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August  2022, 27(8): 4429-4453. doi: 10.3934/dcdsb.2021235

A mathematical model for biodiversity diluting transmission of zika virus through competition mechanics

1. 

Faculty of Science and Mathematics, Sultan Idris Education University, Tanjong Malim, Malaysia

2. 

School of Mathematics and Data Science, Shaanxi University of Science and Technology, Xi'an, China

* Corresponding author: Zongmin Yue

Received  March 2021 Revised  July 2021 Published  August 2022 Early access  September 2021

Whether increasing biodiversity will lead to a promotion (amplification effect) or inhibition (dilution effect) in the transmission of infectious diseases remains to be discovered. In vector-borne infectious diseases, Lyme Disease (LD) and West Nile Virus (WNV) have become typical examples of the dilution effect of biodiversity. Thus, as a vector-borne disease, biodiversity may also play a positive role in the control of the Zika virus. We developed a Zika virus model affected by biodiversity through a competitive mechanism. Through the qualitative analysis of the model, the stability condition of the disease-free equilibrium point and the control threshold of the disease - the basic reproduction number is given. Not only has the numerical analysis verified the inference results, but also it has shown the regulatory effect of the competition mechanism on Zika virus transmission. As competition limits the size of the vector population, the number of final viral infections also decreases. Besides, we also find that under certain parameter conditions, the dilution effect may disappear because of the different initial values. Finally, we emphasized the impact of human activities on biological diversity, to indirectly dilute the abundance of diversity and make the virus continuously spread.

Citation: Zongmin Yue, Fauzi Mohamed Yusof. A mathematical model for biodiversity diluting transmission of zika virus through competition mechanics. Discrete and Continuous Dynamical Systems - B, 2022, 27 (8) : 4429-4453. doi: 10.3934/dcdsb.2021235
References:
[1]

F. B. AgustoS. Bewick and W. F. Fagan, Mathematical model of Zika virus with vertical transmission, Infectious Disease Modelling, 2 (2017), 244-267.  doi: 10.1016/j.idm.2017.05.003.

[2]

A. C. Bartlett and R. T. Staten, Sterile Insect Release Method and Other Genetic Control Strategies, Radcliffe's IPM World Textbook, 1996.

[3]

G. BenelliC. L. Jeffries and T. Walker, Biological control of mosquito vectors: Past, present, and future, Insects, 7 (2016), 52.  doi: 10.3390/insects7040052.

[4]

E. Bonyah, M. A. Khan, K. O. Okosun, et al., A theoretical model for Zika virus transmission, PLoS ONE, 12 (2017), 1-18. doi: 10.1371/journal.pone.0185540.

[5]

G. Bowatte, P. Perera, G. Senevirathne, et al. Tadpoles as dengue mosquito (Aedes aegypti) egg predators, Biological Control, 67 (2013), 469-474. doi: 10.1016/j.biocontrol.2013.10.005.

[6]

V.-M. Cao-LormeauA. BlakeS. MonsS. LastéreC. Roche and J. Vanhomwegen, Guillain-barré syndrome out–break associated with Zika virus infection in French Polynesia: A case-control study, The Lancet., 387 (2016), 1531-1539.  doi: 10.1016/S0140-6736(16)00562-6.

[7]

V.-M. Cao-Lormeau, C. Roche, A. Teissier, E. Robin, A.-L. Berry, H.-P. Mallet, et al., Zika virus, French Polynesia, South Pacific, Emerg Infect Dis., 20 (2014), 1084–1086. doi: 10.3201/eid2006.140138.

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C. Castillo-Chevez and H. R. Thieme, Asymptotically autonomous epidemic models, Mathematical Population Dynamics: Analysis of Heterogeneity. Volume One, (1994), 33–50. Theory of epidemics.

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R. S. de Sousa, L. G. C. de Menezes, J. F. Felizzola, R. de Oliveira Figueiredo, T. D. de Abreu Sá, et al., Water and health in igarapé-aścu, pará, brazil, Saúde Soc. São Paulo, 25 (2016), 1095–1107. https://core.ac.uk/download/pdf/296788278.pdf

[10]

C. Ding, N. Tao and Y. Zhu, A Mathematical Model of Zika Virus and its Optimal Control, Chinese Control Conference, IEEE, 2016.

