Abstract
Mosquitoes are primary vectors of life-threatening diseases such as dengue, malaria, and Zika. A new control method involves releasing mosquitoes carrying bacterium Wolbachia into the natural areas to infect wild mosquitoes and block disease transmission. In this work, we use differential equations to describe Wolbachia spreading dynamics, focusing on the poorly understood effect of imperfect maternal transmission. We establish two useful identities and employ them to prove that the system exhibits monomorphic, bistable, and polymorphic dynamics, and give sufficient and necessary conditions for each case. The results suggest that the largest maternal transmission leakage rate supporting Wolbachia spreading does not necessarily increase with the fitness of infected mosquitoes. The bistable dynamics is defined by the existence of two stable equilibria, whose basins of attraction are divided by the separatrix of a saddle point. By exploring the analytical property of the separatrix with some sharp estimates, we find that Wolbachia in a completely infected population could be wiped out ultimately if the initial population size is small. Surprisingly, when the infection shortens the lifespan of infected females that would impede Wolbachia spreading, such a reversion phenomenon does not occur.
Similar content being viewed by others
References
Bian G, Xu Y, Lu P, Xie Y, Xi Z (2010) The endosymbiotic bacterium Wolbachia induces resistance to dengue virus in Aedes aegypti. PLoS Pathog 6(4):e1000833
Bian G, Joshi D, Dong Y, Lu P, Zhou G, Pan X, Xu Y, Dimopoulos G, Xi Z (2013) Wolbachia invades Anopheles stephensi populations and induces refractoriness to plasmodium infection. Science 340:748–751
Caspari E, Watson GS (1959) On the evolutionary importance of cytoplasmic sterility in mosquitoes. Evolution 13:568–570
Dutra HLC, Rocha MN, Dias FBS, Mansur SB, Caragata EP, Moreira LA (2016) Wolbachia blocks currently circulating Zika virus isolates in Brazilian Aedes aegypti mosquitoes. Cell Host Microbe 19:771–774
Dobson SL, Fox CW, Jiggins FM (2002) The effect of Wolbachia-induced cytoplasmic incompatibility on host population size in natural and manipulated systems. Proc R Soc Lond B Biol Sci 269:437–445
Farkas JZ, Hinow P (2010) Structured and unstructured continuous models for Wolbachia infections. Bull Math Biol 72:2067–2088
Hamm CA, Begun DJ, Vo A, Smith CC, Saelao P, Shaver AO, Jaenike J, Turelli M (2014) Wolbachia do not live by reproductive manipulation alone: infection polymorphism in Drosophila suzukii and D. subpulchrella. Mol Ecol 23:4871–4885
Haygood R, Turelli M (2009) Evolution of incompatibility inducing microbes in subdivided host populations. Evolution 63:432–447
Hirsch MW, Smale S, Devaney RL (2003) Differential equations, dynamical systems, and an introduction to chaos, 2nd edn. Academic Press, San Diego
Hu L, Huang M, Tang M, Yu J, Zheng B (2015) Wolbachia spread dynamics in stochastic environments. Theor Popul Biol 106:32–44
Huang M, Tang M, Yu J (2015) Wolbachia infection dynamics by reaction-diffusion equations. Sci China Math 58:77–96
Huang M, Yu J, Hu L, Zheng B (2016) Qualitative analysis for a Wolbachia infection model with diffusion. Sci China Math 59:1249–1266
Hoffmann AA, Turelli M (1988) Unidirectional incompatibility in Drosophila simulans: inheritance, geographic variation and fitness effects. Genetics 119:435–444
Hoffmann AA, Iturbeormaetxe I, Callahan AG, Phillips BL, Billington K, Axford JK, Montgomery B, Turley AP, O’Neill SL (2014) Stability of the \(w\)Mel Wolbachia infection following Invasion into Aedes aegypti populations. PLoS Negl Trop Dis 8(9):e3115–e3115
Hoffmann AA, Montgomery BL, Popovici J, Iturbe-Ormaetxe I, Johnson PH, Muzzi F, Greenfield M, Durkan M, Leong YS, Dong Y, Cook H, Axford J, Callahan AG, Kenny N, Omodei C, McGraw EA, Ryan PA, Ritchie SA, Turelli M, O’Neill SL (2011) Successful establishment of Wolbachia in Aedes populations to suppress dengue transmission. Nature 476:454–457
