Vector Preference Annihilates Backward Bifurcation and Reduces Endemicity


We propose and analyze a mathematical model of a vector-borne disease that includes vector feeding preference for carrier hosts and intrinsic incubation in hosts. Analysis of the model reveals the following novel results. We show theoretically and numerically that vector feeding preference for carrier hosts plays an important role for the existence of both the endemic equilibria and backward bifurcation when the basic reproduction number \({\mathcal {R}}_0\) is less than one. Moreover, by increasing the vector feeding preference value, backward bifurcation is eliminated and endemic equilibria for hosts and vectors are diminished. Therefore, the vector protects itself and this benefits the host. As an example of these phenomena, we present a case of Andean cutaneous leishmaniasis in Peru. We use parameter values from previous studies, primarily from Peru to introduce bifurcation diagrams and compute global sensitivity of \({\mathcal {R}}_0\) in order to quantify and understand the effects of the important parameters of our model. Global sensitivity analysis via partial rank correlation coefficient shows that \({\mathcal {R}}_0\) is highly sensitive to both sandflies feeding preference and mortality rate of sandflies.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7


  1. Abboubakar H, Buonomo B, Chitnis N (2016) Modelling the effects of malaria infection on mosquito biting behaviour and attractiveness of humans. Ricerche di matematica 65(1):329–346

    MathSciNet  MATH  Article  Google Scholar 

  2. Barradas I, Caja Rivera RM (2018) Cutaneous leishmaniasis in Peru using a vector-host model: backward bifurcation and sensitivity analysis. Math Methods Appl Sci 41:1908–1924

    MathSciNet  MATH  Article  Google Scholar 

  3. Biswas D, Datta A, Roy PK (2016a) Combating leishmaniasis through awareness campaigning: a mathematical study on media efficiency. Int J Math Eng Manag Sci 1(3):139–149

    Google Scholar 

  4. Biswas D, Roy PK, Li XZ, Basir FA, Pal J (2016b) Role of macrophage in the disease dynamics of cutaneous Leishmaniasis: a delay induced mathematical study. Commun Math Biol Neurosci, Article ID, p 4

  5. Biswas D, Kesh DK, Datta A, Chatterjee AN, Roy PK (2014) A mathematical approach to control cutaneous leishmaniasis through insecticide spraying. Sop Trans Appl Math 1(2):44–54

    Article  Google Scholar 

  6. Buonomo B, Vargas-De-Leon C (2013) Stability and bifurcation analysis of a vector-bias model of malaria transmission. Math Biosci 242(1):59–67

    MathSciNet  MATH  Article  Google Scholar 

  7. Cáceres AG et al (2004) Epidemiology of Andean cutaneous leishmaniasis: incrimination of Lutzomyia ayacuchensis (Diptera: Psychodidae) as a vector of Leishmania in geographically isolated, upland valleys of Peru. Am J Trop Med Hyg 70(6):607–612

    Article  Google Scholar 

  8. Caja R (2018) Control policies and vector behavioral effects in modeling vector–host interactions: novel applications for cutaneous leishmaniasis in Peru. Ph.D. thesis, CIMAT, Research Center in Mathematics at Guanajuato-Mexico, Department of Mathematics

  9. Castillo-Chavez BS (2004) Dynamical models of tuberculosis and their applications. Math Biosci Eng 1(2):361–404

    MathSciNet  MATH  Article  Google Scholar 

  10. C.D.C. (Center for Disease Control and Prevention) (2013) Parasites-leishmaniasis

  11. Chamchod F, Britton NF (2011) Analysis of a vector-bias model on malaria transmission. Bull Math Biol 73(3):639–657

    MathSciNet  MATH  Article  Google Scholar 

  12. Coleman RE, Edman JD (1988) Feeding-site selection of Lutzomyia longipalpis (Diptera: Psychodidae) on mice infected with Leishmania mexicana amazonensis. J Med Entomol 25(4):229–233

    Article  Google Scholar 

  13. Diekmann O, Heesterbeek JAP, Metz JA (1990) 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(4):365–382

    MathSciNet  MATH  Article  Google Scholar 

  14. Dujardin JC, Llanos-Cuentas A, Caceres A, Arana M, Dujardin JP, Guerrini F, Hamers R (1993) Molecular karyotype variation in Leishmania (Viannia) peruviana: indication of geographical populations in Peru distributed along a north-south cline. Ann Trop Med Parasitol 87(4):335–347