[11]

S. Escutenaire, P. Chalon, R. Verhagen, et al., Spatial and temporal dynamics of Puumala hantavirus infection in red bank vole (Clethrionomys glareolus) populations in Belgium, Virus Research, 67 (2000), 91-107. doi: 10.1016/S0168-1702(00)00136-2.

[12]

N. M. FergusonZ. M. CucunubáI. DorigattiG. L. Nedjati-GilaniC. A. DonnellyM-G. BasáñezP. Nouvellet and J. Lessler, Countering the Zika epidemic in Latin America, Science (New York, N.Y.), 353 (2016), 353-354.  doi: 10.1126/science.aag0219.

[13]

S. Funk, A. J. Kucharski, A. Camacho, R. M. Eggo, L. Yakob, L. M. Murray and W. J. Edmunds, Comparative analysis of dengue and Zika outbreak sreveals differences by setting and virus, PLOS Neglected Tropical Diseases, 10 (2016), e0005173. doi: 10.1371/journal.pntd.0005173.

[14]

Z. L. Gabriel, I. K. L. P. Paulo, A. K. Roberto, et al., Biodiversity can help prevent malaria outbreaks in tropical forests, PLoS Neglected Tropical Diseases, 7 (2013), e2139. doi: 10.1371/journal.pntd.0002139.

[15]

D. Gao, Y. Lou, D. He, et al., Prevention and control of Zika fever as a mosquito-borne and sexually transmitted disease, Scientific Reports, 6 (2016), 28070.

[16]

L. L. Giatti, A. A. Rocha, F. A. dos Santos, S. C. Bitencourt and E S. Rodrigues de Melo Pieroni, Basic sanitary conditions in Iporanga, São Paulo State, Brazil, Rev Saude Publica, 38 (2004), 1–6. https://www.scielo.br/pdf/rsp/v38n4/en_21088.pdf

[17]

J. Huang, S. Ruan, P. Yu, et al., Bifurcation analysis of a mosquito population model with a saturated release rate of sterile mosquitoes, SIAM J. Appl. Dyn. Syst., 18 (2019), 939-972. doi: 10.1137/18M1208435.

[18]

F. KeesingR. D. Holt and R. S. Ostfeld, Effects of species diversity on disease risk, Ecology Letters, 9 (2006), 485-498.  doi: 10.1111/j.1461-0248.2006.00885.x.

[19]

A. J. Kucharski, S. Funk, R. M. Eggo, H.-P. Mallet, W. J. Edmunds and E. J.Nilles, Transmission dynamics of Zika virus in island populations: A modelling analysis of the 2013-2014 French polynesia outbreak, PLOS Neglected Tropical Diseases, 10 (2016), e0004726. doi: 10.1371/journal.pntd.0004726.

[20]

D. F. A. L, G. González-Parra and T. Benincasa, Mathematical'modeling and numerical simulations of Zika in Colombia considering mutation, Math. Comput. Simulation, 163 (2019), 1–18. doi: 10.1016/j.matcom.2019.02.009.

[21]

Lioyd Wen Feng Lee and Mohd Hafiz Mohd, The biodiversity effect in regulating the prevalence of Sin Nombre virus (SNV), Malaysian Journal of Fundamental and Applied Sciences, 16 (2020), 271-276. 

[22]

J. Li, L. Cai and Y. Li, Stage-structured wild and sterile mosquito population models and their dynamics, Journal of Biological Dynamics, (2016), 79–101. doi: 10.1080/17513758.2016.1159740.

[23]

A. D. LuisA. J. Kuenzi and J. N. Mills, Species diversity concurrently dilutes and amplifies transmission in a zoonotic host–pathogen system through competing mechanisms, Proceedings of the National Academy of Sciences, 115 (2018), 7979-7984.  doi: 10.1073/pnas.1807106115.

