An almost periodic malaria transmission model with time-delayed input of vector

Lizhong Qiang; Bin-Guo Wang

doi:10.3934/dcdsb.2017073

Previous Article
Stability of a viral infection model with state-dependent delay, CTL and antibody immune responses
DCDS-B Home
This Issue
Next Article
Morse indices and symmetry breaking for the Gelfand equation in expanding annuli

June 2017, 22(4): 1525-1546. doi: 10.3934/dcdsb.2017073

An almost periodic malaria transmission model with time-delayed input of vector

Lizhong Qiang and Bin-Guo Wang ^,

School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China

* Corresponding author:Bin-Guo Wang

Received April 2016 Revised December 2016 Published February 2017

An almost periodic malaria transmission model with the time-delayed input of vector is considered. It is shown that the disease is uniformly persistent when the basic reproduction ratio $R_{0}>1$, and it will die out when $R_{0} < 1$ under the assumption that there exists a small invasion. Furthermore, the global stability of the disease-free almost periodic state is obtained provided that the disease-induced death rate is null. Finally, we illustrate the above results by numerical simulations and show that the periodic epidemic models may overestimate or underestimate the malaria risk.

Keywords: Malaria transmission, almost periodicity, delayed input, basic reproduction ratio, skew-product semiflow.

Mathematics Subject Classification: 34C12, 37B55, 92D30.

Citation: Lizhong Qiang, Bin-Guo Wang. An almost periodic malaria transmission model with time-delayed input of vector. Discrete and Continuous Dynamical Systems - B, 2017, 22 (4) : 1525-1546. doi: 10.3934/dcdsb.2017073

References:

[1]	S. Altizer, A. Dobson, P. Hosseini, P. Hudson, M. Pascual and P. Rohani, Seasonality and the dynamics of infectious diseases, Ecology Letters, 9 (2006), 467-484. doi: 10.1111/j.1461-0248.2005.00879.x.
[2]	J. L. Aron, Mathematical modeling of immunity to malaria, Math. Biosci., 90 (1988), 385-396. doi: 10.1016/0025-5564(88)90076-4.
[3]	A. Bomblies, Modeling the role of rainfall patterns in seasonal malaria transmission, Climatic Change, 112 (2012), 673-685. doi: 10.1007/s10584-011-0230-6.
[4]	N. Chitnis and J. M. Hyman, Bifurcation analysis of a mathematical model for malaria transmission, SIAM J. Appl. Math., 67 (2006), 24-45. doi: 10.1137/050638941.
[5]	N. Chitnis, J. M. Hyman and J. M. Cushing, Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model, Bull. Math. Biol., 70 (2008), 1272-1296. doi: 10.1007/s11538-008-9299-0.
[6]	C. Chiyaka, W. Garira and S. Dube, Transmission model of endemic human malaria in a partially immune population, Math. Comput. Modelling, 46 (2007), 806-822. doi: 10.1016/j.mcm.2006.12.010.
[7]	C. Chiyaka, J. M. Tchuenche, W. Garira and S. Dube, A mathematical analysis of the effects of control strategies on the transmission dynamics of malaria, Appl. Math. Comput., 195 (2008), 641-662. doi: 10.1016/j.amc.2007.05.016.
[8]	C. Corduneanu, Almost Periodic Functions Chelsea Publishing Company New York, N. Y. , 1989.
[9]	M. Craig, I. Kleinschmidt, J. Nawn, D. Le Sueur and B. Sharp, Exploring 30 years of malaria case data in kwazulu-natal, south africa: part Ⅰ. the impact of climatic factors, Trop. Med. Int. Health, 9 (2004), 1247-1257. doi: 10.1111/j.1365-3156.2004.01340.x.
[10]	O. Diekmann, J. A. P. Heesterbeek and J. A. Metz, 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 (1990), 365-382. doi: 10.1007/BF00178324.
[11]	A. M. Fink, Almost Periodic Differential Equations Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1974.
[12]	D. Gao, Y. Lou and S. Ruan, A periodic ross-macdonald model in a patchy environment, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 3133-3145. doi: 10.3934/dcdsb.2014.19.3133.
[13]	H. -W. Gao, L. -P. Wang, S. Liang, Y. -X. Liu, S. -L. Tong, J. -J. Wang, Y. -P. Li, X. -F. Wang, H. Yang and J. -Q. Ma, et al. , Change in rainfall drives malaria re-emergence in anhui province china, PLoS ONE, 7 (2012), e43686. doi: 10.1371/journal.pone.0043686.
[14]	J. K. Hale, Asymptotic Behavior of Dissipative Systems Math. Surveys and Monographs 25, Amer. Math. Soc. , Providence, RI, 1988.
[15]	M. B. Hoshen and A. P. Morse, A weather-driven model of malaria transmission, Malar. J., 3 (2004), 32-46. doi: 10.1186/1475-2875-3-32.
[16]	W. Jepson, A. Moutia and C. Courtois, The malaria problem in mauritius: The bionomics of mauritian anophelines, Bulletin of entomological research, 38 (1947), 177-208. doi: 10.1017/S0007485300030273.
[17]	Y. Lou and X.-Q. Zhao, Threshold dynamics in a time-delayed periodic SIS epidemic model, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 169-186. doi: 10.3934/dcdsb.2009.12.169.
[18]	Y. Lou and X.-Q. Zhao, A climate-based malaria transmission model with structured vector population, SIAM J. Appl. Math., 70 (2010), 2023-2044. doi: 10.1137/080744438.
[19]	G. Macdonald et al, The Epidemiology and Control of Malaria Oxford University Press, Oxford, UK, 1957.
[20]	P. Magal and X.-Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM J. Math. Anal., 37 (2005), 251-275. doi: 10.1137/S0036141003439173.
[21]	J. Nedelman, Introductory review some new thoughts about some old malaria models, Math. Biosci., 73 (1985), 159-182. doi: 10.1016/0025-5564(85)90010-0.
[22]	E. Ngarakana-Gwasira, C. Bhunu and E. Mashonjowa, Assessing the impact of temperature on malaria transmission dynamics, Afr. Mat., 25 (2014), 1095-1112. doi: 10.1007/s13370-013-0178-y.
[23]	G. Ngwa, Modelling the dynamics of endemic malaria in growing populations, Discrete Contin. Dyn. Syst. Ser. B, 4 (2004), 1173-1202. doi: 10.3934/dcdsb.2004.4.1173.
[24]	G. A. Ngwa and W. S. Shu, A mathematical model for endemic malaria with variable human and mosquito populations, Math. Comput. Modelling, 32 (2000), 747-763. doi: 10.1016/S0895-7177(00)00169-2.
[25]	P. Reiter, Climate change and mosquito-borne disease, Envir. Hlth. Perspect., 109 (2001), 141-161. doi: 10.2307/3434853.
[26]	R. Ross, The Prevention of Malaria John Murray, London, 1911.
[27]	S. Ruan, D. Xiao and J. C. Beier, On the delayed ross-macdonald model for malaria transmission, Bull. Math. Biol., 70 (2008), 1098-1114. doi: 10.1007/s11538-007-9292-z.
[28]	G. Sell, Topological Dynamics and Ordinary Differential Equations Van Nostrand Reinhold, London, 1971.
[29]	M. Service, Mosquito Ecology: Field Sampling Methods Springer Netherlands, 1993.
[30]	H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative systems Math. Surveys and Monographs, 41, Amer. Math. Soc. , Providence, RI, 1995.
[31]	H. L. Smith and P. Waltman, The Theory of the Chemostat Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511530043.
[32]	P. Van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6.
[33]	H. Wan and H. Zhu, The impact of resource and temperature on malaria transmission, Journal of Biological Systems, 20 (2012), 285-302. doi: 10.1142/S0218339012500118.
[34]	B. G. Wang, W. T. Li and L. Zhang, An almost periodic epidemic model with age structure in a patchy environment, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 291-311. doi: 10.3934/dcdsb.2016.21.291.
[35]	B.-G. Wang and X.-Q. Zhao, Basic reproduction ratios for almost periodic compartmental epidemic models, J. Dyn. Diff. Equ., 25 (2013), 535-562. doi: 10.1007/s10884-013-9304-7.
[36]	W. Wang and X.-Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments, J. Dynam. Differential Equations, 20 (2008), 699-717. doi: 10.1007/s10884-008-9111-8.
[37]	The world health report, 2014. Available from: http://www.who.int/malaria/media/en/.
[38]	X. -Q. Zhao, Dynamical Systems in Population Biology Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21761-1.
[39]	X.-Q. Zhao, Global attractivity in monotone and subhomogeneous almost periodic systems, J. Differential Equations, 187 (2003), 494-509. doi: 10.1016/S0022-0396(02)00054-2.