Keeling MJ, Jiggins FM, Read JM (2003) The invasion and coexistence of competing Wolbachia strains. Heredity 91:382–388
Kriesner P, Hoffmann AA, Lee SF, Turelli M, Weeks AR (2013) Rapid sequential spread of two Wolbachia variants in Drosophila simulans. PLoS Pathog 9(9):e1003607
Mcmeniman CJ, Lane RV, Cass BN, Fong AWC, Sidhu M, Wang YF, O’Neill SL (2009) Stable introduction of a life-shortening Wolbachia infection into the mosquito Aedes aegypi. Science 323:141–144
Nisbet RM, Gurney WSC (1983) The systematic formulation of population models for insects with dynamically varying instar duration. Theor Popul Biol 23:114–135
Rasgon JL, Styer LM, Scott TW (2003) Wolbachia-induced mortality as a mechanism to modulate pathogen transmission by vector arthropods. J Med Entomol 40:125–132
Turelli M, Hoffmann AA (1991) Rapid spread of an inherited incompatibility factor in California Drosophila. Nature 353:440–442
Turelli M, Hoffmann AA (1995) Cytoplasmic incompatibility in Drosophila simulans: dynamics and parameter estimates from natural populations. Genetics 140:1319–1338
Turelli M (2010) Cytoplasmic incompatibility in populations with overlapping generations. Evolution 64:232–241
Waltz E (2016) US reviews plan to infect mosquitoes with bacteria to stop disease. Nature 533:450–451
Walker T, Johnson PH, Moreira LA, Iturbe-Ormaetxe I, Frentiu FD, McMeniman CJ, Leong YS, Dong Y, Axford J, Kriesner P, Lloyd AL, Ritchie SA, O’Neill SL, Hoffmann AA (2011) The \(w\)Mel Wolbachia strain blocks dengue and invades caged Aedes aegypti populations. Nature 476:450–453
Weeks AR, Turelli M, Harcombe WR, Reynolds KT, Hoffmann AA (2007) From parasite to mutualist: rapid evolution of Wolbachia in natural populations of Drosophila. PLoS Biol 5(5):e114
Xi Z, Khoo CC, Dobson SL (2005) Wolbachia establishment and invasion in an Aedes aegypti laboratory population. Science 310:326–328
Zhang Z, Ding T, Huang W, Dong Z (2006) Qualitative theory of differential equations, vol 101. American Mathematical Soc, Providence
Zheng B, Tang M, Yu J (2014) Modeling Wolbachia spread in mosquitoes through delay differential equations. SIAM J Appl Math 74:743–770
Acknowledgements
This work was supported by China Scholarship Council (No. 201409945004), National Natural Science Foundation of China (11301103, 11631005, 11626246), Program for Changjiang Scholars and Innovative Research Team in University (IRT_16R16), and Guangdong Innovative Research Team program (2011S009). We thank Glenn Webb, Michael Turelli, and Zhiyong Xi for their suggestions and encouragements. We are also indebted to the two anonymous reviewers for their careful reading of the manuscript and constructive criticism.
Author information
Authors and Affiliations
Corresponding author
Appendices
Appendices
A The trace of the Jacobian matrix
Lemma A.1
Let (1.6) hold. Then tr \(\mathbf {J}<0\) at the interior equilibrium point of (1.4)–(1.5) if it is the unique one, or the one with a smaller y-coordinate.
Proof
In view of Lemma 2.2 and the labeling of equilibrium points following its proof, the current lemma assures tr \(\mathbf {J}<0\) at \(E^*(x^*, y^*)\) in Case (C2), or at \(E_2^*(x_2^*, y_2^*)\) in Cases (C3), (C4) (i), or (C5). For notational simplicity, we let \((x_2,y_2)\) stand for \((x_2^*, y_2^*)\). By the second part of (3.3), it is equivalent to prove \(H(y^*)>0\) or \(H(y_2)>0\).
Case (C2): \(\kappa < 1\), \(\beta \mu <1\), and \((\kappa +\beta \mu )^2=4\beta \mu \). The system has a unique interior equilibrium point \(E^*(x^*, y^*)\). By substituting \(y=y^*\) into (3.4), changing \((1-\delta )\kappa \) to \(-\delta \kappa \), and replacing \((\kappa +\beta \mu )^2\) with \(4\beta \mu \) when applicable, we obtain
where the last inequality follows from the assumptions \(\kappa <1\) and \(\beta \mu <1\).