[24] Z. Ma and Y. Zhou, Qualitative and Stability Methods for Ordinary Differential Equations, Science Press, Beijing China, 2001. 
[25]

C. A. Manore, K. S. Hickmann, S. Xu, et al., Comparing dengue and chikungunya emergence and endemic transmission in A. aegypti and A. albopictus, J. Theoret. Biol., 356 (2014), 174-191. doi: 10.1016/j.jtbi.2014.04.033.

[26]

T. Y. MiyaokaS. Lenhart and J. F. C. A. Meyer, Optimal control of vaccination in a vector–borne reaction–diffusion model applied to Zika virus, J. Math. Biol., 79 (2019), 1077-1104.  doi: 10.1007/s00285-019-01390-z.

[27]

United Nations, Convention on Biological Diversity; 1992., Available from: https://www.cbd.int/convention/text/default.shtml. Accessed July 27, 2014.

[28]

A. S. Oliveira MeloG. MalingerR. XimenesP. O. SzejnfeldS. Alves Sampaio and A. M. Bispo de Filippis, Zika virus intrauterine infection causes fetal brain abnormality and microcephaly: Tip of the iceberg?, Ultrasound in Obstetrics & Gynecology, 47 (2016), 6-7.  doi: 10.1002/uog.15831.

[29]

R. R. PatilCh. Satish Kumar and M. Bagvandas, Biodiversity loss: Public health risk of disease spread and epidemics, Annals of Tropical Medicine and Public Health, 23 (2017), 1432-1438. 

[30]

I. D. Peixoto and G. Abramson, The effect of biodversity on the Hantavirus epizootic, The Ecological Society of America, 87 (2006), 873-879. 

[31]

P. Suparit, A. Wiratsudakul and C. Modchang, A mathematical model for Zika virus transmission dynamics with a time dependent mosquito biting rate, Theoretical Biology and Medical Modelling 2018, 15 (2018), Article number: 11. doi: 10.1186/s12976-018-0083-z.

[32]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibrium for compartmental models of disease transmission, Mathematical Biosciences, 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6.

[33]

Z. WenH. Song and G.-L. Ming, How does Zika virus cause microcephaly?, Genns & Development, 31 (2017), 849-861.  doi: 10.1101/gad.298216.117.

[34]

WHO declared the Zika epidemic an "International Public Health Emergency".

[35]

World Health Organization, Neurological Syndrome, Congenital Malformations, and Zika Virus Infection, implications for public health in the Americas. Epidemiological Alert. 2015. http://www.paho.org/hq/index.php?option=com_docman&task=doc_view&Itemid=270&gid=32405&lang=en

[36]

H. YinC. YangX. Zhang and J. Li, Dynamics of malaria transmission model with sterile mosquitoes, J. Biol. Dyn., 12 (2018), 577-595.  doi: 10.1080/17513758.2018.1498983.

[37]

F. M. YusofF. A. Abdullah and A. I. M. Ismail, Modeling and optimal control on the spread of Hantavirus infection, Mathematics., 7 (2019), 1-11.  doi: 10.3390/math7121192.

[38]

F. M. Yusof, A. Azmi, M. H. Mohd and A. I. M. Ismail, Effect of biodiversity on the spread of leptospirosis infection, In Proceedings of the International Conference on Mathematical Sciences and Technology 2018 (MathTech 2018), The Hotel Equatorial Penang, Malaysia, (2018), 10–12.

[39]

Zika virus introduced by WHO, https://www.who.int/mediacentre/factsheets/Zika/en/.

show all references

References:
[1]

F. B. AgustoS. Bewick and W. F. Fagan, Mathematical model of Zika virus with vertical transmission, Infectious Disease Modelling, 2 (2017), 244-267.  doi: 10.1016/j.idm.2017.05.003.

[2]

A. C. Bartlett and R. T. Staten, Sterile Insect Release Method and Other Genetic Control Strategies, Radcliffe's IPM World Textbook, 1996.