show all references

References:

[1]	S. Altizer, A. Dobson, P. Hosseini, P. Hudson, M. Pascual and P. Rohani, Seasonality and the dynamics of infectious diseases, Ecology Letters, 9 (2006), 467-484. doi: 10.1111/j.1461-0248.2005.00879.x.
[2]	J. L. Aron, Mathematical modeling of immunity to malaria, Math. Biosci., 90 (1988), 385-396. doi: 10.1016/0025-5564(88)90076-4.
[3]	A. Bomblies, Modeling the role of rainfall patterns in seasonal malaria transmission, Climatic Change, 112 (2012), 673-685. doi: 10.1007/s10584-011-0230-6.
[4]	N. Chitnis and J. M. Hyman, Bifurcation analysis of a mathematical model for malaria transmission, SIAM J. Appl. Math., 67 (2006), 24-45. doi: 10.1137/050638941.
[5]	N. Chitnis, J. M. Hyman and J. M. Cushing, Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model, Bull. Math. Biol., 70 (2008), 1272-1296. doi: 10.1007/s11538-008-9299-0.
[6]	C. Chiyaka, W. Garira and S. Dube, Transmission model of endemic human malaria in a partially immune population, Math. Comput. Modelling, 46 (2007), 806-822. doi: 10.1016/j.mcm.2006.12.010.
[7]	C. Chiyaka, J. M. Tchuenche, W. Garira and S. Dube, A mathematical analysis of the effects of control strategies on the transmission dynamics of malaria, Appl. Math. Comput., 195 (2008), 641-662. doi: 10.1016/j.amc.2007.05.016.
[8]	C. Corduneanu, Almost Periodic Functions Chelsea Publishing Company New York, N. Y. , 1989.
[9]	M. Craig, I. Kleinschmidt, J. Nawn, D. Le Sueur and B. Sharp, Exploring 30 years of malaria case data in kwazulu-natal, south africa: part Ⅰ. the impact of climatic factors, Trop. Med. Int. Health, 9 (2004), 1247-1257. doi: 10.1111/j.1365-3156.2004.01340.x.
[10]	O. Diekmann, J. A. P. Heesterbeek and J. A. Metz, 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 (1990), 365-382. doi: 10.1007/BF00178324.
[11]	A. M. Fink, Almost Periodic Differential Equations Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1974.
[12]	D. Gao, Y. Lou and S. Ruan, A periodic ross-macdonald model in a patchy environment, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 3133-3145. doi: 10.3934/dcdsb.2014.19.3133.
[13]	H. -W. Gao, L. -P. Wang, S. Liang, Y. -X. Liu, S. -L. Tong, J. -J. Wang, Y. -P. Li, X. -F. Wang, H. Yang and J. -Q. Ma, et al. , Change in rainfall drives malaria re-emergence in anhui province china, PLoS ONE, 7 (2012), e43686. doi: 10.1371/journal.pone.0043686.
[14]	J. K. Hale, Asymptotic Behavior of Dissipative Systems Math. Surveys and Monographs 25, Amer. Math. Soc. , Providence, RI, 1988.
[15]	M. B. Hoshen and A. P. Morse, A weather-driven model of malaria transmission, Malar. J., 3 (2004), 32-46. doi: 10.1186/1475-2875-3-32.
[16]	W. Jepson, A. Moutia and C. Courtois, The malaria problem in mauritius: The bionomics of mauritian anophelines, Bulletin of entomological research, 38 (1947), 177-208. doi: 10.1017/S0007485300030273.
[17]	Y. Lou and X.-Q. Zhao, Threshold dynamics in a time-delayed periodic SIS epidemic model, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 169-186. doi: 10.3934/dcdsb.2009.12.169.
[18]	Y. Lou and X.-Q. Zhao, A climate-based malaria transmission model with structured vector population, SIAM J. Appl. Math., 70 (2010), 2023-2044. doi: 10.1137/080744438.