    Article  Google Scholar 

  15. Dushoff J, Huang W, Castillo-Chavez C (1998) Backwards bifurcations and catastrophe in simple models of fatal diseases. J Math Biol 36(3):227–248

    MathSciNet  MATH  Article  Google Scholar 

  16. Gorahava K, Rosenberger JM, Mubayi A (2015) Optimizing insecticide allocation strategies based on houses and livestock shelters for visceral leishmaniasis control in Bihar. India. Am J Trop Med Hygiene 93(1):114–122

    Article  Google Scholar 

  17. INEI (Peruvian National Institute of Statistics and Informatics). Peruvian population 1995–2015

  18. Knols BG, Meuerink J (1997) Odors influence mosquito behavior. Sci Med 4:56–63

    Google Scholar 

  19. Koella JC et al (1998) The malaria parasite, Plasmodium falciparum, increases the frequency of multiple feeding of its mosquito vector, Anopheles gambiae. Proc R Soc Lond B Biol Sci 265(1398):763–768

    Article  Google Scholar 

  20. Lainson R, Shaw JJ (1978) Epidemiology and ecology of leishmaniasis in Latin-America. Nature 273:595–600

    Article  Google Scholar 

  21. Lemon, Stanley M et al (2008) Vector-borne diseases: understanding the environmental, human health, and ecological connections. Workshop summary. En Vector-borne diseases: understanding the environmental, human health, and ecological connections. Workshop summary. National Academies Press

  22. Leonardo S et al (2004) Leishmaniasis. Peruvian. J Dermatol 14(2):82–98

    Google Scholar 

  23. Lyimo IN, Ferguson HM (2009) Ecological and evolutionary determinants of host species choice in mosquito vectors. Trends Parasitol 25(4):189–196

    Article  Google Scholar 

  24. Llanos-Cuentas EA, Davies CR, Pyke SDM, Dye C (1995) Cutaneous leishmaniasis in the Peruvian Andes: an epidemiological study of infection and immunity. Epidemiol Infect 114(02):297–318

    Article  Google Scholar 

  25. Marino S, Hogue IB, Ray CJ, Kirschner DE (2008) A methodology for performing global uncertainty and sensitivity analysis in systems biology. J Theoret Biol 254(1):178–196

    MathSciNet  MATH  Article  Google Scholar 

  26. Ministry of Health of Peru (MINSA) (2015) Bulletin of Epidemiology 2013, 2014, 2015

  27. McKay MD, Beckman RJ, Conover WJ (1979) Comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics 21(2):239–245

    MathSciNet  MATH  Google Scholar 

  28. Moore J (1995) The behavior of parasitized animals. Bioscience 45(2):89–96

    MathSciNet  Article  Google Scholar 

  29. ORegan SM, Lillie JW, Drake JM (2016) Leading indicators of mosquito-borne disease elimination. Theor Ecol 9(3):269–286

    Article  Google Scholar 

  30. O’Shea B, Rebollar-Tellez E, Ward RD, Hamilton JGC, El Naiem D, Polwart A (2002) Enhanced sandfly attraction to Leishmania-infected hosts. Trans R Soc Trop Med Hyg 96(2):117–118

    Article  Google Scholar 

  31. Pan American Health Organization-World Health Organization (2014) Leishmaniasis: small bites big threats

  32. Pedro SA, Tonnang HEZ, Abelman S (2016) Uncertainty and sensitivity analysis of a Rift Valley fever model. Appl Math Comput 279:170–186

    MathSciNet  MATH  Google Scholar 

  33. Pérez JE, Ogusuku E, Inga R, Lopez M, Monje J, Paz L, Guerra H (1994) Natural Leishmania infection of Lutzomyia spp. in Peru. Trans R Soc Trop Med Hyg 88(2):161–164

    Article  Google Scholar 

  34. Penn D, Potts WK (1998) Chemical signals and parasite-mediated sexual selection. Trends Ecol Evolut 13(10):391–396

    Article  Google Scholar 

  35. Rabinovich JE, Feliciangeli MD (2004) Parameters of Leishmania braziliensis transmission by indoor Lutzomyia ovallesi in Venezuela. Am J Trop Med Hyg 70(4):373–382

    Article  Google Scholar 

  36. Rogers ME, Bates PA (2007) Leishmania manipulation of sand fly feeding behavior results in enhanced transmission. PLoS Pathog 3(6):818–826

    Article  Google Scholar 

  37. Takken W, Verhulst NO (2013) Host preferences of blood-feeding mosquitoes. Annu Review Entomol 58:433–453

    Article  Google Scholar 

  38. The Center for Food Security and Public Health (2009) Leishmaniasis (Cutaneous and Visceral)

  39. Van den Driessche P, Watmough J (2002) Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math Biosci 180(1):29–48

    MathSciNet  MATH  Article  Google Scholar 

  40. Villaseca P, Llanos-Cuentas A, Perez E (1993) A comparative field study of the relative importance of Lutzomyia peruensis and Lutzomyia verrucarum as vectors of cutaneous leishmaniasis in the Peruvian Andes. Am J Trop Med Hyg 49(2):260–269

    Article  Google Scholar 

  41. Villavicencio Pulido G, Barradas I, Luna B (2016) Backward bifurcation for some general recovery functions. Mathematical methods in the applied sciences. Addison-Wesley, Boston

    Google Scholar 

  42. Votypka J, Pruzinova K, Hlavacova J, Volf P (2015) The effect of temperature and avian blood on Leishmania development in sand flies bulletin of SEA, vol 26

  43. W.H.O. (2016) Vector-borne diseases.

  44. Zheng Y, Rundell A (2006) Comparative study of parameter sensitivity analyses of the TCR-activated Erk-MAPK signalling pathway. IEE Proc Syst Biol 153(4):201–211

    Article  Google Scholar 

Download references


Rocío Marilyn Caja Rivera acknowledges fruitful conversations to Dr. Linda Allen (TEXAS TECH UNIVERSITY), Dr. Abraham Cáceres (UNMSM-PERU), Dr. Sergio Ibañez (INECOL-MEXICO) and she expresses gratefulness to anonymous reviewers for careful reading and valuable comments to this research.

Author information



Corresponding author

Correspondence to Rocio Caja Rivera.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.


Appendix 1

Proof of Theorem 1


As \(f(\lambda _h)\) in Eq. (10) is a quadratic function with \(A>0\), it follows that the minimum of f occurs at \(\hat{\lambda }_h= -\frac{B}{2A}>0\) with \(f(\hat{\lambda _h})=-\frac{B^{2}-4AC}{4A}\). Thus if \(B^2-4AC=0\), then system (1) has a unique endemic equilibrium.


As \(f(\lambda _h)\) in Eq. (10) is a quadratic function with \(B<0\), \(A>0\), \(C>0\), \(B^{2}-4AC>0\), it follows that the minimum of f occurs at \(\hat{\lambda }_h= -\frac{B}{2A}>0\) with \(f(\hat{\lambda _h})=-\frac{B^{2}-4AC}{4A}\). Thus if \(B^{2}-4AC>0\), then system (1) has two endemic equilibria.


For \(\alpha _v<\alpha _v^{*}\) and \({\mathcal {R}}_{0}<\sqrt{H}<1\). By hypothesis, \(A>0\), \(B>0\) and \(C>0\). Then, Eq. (10) does not have any positive root. Thus, conclusion 1.c) holds.


For \(\alpha _v<\alpha _v^{*}\) and \({\mathcal {R}}_{0} \ge 1\) we get \(B<0\) and \(C\le 0\). Then conclusion 1.d holds.


For \(\alpha _v \ge \alpha _v^{*}\) and \({\mathcal {R}}_{0} >1\) we get \(C<0\). Then system (1) has a unique endemic equilibrium.


For \(\alpha _v \ge \alpha _v^{*}\) and \({\mathcal {R}}_{0} \le 1\) we get \(C\ge 0\) and \(H>{\mathcal {R}}_{0}^2\). Then system (1) has no endemic equilibrium. \(\square \)

Appendix 2

Proof of Theorem 2

The Jacobian matrix of system (1), computed at \(E_{0}\) for \(b_1^*\), is given by :

$$\begin{aligned} J(E_0,b_1^{*}) = \begin{bmatrix} -\mu _h&\quad 0&\quad \omega _h&\quad 0&\quad -b_1^*\beta _1 \\ 0&\quad -(\sigma _h+\mu _h)&\quad 0&\quad 0&\quad b_1^*\beta _1\\ 0&\quad \sigma _h&\quad -(\mu _h+\delta _h+\omega _h)&\quad 0&\quad 0 \\ 0&\quad 0&\quad -\frac{b_2\beta _2\alpha _v\varLambda _v\mu _h}{\mu _v\varLambda _h}&\quad -\mu _v&\quad 0\\ 0&\quad 0&\quad \frac{b_2\beta _2\alpha _v\varLambda _v\mu _h}{\mu _v\varLambda _h}&\quad 0&\quad -\mu _v\\ \end{bmatrix}. \end{aligned}$$

The characteristic polynomial of the Jacobian matrix is:

$$\begin{aligned} (\lambda _h+\mu _h)(\lambda _h+\mu _v)(\lambda _h^3+\lambda _h^2 d_1+\lambda _hd_2+ d_3) \end{aligned}$$


$$\begin{aligned} d_1= & {} \mu _v+\mu _h+\delta _h+\omega _h+\sigma _h+\mu _h\\ d_2= & {} (\mu _h+\delta _h+ \omega _h)\mu _v+(\sigma _h+\mu _h)(\mu _h+\delta _h+ \omega _h)+(\sigma _h+\mu _h)\mu _v\\ d_3= & {} (\sigma _h+\mu _h)(\mu _h+\delta _h+\omega _h)\mu _v +\frac{b_1^*\beta _1\sigma _h\beta _2b_2\alpha _v\mu _h\varLambda _v}{\mu _v\varLambda _h} \end{aligned}$$

We replace the value of \(b_1^*\) in \(d_3\), and we get

$$\begin{aligned} \lambda _h(\lambda _h+\mu _h)(\lambda _h+\mu _v)(\lambda _h^2+\lambda _h d_1+ d_2)=0 \end{aligned}$$

The Jacobian matrix admits a zero eigenvalue and the other eigenvalues are real and negative. Thus, the disease-free equilibrium \(E_0\) is a non-hyperbolic equilibrium and assumption (A1) of Theorem (Castillo-Chavez 2004) is demonstrated. We indicate by \(v=(v_1, v_2, v_3,v_4,v_5)\) and \(w=(w_1,w_2,w_3,w_4,w_5)^T\), a right and a left eigenvector associated with the zero eigenvalue, respectively, such that their dot product is one \(v.w=1\). Multiplying vJ and Jw and setting each of them equal to zero yields:

$$\begin{aligned}&-\mu _h w_1+\omega _h w_3-b_1^*\beta _1 w_5=0\quad -(\sigma _h+\mu _h) w_2+b_1^*\beta _1 w_5=0\\&\sigma _h w_2-(\mu _h+\delta _h+\omega _h) w_3=0 \quad -\frac{b_2\beta _2 \alpha _v\varLambda _v\mu _h }{\mu _v\varLambda _h}w_3-\mu _vw_4=0\\&\frac{b_2\beta _2 \alpha _v\varLambda _v\mu _h }{\mu _v\varLambda _h}w_3-\mu _vw_5 =0\quad -\mu _h v_1=0\quad -(\sigma _h+\mu _h)v_2+\sigma _hv_3=0 \\&-(\mu _h +\delta _h+\omega _h)v_3+\frac{b_2\beta _2\alpha _v\varLambda _v\mu _h}{\mu _v\varLambda _h}=0 -\mu _v v_4=0 \quad b_1^{*}\beta _1v_2-\mu _vv_5=0 \\ \end{aligned}$$


$$\begin{aligned} v= & {} \left( 0,\sigma _h,\sigma _h+\mu _h,0,\frac{\mu _v\varLambda _h(\sigma _h+\mu _h)(\mu _h+\delta _h+\omega _h)}{\beta _1b_2\beta _2\alpha _v\varLambda _v\mu _h}\right) \\ w= & {} \left( -\frac{((\mu _h+\delta _h)(\mu _h+\sigma _h)+\mu _h\omega _h) \beta _1\mu _v}{\mu _h\sigma _h\theta }, \frac{(\mu _h+\delta _h+\omega _h)\beta _1\mu _v}{\theta },\right. \\&\left. \frac{\beta _1\mu _v}{\theta },-\frac{b_2\beta _2\alpha _v\mu _h\beta _1\varLambda _v}{\mu _v\varLambda _h\theta },\frac{b_2\beta _2\alpha _v\mu _h\beta _1\varLambda _v}{\mu _v\varLambda _h\theta }\right) ^T \end{aligned}$$


$$\begin{aligned} \theta =(\mu _h+\delta _h+\omega _h)(\sigma _h+\mu _h+\sigma _h\beta _1\mu _v)+\beta _1\mu _v(\sigma _h+\mu _h) \end{aligned}$$