[3]

G. BenelliC. L. Jeffries and T. Walker, Biological control of mosquito vectors: Past, present, and future, Insects, 7 (2016), 52.  doi: 10.3390/insects7040052.

[4]

E. Bonyah, M. A. Khan, K. O. Okosun, et al., A theoretical model for Zika virus transmission, PLoS ONE, 12 (2017), 1-18. doi: 10.1371/journal.pone.0185540.

[5]

G. Bowatte, P. Perera, G. Senevirathne, et al. Tadpoles as dengue mosquito (Aedes aegypti) egg predators, Biological Control, 67 (2013), 469-474. doi: 10.1016/j.biocontrol.2013.10.005.

[6]

V.-M. Cao-LormeauA. BlakeS. MonsS. LastéreC. Roche and J. Vanhomwegen, Guillain-barré syndrome out–break associated with Zika virus infection in French Polynesia: A case-control study, The Lancet., 387 (2016), 1531-1539.  doi: 10.1016/S0140-6736(16)00562-6.

[7]

V.-M. Cao-Lormeau, C. Roche, A. Teissier, E. Robin, A.-L. Berry, H.-P. Mallet, et al., Zika virus, French Polynesia, South Pacific, Emerg Infect Dis., 20 (2014), 1084–1086. doi: 10.3201/eid2006.140138.

[8]

C. Castillo-Chevez and H. R. Thieme, Asymptotically autonomous epidemic models, Mathematical Population Dynamics: Analysis of Heterogeneity. Volume One, (1994), 33–50. Theory of epidemics.

[9]

R. S. de Sousa, L. G. C. de Menezes, J. F. Felizzola, R. de Oliveira Figueiredo, T. D. de Abreu Sá, et al., Water and health in igarapé-aścu, pará, brazil, Saúde Soc. São Paulo, 25 (2016), 1095–1107. https://core.ac.uk/download/pdf/296788278.pdf

[10]

C. Ding, N. Tao and Y. Zhu, A Mathematical Model of Zika Virus and its Optimal Control, Chinese Control Conference, IEEE, 2016.

[11]

S. Escutenaire, P. Chalon, R. Verhagen, et al., Spatial and temporal dynamics of Puumala hantavirus infection in red bank vole (Clethrionomys glareolus) populations in Belgium, Virus Research, 67 (2000), 91-107. doi: 10.1016/S0168-1702(00)00136-2.

[12]

N. M. FergusonZ. M. CucunubáI. DorigattiG. L. Nedjati-GilaniC. A. DonnellyM-G. BasáñezP. Nouvellet and J. Lessler, Countering the Zika epidemic in Latin America, Science (New York, N.Y.), 353 (2016), 353-354.  doi: 10.1126/science.aag0219.

[13]

S. Funk, A. J. Kucharski, A. Camacho, R. M. Eggo, L. Yakob, L. M. Murray and W. J. Edmunds, Comparative analysis of dengue and Zika outbreak sreveals differences by setting and virus, PLOS Neglected Tropical Diseases, 10 (2016), e0005173. doi: 10.1371/journal.pntd.0005173.

[14]

Z. L. Gabriel, I. K. L. P. Paulo, A. K. Roberto, et al., Biodiversity can help prevent malaria outbreaks in tropical forests, PLoS Neglected Tropical Diseases, 7 (2013), e2139. doi: 10.1371/journal.pntd.0002139.

[15]

D. Gao, Y. Lou, D. He, et al., Prevention and control of Zika fever as a mosquito-borne and sexually transmitted disease, Scientific Reports, 6 (2016), 28070.

[16]

L. L. Giatti, A. A. Rocha, F. A. dos Santos, S. C. Bitencourt and E S. Rodrigues de Melo Pieroni, Basic sanitary conditions in Iporanga, São Paulo State, Brazil, Rev Saude Publica, 38 (2004), 1–6. https://www.scielo.br/pdf/rsp/v38n4/en_21088.pdf

[17]

J. Huang, S. Ruan, P. Yu, et al., Bifurcation analysis of a mosquito population model with a saturated release rate of sterile mosquitoes, SIAM J. Appl. Dyn. Syst., 18 (2019), 939-972. doi: 10.1137/18M1208435.