[19]	G. Macdonald et al, The Epidemiology and Control of Malaria Oxford University Press, Oxford, UK, 1957.
[20]	P. Magal and X.-Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM J. Math. Anal., 37 (2005), 251-275. doi: 10.1137/S0036141003439173.
[21]	J. Nedelman, Introductory review some new thoughts about some old malaria models, Math. Biosci., 73 (1985), 159-182. doi: 10.1016/0025-5564(85)90010-0.
[22]	E. Ngarakana-Gwasira, C. Bhunu and E. Mashonjowa, Assessing the impact of temperature on malaria transmission dynamics, Afr. Mat., 25 (2014), 1095-1112. doi: 10.1007/s13370-013-0178-y.
[23]	G. Ngwa, Modelling the dynamics of endemic malaria in growing populations, Discrete Contin. Dyn. Syst. Ser. B, 4 (2004), 1173-1202. doi: 10.3934/dcdsb.2004.4.1173.
[24]	G. A. Ngwa and W. S. Shu, A mathematical model for endemic malaria with variable human and mosquito populations, Math. Comput. Modelling, 32 (2000), 747-763. doi: 10.1016/S0895-7177(00)00169-2.
[25]	P. Reiter, Climate change and mosquito-borne disease, Envir. Hlth. Perspect., 109 (2001), 141-161. doi: 10.2307/3434853.
[26]	R. Ross, The Prevention of Malaria John Murray, London, 1911.
[27]	S. Ruan, D. Xiao and J. C. Beier, On the delayed ross-macdonald model for malaria transmission, Bull. Math. Biol., 70 (2008), 1098-1114. doi: 10.1007/s11538-007-9292-z.
[28]	G. Sell, Topological Dynamics and Ordinary Differential Equations Van Nostrand Reinhold, London, 1971.
[29]	M. Service, Mosquito Ecology: Field Sampling Methods Springer Netherlands, 1993.
[30]	H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative systems Math. Surveys and Monographs, 41, Amer. Math. Soc. , Providence, RI, 1995.
[31]	H. L. Smith and P. Waltman, The Theory of the Chemostat Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511530043.
[32]	P. Van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6.
[33]	H. Wan and H. Zhu, The impact of resource and temperature on malaria transmission, Journal of Biological Systems, 20 (2012), 285-302. doi: 10.1142/S0218339012500118.
[34]	B. G. Wang, W. T. Li and L. Zhang, An almost periodic epidemic model with age structure in a patchy environment, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 291-311. doi: 10.3934/dcdsb.2016.21.291.
[35]	B.-G. Wang and X.-Q. Zhao, Basic reproduction ratios for almost periodic compartmental epidemic models, J. Dyn. Diff. Equ., 25 (2013), 535-562. doi: 10.1007/s10884-013-9304-7.
[36]	W. Wang and X.-Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments, J. Dynam. Differential Equations, 20 (2008), 699-717. doi: 10.1007/s10884-008-9111-8.
[37]	The world health report, 2014. Available from: http://www.who.int/malaria/media/en/.
[38]	X. -Q. Zhao, Dynamical Systems in Population Biology Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21761-1.
[39]	X.-Q. Zhao, Global attractivity in monotone and subhomogeneous almost periodic systems, J. Differential Equations, 187 (2003), 494-509. doi: 10.1016/S0022-0396(02)00054-2.

Figure 1. Compartmental model for malaria.

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2. The total number of vector.

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3. The graph of $\|u(t)\|$ and $\ln\|u(t)\|$ of (26) when $k=2.5$.

