\frac{\mu}{F})$, where the solid and dashed line represent the solution in $u-u'$ and $v-v'$ space respectively
A diagram showing the uniqueness of $c_*$ by dimension counting, where $\tilde{\mathbf{e}}_-$, $\tilde{\mathbf{e}}_+$ denote the equilibria at the end points of the heteroclinic orbit in (33). In the illustration, $W^{u, s}(\tilde{\mathbf{e}}_{-, +}) \times \mathcal{C}_\varepsilon$ are shown as 2-dimensional manifolds which intersect transversally in 3-dimensional space. The one dimensional intersection corresponds to the heteroclinic connection between $\tilde{\mathbf{e}}_-$ and $\tilde{\mathbf{e}}_+$ at $c_*$
Simulations corresponding to parameter set three, listed in Table 2
Simulations corresponding to parameter set two, listed in Table 3
Simulations corresponding to parameter set three, listed in Table 4
Simulations corresponding to parameter set four, listed in Table 5
Parameter values and definitions
 Symbol Definition Value $F$ Relative fecundity of uninfected to infected females $1.0526$ $s_h$ Probability of embryo death due to CI $0.45$ $\mu$ Mortality rate $0.0162$ $\alpha$ Reduction in lifespan due to infection $1.1$
 Symbol Definition Value $F$ Relative fecundity of uninfected to infected females $1.0526$ $s_h$ Probability of embryo death due to CI $0.45$ $\mu$ Mortality rate $0.0162$ $\alpha$ Reduction in lifespan due to infection $1.1$
Parameter set one
 Symbol Definition Value $F$ Relative fecundity of uninfected to infected females $1.05$ $s_h$ Probability of embryo death due to CI $0.7$ $\mu$ Mortality rate $0.03$ $\alpha$ Reduction in lifespan due to infection $1.2$
 Symbol Definition Value $F$ Relative fecundity of uninfected to infected females $1.05$ $s_h$ Probability of embryo death due to CI $0.7$ $\mu$ Mortality rate $0.03$ $\alpha$ Reduction in lifespan due to infection $1.2$
Parameter set two
 Symbol Definition Value $F$ Relative fecundity of uninfected to infected females $1.1$ $s_h$ Probability of embryo death due to CI $0.9$ $\mu$ Mortality rate $0.02$ $\alpha$ Reduction in lifespan due to infection $1.3$
 Symbol Definition Value $F$ Relative fecundity of uninfected to infected females $1.1$ $s_h$ Probability of embryo death due to CI $0.9$ $\mu$ Mortality rate $0.02$ $\alpha$ Reduction in lifespan due to infection $1.3$
Parameter set three
 Symbol Definition Value $F$ Relative fecundity of uninfected to infected females $1.1$ $s_h$ Probability of embryo death due to CI $0.8$ $\mu$ Mortality rate $0.05$ $\alpha$ Reduction in lifespan due to infection $1.2$
 Symbol Definition Value $F$ Relative fecundity of uninfected to infected females $1.1$ $s_h$ Probability of embryo death due to CI $0.8$ $\mu$ Mortality rate $0.05$ $\alpha$ Reduction in lifespan due to infection $1.2$
Parameter set four
 Symbol Definition Value $F$ Relative fecundity of uninfected to infected females $1.4$ $s_h$ Probability of embryo death due to CI $0.9$ $\mu$ Mortality rate $0.1$ $\alpha$ Reduction in lifespan due to infection $1.2$
 Symbol Definition Value $F$ Relative fecundity of uninfected to infected females $1.4$ $s_h$ Probability of embryo death due to CI $0.9$ $\mu$ Mortality rate $0.1$ $\alpha$ Reduction in lifespan due to infection $1.2$
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