[18]

F. KeesingR. D. Holt and R. S. Ostfeld, Effects of species diversity on disease risk, Ecology Letters, 9 (2006), 485-498.  doi: 10.1111/j.1461-0248.2006.00885.x.

[19]

A. J. Kucharski, S. Funk, R. M. Eggo, H.-P. Mallet, W. J. Edmunds and E. J.Nilles, Transmission dynamics of Zika virus in island populations: A modelling analysis of the 2013-2014 French polynesia outbreak, PLOS Neglected Tropical Diseases, 10 (2016), e0004726. doi: 10.1371/journal.pntd.0004726.

[20]

D. F. A. L, G. González-Parra and T. Benincasa, Mathematical'modeling and numerical simulations of Zika in Colombia considering mutation, Math. Comput. Simulation, 163 (2019), 1–18. doi: 10.1016/j.matcom.2019.02.009.

[21]

Lioyd Wen Feng Lee and Mohd Hafiz Mohd, The biodiversity effect in regulating the prevalence of Sin Nombre virus (SNV), Malaysian Journal of Fundamental and Applied Sciences, 16 (2020), 271-276. 

[22]

J. Li, L. Cai and Y. Li, Stage-structured wild and sterile mosquito population models and their dynamics, Journal of Biological Dynamics, (2016), 79–101. doi: 10.1080/17513758.2016.1159740.

[23]

A. D. LuisA. J. Kuenzi and J. N. Mills, Species diversity concurrently dilutes and amplifies transmission in a zoonotic host–pathogen system through competing mechanisms, Proceedings of the National Academy of Sciences, 115 (2018), 7979-7984.  doi: 10.1073/pnas.1807106115.

[24] Z. Ma and Y. Zhou, Qualitative and Stability Methods for Ordinary Differential Equations, Science Press, Beijing China, 2001. 
[25]

C. A. Manore, K. S. Hickmann, S. Xu, et al., Comparing dengue and chikungunya emergence and endemic transmission in A. aegypti and A. albopictus, J. Theoret. Biol., 356 (2014), 174-191. doi: 10.1016/j.jtbi.2014.04.033.

[26]

T. Y. MiyaokaS. Lenhart and J. F. C. A. Meyer, Optimal control of vaccination in a vector–borne reaction–diffusion model applied to Zika virus, J. Math. Biol., 79 (2019), 1077-1104.  doi: 10.1007/s00285-019-01390-z.

[27]

United Nations, Convention on Biological Diversity; 1992., Available from: https://www.cbd.int/convention/text/default.shtml. Accessed July 27, 2014.

[28]

A. S. Oliveira MeloG. MalingerR. XimenesP. O. SzejnfeldS. Alves Sampaio and A. M. Bispo de Filippis, Zika virus intrauterine infection causes fetal brain abnormality and microcephaly: Tip of the iceberg?, Ultrasound in Obstetrics & Gynecology, 47 (2016), 6-7.  doi: 10.1002/uog.15831.

[29]

R. R. PatilCh. Satish Kumar and M. Bagvandas, Biodiversity loss: Public health risk of disease spread and epidemics, Annals of Tropical Medicine and Public Health, 23 (2017), 1432-1438. 

[30]

I. D. Peixoto and G. Abramson, The effect of biodversity on the Hantavirus epizootic, The Ecological Society of America, 87 (2006), 873-879. 

[31]

P. Suparit, A. Wiratsudakul and C. Modchang, A mathematical model for Zika virus transmission dynamics with a time dependent mosquito biting rate, Theoretical Biology and Medical Modelling 2018, 15 (2018), Article number: 11. doi: 10.1186/s12976-018-0083-z.

[32]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibrium for compartmental models of disease transmission, Mathematical Biosciences, 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6.

[33]

Z. WenH. Song and G.-L. Ming, How does Zika virus cause microcephaly?, Genns & Development, 31 (2017), 849-861.  doi: 10.1101/gad.298216.117.