Case (C3): \(\kappa <1\), \(\beta \mu <1\), and \((\kappa +\beta \mu )^2>4\beta \mu \). By changing the equality in (A.1) to “>”, we see that \(H(y^*)>0\) still holds. Instead of computing \(H(y_2)\) directly, we proceed by showing that \(H(y)-G(y)\) is positive at \(y=y_2\) as \(G(y_2)=0\). Subtracting (3.4) by (2.10) gives
with the constant coefficients
We claim \(c_2>0\). To confirm it, we first show that Condition (C3) implies
Indeed, by using (2.2) we may rewrite \(\kappa +\beta \mu \) as
Then by applying Condition (C3) we obtain
Furthermore, \(\gamma >1\) gives \((1-\mu )/\delta >\mu \), and so \(\kappa =\beta (1-\mu )/\delta >\beta \mu \). This completes the proof of (A.4). Consequently,
Of course, if \(c_1\ge 0\), then \(H(y)>G(y)\) for all \(y>0\) and there remains nothing to prove. We note that \(c_1<0\) is possible as shown by an example with \(\delta =1\), \(\kappa +\beta \mu =\beta <2\), and \(c_1=\beta -2<0\). Since \(y_2\) is the smaller root between the two distinct roots of G(y) in \((0,\kappa )\), \(y_2\) is less than the critical point of G where it takes the minimum value. Hence
Recall that \(H(y^*)>0\). Now, when \(c_1<0\), we have
Cases (C4) (i) and (C5): By changing \((1-\delta )\kappa \) to \(-\delta \kappa \) in (3.4), and then replacing \(y_2\) with \(\kappa -x_2\), we find
Since \(\kappa \ge 1\) in these cases, it follows immediately that \(H(y_2)> \kappa (\kappa -1)\ge 0\). \(\square \)
B Proof of Lemma 3.1
Proof
In Case (C2), \(E^*(x^*, y^*)\in \Gamma _y\). In Case (C3), the intersection at a point \((x,y^*)\) with \(x_1^*<x<x_2^*\) is also clear in view of (2.13). To show the uniqueness of the intersection, we first prove
when either (C2) or (C3) holds. Recall from (A.4) that \(\kappa >\beta \mu \) in Case (C3). By checking the proof of (A.4) it is easy to see that it again goes through for Case (C2) and that \(\kappa >\beta \mu \) remains valid. Now we have
As \((\kappa +\beta \mu )^2\ge 4\beta \mu \) implies \(\kappa +\beta \mu \ge 2 \sqrt{\beta \mu }\), (B.1) follows at once.
Let \((x, y^*)\in \Gamma _y\). Then \(g(x, y^*)=0\), or
We may rewrite it as a quadratic equation of x in the form
with the leading coefficient \(\beta \mu -y^*<0\) by (B.1), and \((y^*)^2- (y^*)^3>0\) as \(y^*<\kappa <1\). Hence it has at least one negative root and no more than one positive root. \(\square \)
C Proof of Lemma 3.2
Proof
Let \(a\ge \delta \). When \(x>x^*\), it is apparently true by (3.10) that \(y=h_a(x)>y^*\) in Case (C2). For Case (C3), we see from (2.13) that \(x_1^*<x^*\), \(y_1^*>y^*\), and therefore
This and (B.1) help us derive
provided that
Next we prove (3.11) for x close to \(x^*\) and \(x>x^*\). First let (C3) hold. In this case, we have \(y=h_a(x)>y_1^*\) for all \(x>x_1^*\). In the domain where \(x>x_1^*\) and \(y>y_1^*\), we have \(f(x,y)<0\) and therefore for all \(a\ge \delta \)
When \(x\in (x_1^*, x_2^*]\), the shadow point \((x_s,y_s)\) of \((x,h_a(x))\) satisfies \(y_s\in (y_2^*, y_1^*)\) in which \(G<0\). It follows that \(F_a(x, h_a(x))>0\) for \(x\in (x_1^*, x_2^*]\) which includes \((x^*, x_2^*]\) as a subinterval. This verifies (3.11) for all \(a\ge \delta \) and \(x\in (x^*, x_2^*]\) in Case (C3). Now let (C2) hold. We need a more delicate approach since \(G(y)>0\) for all \(y>y^*\) now. As \(f(x^*,y^*)=g(x^*,y^*)=0\), we have \(F_a(x^*,y^*)=0\) and so
In Case (C2), \(\kappa +\beta \mu =2\sqrt{\beta \mu }\), \(y^*=\kappa (\kappa +\beta \mu )/2=\kappa \sqrt{\beta \mu }\), and therefore
and
Let \(a\ge 2\delta \), there is \({\bar{x}}>x^*\) such that for all \(x\in (x^*, {\bar{x}}]\),
and consequently
Finally, let \(I_x=[{\bar{x}}, {{\bar{a}}}/\delta ]\) in Case (C2), or \(I_x=[x_2^*, {{\bar{a}}}/\delta ]\) in Case (C3), where \({\bar{a}}\) is given in (C.1). For all \(x\in I_x\) and \(y\in [y^*, h_\delta (x)]\), \(f(x, y)<0\) as \(x+y>\kappa \), and \(g(x, y)<0\) by Lemma 3.1. Let
Then for any \(a> M_f/m_g\) and \(x\in I_x\), we have
In conclusion, we find that (3.11) is valid if \(a>\max \{{\bar{a}}, 2\delta , M_f/m_g\}\). \(\square \)
Rights and permissions
About this article
Cite this article
Zheng, B., Tang, M., Yu, J. et al. Wolbachia spreading dynamics in mosquitoes with imperfect maternal transmission. J. Math. Biol. 76, 235–263 (2018). https://doi.org/10.1007/s00285-017-1142-5
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00285-017-1142-5
Keywords
- Population dynamics
- Wolbachia
- Imperfect maternal transmission
- Cytoplasmic incompatibility
- Monomorphism, polymorphism, bistability