The functions \(f_k\), \(k=1,\ldots ,5\) are the right side of the differential equations in (1a)-(1e). We define two quantities important for verification of the subcritical bifurcation

$$\begin{aligned} a=\sum _{k,i,j=1}^{5}v_kw_iw_j\frac{\partial ^2f_k}{\partial x_i\partial x_j}(E_0,b_1^*)\quad b=\sum _{k,i=1}^{5}v_kw_i\frac{\partial ^2f_k}{\partial x_i\partial b_1}(E_0,b_1^*) \end{aligned}$$

It can be checked that:

$$\begin{aligned} \frac{\partial ^2f_5}{\partial x_1 \partial x_3}= & {} \frac{\partial ^2f_5}{\partial S_h \partial I_h}=-\frac{b_2\beta _2\alpha _v\varLambda _v\mu _h^2}{\mu _v\varLambda _h^2} \frac{\partial ^2f_2}{\partial x_3 \partial x_5}=\frac{\partial ^2f_2}{\partial I_h \partial I_v}\\= & {} -\frac{\mu _v^2(\sigma _h+\mu _h)(\mu _h+\delta _h+\omega _h)}{b_2\sigma _h\beta _2\varLambda _v}\\ \frac{\partial ^2f_2}{\partial x_3 \partial x_5}= & {} \frac{\partial ^2f_2}{\partial I_h \partial I_v}=-\frac{\mu _v^{2}(\sigma _h+\mu _h)(\mu _h+\delta _h+\omega _h)}{b_2\sigma _h\beta _2\varLambda _v} \quad \frac{\partial ^2f_5}{\partial x_3^2}=\frac{\partial ^2f_5}{\partial I_h^2}\\= & {} -\frac{2b_2\beta _2\alpha _v^2\varLambda _v\mu _h^2}{\mu _v\varLambda _h^2}\\ \frac{\partial ^2f_5}{\partial x_3 \partial x_4}= & {} \frac{-b_2\beta _2\alpha _v \mu _h}{\varLambda _h} \quad \frac{\partial ^2f_5}{\partial x_5 \partial b_1}= \frac{\partial ^2f_5}{\partial I_v \partial b_1}=\beta _1. \end{aligned}$$

Accordingly to the coefficients a and b described in Theorem 4.1 of (Castillo-Chavez 2004), it follows:

$$\begin{aligned} b=\beta _1\;\hbox {is}\;\hbox {positive}, \end{aligned}$$


$$\begin{aligned} a= & {} \frac{2(\sigma _h+\mu _h)(\mu _h+\delta _h+\omega _h) \beta _1\mu _v(2\mu _v+b_2\beta _2)}{\theta ^2}\\&\times (\frac{\mu _v((\mu _h+\delta _h)(\mu _h+\sigma _h)+\mu _h\omega _h}{\mu _h\sigma _h(2\mu _v+b_2\beta _2)}-\alpha _v). \end{aligned}$$

Then, a is positive when \(\alpha _v<\frac{\mu _v((\mu _h+\delta _h)(\mu _h+\sigma _h)+\mu _h\omega _h)}{\mu _h\sigma _h(2\mu _v+b_2\beta _2)} =\alpha _v^*\). Consequently, system (1) shows backward bifurcation at \({\mathcal {R}}_0\) when \(\alpha _v< \alpha _v^*\).

On the other hand, a is always negative when

$$\begin{aligned} \alpha _v>\frac{\mu _v((\mu _h+\delta _h)(\mu _h+\sigma _h)+\mu _h\omega _h)}{\mu _h\sigma _h(2\mu _v+b_2\beta _2)} =\alpha _v^*. \end{aligned}$$

Therefore, system (1) exhibits a forward bifurcation at \({\mathcal {R}}_{0}=1\) when \(\alpha _v>\alpha _v^*\). \(\square \)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Caja Rivera, R., Barradas, I. Vector Preference Annihilates Backward Bifurcation and Reduces Endemicity. Bull Math Biol 81, 4447–4469 (2019).

Download citation


  • Vector-borne disease
  • Feeding preference
  • Backward bifurcation
  • Cutaneous leishmaniasis
  • Global sensitivity analysis