[34]

WHO declared the Zika epidemic an "International Public Health Emergency".

[35]

World Health Organization, Neurological Syndrome, Congenital Malformations, and Zika Virus Infection, implications for public health in the Americas. Epidemiological Alert. 2015. http://www.paho.org/hq/index.php?option=com_docman&task=doc_view&Itemid=270&gid=32405&lang=en

[36]

H. YinC. YangX. Zhang and J. Li, Dynamics of malaria transmission model with sterile mosquitoes, J. Biol. Dyn., 12 (2018), 577-595.  doi: 10.1080/17513758.2018.1498983.

[37]

F. M. YusofF. A. Abdullah and A. I. M. Ismail, Modeling and optimal control on the spread of Hantavirus infection, Mathematics., 7 (2019), 1-11.  doi: 10.3390/math7121192.

[38]

F. M. Yusof, A. Azmi, M. H. Mohd and A. I. M. Ismail, Effect of biodiversity on the spread of leptospirosis infection, In Proceedings of the International Conference on Mathematical Sciences and Technology 2018 (MathTech 2018), The Hotel Equatorial Penang, Malaysia, (2018), 10–12.

[39]

Zika virus introduced by WHO, https://www.who.int/mediacentre/factsheets/Zika/en/.

Figure 1.  The phase portraits of model (3). The parameters are taken as $ \Lambda_M = 0.5, K = 4, \kappa = 4 $. $ a $ and $ q $ are marked under each image. Fig. 1(A) represents $ \overline{E}_{2*}^{{}} = (1.5087,2.9826) $is a nodal sink, $ \overline{E}_{1*}^{{}} = (2.6513,0.69737) $is a saddle point and $ \overline{E}_{0}^{{}} = (3.2861,0) $is a nodal sink. Fig. 1(B) shows is a nodal sink and $ \overline{E}_{0}^{{}} = (3.2861,0) $is a saddle point. A asymptotically stable non-double node $ \overline{E}_{0*}^{{}} = (2.0001,1.7331) $ is shown in Fig. 1(C). In Fig. 1(D), there exist a globally stable node $ \overline{E}_{0*}^{{}} = (2.0767,4.754) $. The dotted curve is the two nullclines
Figure 2.  Take $ a = 0.2,\varepsilon = 2,q = 3,{{\Lambda }_{M}} = 0.5,{{\Lambda }_{H}} = 0.2 $. (A) Initial value is ($ 1, 0.5, 0.6, 0.2, 1.5, 0.05, 0.05, 1 $), a stable positive equilibrium point exists. (B) Change the initial value to ($ 0.02, 0.02, 0.01, 0.01, 0.15, 0.05, 0.05, 0.05 $), then the aliens $ Z(t) $ convert to zero. The system stabilizes to its boundary equilibrium point
Figure 3.  Taking $ a = 0.2,\varepsilon = 0.6, q = 1, \Lambda_M = 0.5,\Lambda _H = 0.2 $, then $ \varepsilon q<1 $. Interior positive equilibrium point$ E_{{}}^{*} $is stable
Figure 4.  Taking $ a = 0.6, \varepsilon = 2, q = 3, \Lambda_M = 0.5, \Lambda_H = 0.02 $, $ E_0 $ is stable globally
Figure 5.  Taking $ a = 0.2,\varepsilon = 2,q = 3,{{\Lambda }_{M}} = 0.5,{{\Lambda }_{H}} = 0.2 $. Initial value is ($ 1, 0.5, 0.6, 0.2, 1.5, 0.05, 0.05, 1 $). The trends of each variable during a same period
Figure 6.  Take $ {{\Lambda }_{M}} = 0.5 $ and $ {{\Lambda }_{H}} = 0.05,{{\Lambda }_{H}} = 0.02,{{\Lambda }_{H}} = 0.002 $, respectively. $ {{R}_{0}} $ increases as $ {{N}_{M}} $ increases