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4. The long-term behavior of vector and human populations when $R_{0}<1$.

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5. The graph of $\|u(t)\|$ and $\ln\|u(t)\|$ of (26) when $k=5$.

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6. The long-term behavior of vector and human populations when $R_{0}>1$.

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 7. Relationship between $\widehat{p}$ and $R_{0}$.

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 8. The long-term trend of $\ln\|u(t)\|$ of (27) with $m=2.4$ and $m=2.5$.

Figure Options

Download full-size image

Download as PowerPoint slide

Table 1. The parameters of model and their means.

Parameters	Meaning
$a$	Infection probability from infective humans to susceptible mosquitoes.
$\eta$	Transfer rate of mosquitoes from the exposed class to the infectious class.
$\lambda$	The proportion between the probability of transmission from recovered humans to susceptible vectors and the probability of transmission from infectious humans.
$\gamma_{R}$	Per capita loss rate of immunity for humans.
$\gamma_{E}$	Transfer ratio of humans from the exposed class to the infectious class.
$\gamma_{I}$	Human recovery rate.
$c$	Infection probability from infectious mosquitoes to susceptible humans.
$d_{h}$	Death rate of humans.
$\sigma_{h}$	The diseased-induced rate of humans.
$\Lambda_{h}$	Supplement rate of humans.

Table Options

Download as excel

Parameters	Meaning
$a$	Infection probability from infective humans to susceptible mosquitoes.
$\eta$	Transfer rate of mosquitoes from the exposed class to the infectious class.
$\lambda$	The proportion between the probability of transmission from recovered humans to susceptible vectors and the probability of transmission from infectious humans.
$\gamma_{R}$	Per capita loss rate of immunity for humans.
$\gamma_{E}$	Transfer ratio of humans from the exposed class to the infectious class.
$\gamma_{I}$	Human recovery rate.
$c$	Infection probability from infectious mosquitoes to susceptible humans.
$d_{h}$	Death rate of humans.
$\sigma_{h}$	The diseased-induced rate of humans.
$\Lambda_{h}$	Supplement rate of humans.