Figure 7.  Co-dimension 2 bifurcation diagrams show the distribution of equilibrium points of system (3) and their stability, where (A)$ \varepsilon = 2,q = 3 $(B)$ \varepsilon = 0.5,q = 0.5 $
Figure 8.  Take $ a = 0.31 $, $ q = 3,{{\Lambda }_{M}} = 0.5,{{\Lambda }_{H}} = 0.02 $. By decreasing $ \varepsilon $ from 3 to 1 unit, mosquitoes become less competitive
Figure 9.  Take $ a = 0.31 $, $ \varepsilon = 2,{{\Lambda }_{M}} = 0.5,{{\Lambda }_{H}} = 0.02 $. With $ q $ decreasing from 3 to 1 units, mosquitoes become more competitive
Figure 10.  Taking $ \varepsilon = 2,a = 0.31 $. By increasing the maximum capacity of the environment for mosquitoes $ K $, the total mosquito size increases with the same $ q $
Table 1.  Description of the parameters used in model (2)
Symbol Description
$ {\Lambda _H} $ Recruitment rate of susceptible humans
$ {\Lambda _M} $ Recruitment rate of susceptible mosquitoes
$ {\mu _H} $ Natural death rate in humans
$ {\mu _M} $ Natural death rate in mosquitoes
$ {\beta _H} $ Mosquito-to-human transmission rate
$ {\beta _M} $ Human-to-mosquito transmission rate
$ {\alpha _H} $ The rate of exposed humans moving into infectious class
$ {\rho _{}} $ Human factor transmission rate
$ {r_{}} $ Human recovery rate
$ {\delta _M} $ The rate flow from $ E_M $ to $ I_M $
$ K $ The maximum environmental capacity for mosquitoes without alien
$ \kappa $ The maximum environmental capacity for aliens without mosquitoes
$ q $ and $ \varepsilon $ The inhibition between aliens and mosquitoes
$ a $ Alien's natural growth rate
Symbol Description
$ {\Lambda _H} $ Recruitment rate of susceptible humans
$ {\Lambda _M} $ Recruitment rate of susceptible mosquitoes
$ {\mu _H} $ Natural death rate in humans
$ {\mu _M} $ Natural death rate in mosquitoes
$ {\beta _H} $ Mosquito-to-human transmission rate
$ {\beta _M} $ Human-to-mosquito transmission rate
$ {\alpha _H} $ The rate of exposed humans moving into infectious class
$ {\rho _{}} $ Human factor transmission rate
$ {r_{}} $ Human recovery rate
$ {\delta _M} $ The rate flow from $ E_M $ to $ I_M $
$ K $ The maximum environmental capacity for mosquitoes without alien
$ \kappa $ The maximum environmental capacity for aliens without mosquitoes
$ q $ and $ \varepsilon $ The inhibition between aliens and mosquitoes
$ a $ Alien's natural growth rate
Table 2.  Different expressions of positive equilibrium points of model (3)
$\varepsilon q > 1$ $\varepsilon q = 1$ $\varepsilon q < 1$
$\Delta > 0$ $\Delta = 0$ —— $\Delta < 0$
$a > \frac{{\varepsilon N_{M1}^*}}{\kappa }$ $\frac{{\varepsilon N_{M2}^*}}{\kappa } < a < \frac{{\varepsilon N_{M1}^*}}{\kappa }$ $a > \frac{{\varepsilon N_{M0}^*}}{\kappa }$ $a > \frac{{\varepsilon N_{M0}^*}}{\kappa }$ $a > \frac{{\varepsilon N_{M0}^*}}{\kappa }$
Two equilibrium One equilibrium one equilibrium One equilibrium One equilibrium
$\begin{array}{l} N_{M1}^* = N_{M1}^{}\\N_{M2}^* = N_{M2}^{} \end{array}$ $N_{M2}^* = N_{M2}^{}$ $\begin{array}{c} N_{M0}^* = \frac{{K{\mu _M} + \kappa aq}}{{2(\varepsilon q - 1)}} \end{array} $ $\begin{array}{c} N_{M0}^* =\frac{{K{\Lambda _M}}}{{K{\mu _M} + \kappa aq}} \end{array}$ $N_{M0}^* = N_{M2}^{}$
$\varepsilon q > 1$ $\varepsilon q = 1$ $\varepsilon q < 1$
$\Delta > 0$ $\Delta = 0$ —— $\Delta < 0$
$a > \frac{{\varepsilon N_{M1}^*}}{\kappa }$ $\frac{{\varepsilon N_{M2}^*}}{\kappa } < a < \frac{{\varepsilon N_{M1}^*}}{\kappa }$ $a > \frac{{\varepsilon N_{M0}^*}}{\kappa }$ $a > \frac{{\varepsilon N_{M0}^*}}{\kappa }$ $a > \frac{{\varepsilon N_{M0}^*}}{\kappa }$
Two equilibrium One equilibrium one equilibrium One equilibrium One equilibrium
$\begin{array}{l} N_{M1}^* = N_{M1}^{}\\N_{M2}^* = N_{M2}^{} \end{array}$ $N_{M2}^* = N_{M2}^{}$ $\begin{array}{c} N_{M0}^* = \frac{{K{\mu _M} + \kappa aq}}{{2(\varepsilon q - 1)}} \end{array} $ $\begin{array}{c} N_{M0}^* =\frac{{K{\Lambda _M}}}{{K{\mu _M} + \kappa aq}} \end{array}$ $N_{M0}^* = N_{M2}^{}$
Table 3.  Fixed parameters values in the numerical simulation of the system (2)
Parameter Description Value Ref
$ \beta_H $ Mosquito-to-human transmission rate 0.2 per day Bonyah et al. [4]
$ {\beta _M} $ Human-to-mosquito Transmission rate 0.09 per day Bonyah et al. [4]
$ {\alpha _H} $ The rate of exposed humans moving into infectious class $ \frac{1}{{5.5}} \approx 0.18 $ per day Ferguson et al. [12]
$ {\rho _{}} $ Human factor transmission rate 0.0029 assumed
$ {\delta _M} $ The rate flow from $ {E_M} $ to $ {I_M} $ $ \frac{1}{{8.2}} \approx 0.12 $ per day Ferguson et al. [12]
$ {r_{}} $ Human recovery rate $ \frac{1}{6} \approx 0.17 $ per day Ferguson et al. [12]
$ {\mu _H} $ Natural death rate in humans $ \frac{1}{{360 \times 60}} \approx 0.00005 $ per day Manore et al. [25]
$ {\mu _M} $ Natural death rate in mosquitoes $ \frac{1}{{14}} \approx 0.07 $ per day Manore et al. [25]
$ K $ The maximum environmental capacity for mosquitoes without aliens 40 assumed
$ k $ The maximum environmental capacity for aliens without mosquitoes 30 assumed
Parameter Description Value Ref
$ \beta_H $ Mosquito-to-human transmission rate 0.2 per day Bonyah et al. [4]
$ {\beta _M} $ Human-to-mosquito Transmission rate 0.09 per day Bonyah et al. [4]
$ {\alpha _H} $ The rate of exposed humans moving into infectious class $ \frac{1}{{5.5}} \approx 0.18 $ per day Ferguson et al. [12]
$ {\rho _{}} $ Human factor transmission rate 0.0029 assumed
$ {\delta _M} $ The rate flow from $ {E_M} $ to $ {I_M} $ $ \frac{1}{{8.2}} \approx 0.12 $ per day Ferguson et al. [12]
$ {r_{}} $ Human recovery rate $ \frac{1}{6} \approx 0.17 $ per day Ferguson et al. [12]
$ {\mu _H} $ Natural death rate in humans $ \frac{1}{{360 \times 60}} \approx 0.00005 $ per day Manore et al. [25]
$ {\mu _M} $ Natural death rate in mosquitoes $ \frac{1}{{14}} \approx 0.07 $ per day Manore et al. [25]
$ K $ The maximum environmental capacity for mosquitoes without aliens 40 assumed
$ k $ The maximum environmental capacity for aliens without mosquitoes 30 assumed
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