[1]	Bogdan Sasu, A. L. Sasu. Input-output conditions for the asymptotic behavior of linear skew-product flows and applications. Communications on Pure and Applied Analysis, 2006, 5 (3) : 551-569. doi: 10.3934/cpaa.2006.5.551
[2]	P.E. Kloeden, Victor S. Kozyakin. The perturbation of attractors of skew-product flows with a shadowing driving system. Discrete and Continuous Dynamical Systems, 2001, 7 (4) : 883-893. doi: 10.3934/dcds.2001.7.883
[3]	Saša Kocić. Reducibility of skew-product systems with multidimensional Brjuno base flows. Discrete and Continuous Dynamical Systems, 2011, 29 (1) : 261-283. doi: 10.3934/dcds.2011.29.261
[4]	Tomás Caraballo, Alexandre N. Carvalho, Henrique B. da Costa, José A. Langa. Equi-attraction and continuity of attractors for skew-product semiflows. Discrete and Continuous Dynamical Systems - B, 2016, 21 (9) : 2949-2967. doi: 10.3934/dcdsb.2016081
[5]	Jing Feng, Bin-Guo Wang. An almost periodic Dengue transmission model with age structure and time-delayed input of vector in a patchy environment. Discrete and Continuous Dynamical Systems - B, 2021, 26 (6) : 3069-3096. doi: 10.3934/dcdsb.2020220
[6]	Tianhui Yang, Ammar Qarariyah, Qigui Yang. The effect of spatial variables on the basic reproduction ratio for a reaction-diffusion epidemic model. Discrete and Continuous Dynamical Systems - B, 2022, 27 (6) : 3005-3017. doi: 10.3934/dcdsb.2021170
[7]	Juan A. Calzada, Rafael Obaya, Ana M. Sanz. Continuous separation for monotone skew-product semiflows: From theoretical to numerical results. Discrete and Continuous Dynamical Systems - B, 2015, 20 (3) : 915-944. doi: 10.3934/dcdsb.2015.20.915
[8]	Sylvia Novo, Carmen Núñez, Rafael Obaya, Ana M. Sanz. Skew-product semiflows for non-autonomous partial functional differential equations with delay. Discrete and Continuous Dynamical Systems, 2014, 34 (10) : 4291-4321. doi: 10.3934/dcds.2014.34.4291
[9]	Chuangxia Huang, Hua Zhang, Lihong Huang. Almost periodicity analysis for a delayed Nicholson's blowflies model with nonlinear density-dependent mortality term. Communications on Pure and Applied Analysis, 2019, 18 (6) : 3337-3349. doi: 10.3934/cpaa.2019150
[10]	Chuangxia Huang, Hedi Yang, Jinde Cao. Weighted pseudo almost periodicity of multi-proportional delayed shunting inhibitory cellular neural networks with D operator. Discrete and Continuous Dynamical Systems - S, 2021, 14 (4) : 1259-1272. doi: 10.3934/dcdss.2020372
[11]	Cruz Vargas-De-León. Global analysis of a delayed vector-bias model for malaria transmission with incubation period in mosquitoes. Mathematical Biosciences & Engineering, 2012, 9 (1) : 165-174. doi: 10.3934/mbe.2012.9.165
[12]	Hui Wan, Jing-An Cui. A model for the transmission of malaria. Discrete and Continuous Dynamical Systems - B, 2009, 11 (2) : 479-496. doi: 10.3934/dcdsb.2009.11.479
[13]	Hui Cao, Yicang Zhou. The basic reproduction number of discrete SIR and SEIS models with periodic parameters. Discrete and Continuous Dynamical Systems - B, 2013, 18 (1) : 37-56. doi: 10.3934/dcdsb.2013.18.37
[14]	Tianhui Yang, Lei Zhang. Remarks on basic reproduction ratios for periodic abstract functional differential equations. Discrete and Continuous Dynamical Systems - B, 2019, 24 (12) : 6771-6782. doi: 10.3934/dcdsb.2019166
[15]	Noriaki Yamazaki. Almost periodicity of solutions to free boundary problems. Conference Publications, 2001, 2001 (Special) : 386-397. doi: 10.3934/proc.2001.2001.386
[16]	Hongyan Yin, Cuihong Yang, Xin'an Zhang, Jia Li. The impact of releasing sterile mosquitoes on malaria transmission. Discrete and Continuous Dynamical Systems - B, 2018, 23 (9) : 3837-3853. doi: 10.3934/dcdsb.2018113
[17]	Zuowei Cai, Jianhua Huang, Liu Yang, Lihong Huang. Periodicity and stabilization control of the delayed Filippov system with perturbation. Discrete and Continuous Dynamical Systems - B, 2020, 25 (4) : 1439-1467. doi: 10.3934/dcdsb.2019235
[18]	Dongfeng Zhang, Junxiang Xu, Xindong Xu. Reducibility of three dimensional skew symmetric system with Liouvillean basic frequencies. Discrete and Continuous Dynamical Systems, 2018, 38 (6) : 2851-2877. doi: 10.3934/dcds.2018123
[19]	Peng Sun. Measures of intermediate entropies for skew product diffeomorphisms. Discrete and Continuous Dynamical Systems, 2010, 27 (3) : 1219-1231. doi: 10.3934/dcds.2010.27.1219
[20]	Nicolas Bacaër, Xamxinur Abdurahman, Jianli Ye, Pierre Auger. On the basic reproduction number $R_0$ in sexual activity models for HIV/AIDS epidemics: Example from Yunnan, China. Mathematical Biosciences & Engineering, 2007, 4 (4) : 595-607. doi: 10.3934/mbe.2007.4.595

2020 Impact Factor: 1.327

Tools

Metrics

Tools

Article outline

Figures and Tables

2020 Impact Factor: 1.327

Tools

Figures and Tables

2020 Impact Factor: 1.327

American Institute of Mathematical Sciences

An almost periodic malaria transmission model with time-delayed input of vector

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

An almost periodic malaria transmission model with time-delayed input of vector

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors