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A mathematical model of dengue transmission with memory. (English) Zbl 1329.92140
Summary: We propose and analyze a new compartmental model of dengue transmission with memory between human-to-mosquito and mosquito-to-human. The memory is incorporated in the model by using a fractional differential operator. A threshold quantity \(R_0\), similar to the basic reproduction number, is worked out. We determine the stability condition of the disease-free equilibrium (DFE) \(E_0\) with respect to the order of the fractional derivative \(\alpha\) and \(R_0\). We determine \(\alpha\) dependent threshold values for \(R_0\), below which DFE (\(E_0\)) is always stable, above which DFE is always unstable, and at which the system exhibits a Hopf-type bifurcation. It is shown that even though \(R_0\) is less than unity, the DFE may not be always stable, and the system exhibits a Hopf-type bifurcation. Thus, making \(R_0 < 1\) for controlling the disease is no longer a sufficient condition. This result is synergistic with the concept of backward bifurcation in dengue ODE models. It is also shown that \(R_0 > 1\) may not be a sufficient condition for the persistence of the disease. For a special case, when \(\alpha = \frac{1}{2}\), we analytically verify our findings and determine the critical value of \(R_0\) in terms of some important model parameters. Finally, we discuss about some dengue control strategies in light of the threshold quantity \(R_0\).

MSC:
92D30 Epidemiology
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[1] Abdelouahab, M.; Hamri, N.-E.; Wang, J., Hopf bifurcation and chaos in fractional-order modified hybrid optical system, Nonlinear Dyn, 69, 275-284, (2012) · Zbl 1254.37034
[2] Acharya, A.; Goswami, K.; Srinath, S.; Goswami, A., Awareness about dengue syndrome and related preventive practices amongst residents of an urban resettlement colony of south Delhi, J Vector Borne Dis, 42, 122-127, (2005)
[3] Agarwal, R.; Ntouyas, S.; Ahmad, B.; Alhothuali, M., Existence of solutions for integro-differential equations of fractional order with nonlocal three-point fractional boundary conditions, Adv Difference Equ, 128, 1-9, (2013) · Zbl 1390.34056
[4] Ahmed, E.; Elgazzar, A., On fractional order differential equations model for nonlocal epidemics, Physica A, 379, 2, 607-614, (2007)
[5] Andraud, M.; Hens, N.; Marais, C.; Beutels, P., Dynamic epidemiological models for dengue transmission: a systematic review of structural approaches, PLoS ONE, 7, 11, e49085, (2012)
[6] Atangana, A.; Secer, A., A note on fractional order derivatives and table of fractional derivatives of some special functions, Abstr Appl Anal, 2013, 1-8, (2013) · Zbl 1276.26010
[7] Bartley, L.; Donnelly, C.; Garnett, G., The seasonal pattern of dengue in endemic areas: mathematical models of mechanisms, Trans R Soc Trop Med Hyg, 96, 4, 387-397, (2002)
[8] Blaney, J.; Matro, J.; Murphy, B.; Whitehead, S., Recombinant, live attenuated tetravalent dengue virus vaccine formulations induce a balanced, broad, and protective neutralizing antibody response against each of the four serotypes in rhesus monkeys, J Virol, 79, 9, 5516-5528, (2005)
[9] Center For Disease Control. Dengue fact sheet; 2007. <http://www.cdc.gov/ncidod/dvbid/dengue/resources/DengueFactSheet.pdf>. [accessed 21.05.14].
[10] Center For Vaccine Development. Live attenuated tetravalent den vaccine; 2007. <http://www.denguevaccines.org/live-attenuated-vaccines>. [ accessed 21.05.2014].
[11] Chamberlain, R.; Sudia, W., Mechanism of transmission of viruses by mosquitoes, Ann Rev Entomol, 6, 371-390, (1961)
[12] Chaves, L.; Harrington, L.; Keogh, C.; Nguyen, A.; Kitron, U., Blood feeding patterns of mosquitoes: random or structured?, Front Zool, 7, 3, 1-11, (2010)
[13] Chen, A.; Huang, L.; Liu, Z.; Cao, J., Periodic bidirectional associative memory neural networks with distributed delays, J Math Anal Appl, 317, 1, 80-102, (2006) · Zbl 1086.68111
[14] Chikrii, A.; Matychyn, I., Riemann-Liouville, Caputo, and Sequential fractional derivatives in differential games, vol. 11, (2011), Birkhuser Boston · Zbl 1221.49070
[15] Chilaka, N.; Perkins, E.; Tripet, F., Visual and olfactory associative learning in the malaria vector anopheles gambiae sensu stricto, Malaria J, 11, 27, 1-11, (2012)
[16] Chowell, G.; Diaz-Duen¯as, P.; Miller, J.; Alcazar-Velazco, A.; Hyman, J., Estimation of the reproduction number of dengue fever from spatial epidemic data, Math Biosci, 208, 2, 571-589, (2007) · Zbl 1119.92055
[17] Diekmann, O.; Heesterbeek, J., Mathematical epidemiology of infectious diseases: model building, analysis and interpretation, (2000), Wiley New York · Zbl 0997.92505
[18] Diekmann, O.; Heesterbeek, J.; Metz, J., 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, 365-382, (1990) · Zbl 0726.92018
[19] Diethelm, K., A fractional calculus based model for the simulation of an outbreak of dengue fever, Nonlinear Dyn, 71, 4, 613-619, (2013)
[20] Dietz, K., Transmission and control of arbovirus diseases, (Ludwig, D.; Cooke, K., Epidemiology, (1975), SIAM Philadelphia), 104-121 · Zbl 0322.92023
[21] Dokoumetzidis, A.; Magin, R.; Macheras, P., A commentary on fractionalization of multi-compartmental models, J Pharmacokinet Pharmacodyn, 37, 203-207, (2010)
[22] Dokoumetzidis, A.; Magin, R.; Macheras, P., Fractional kinetics in multi-compartmental systems, J Pharmacokinet Pharmacodyn, 37, 507-524, (2010)
[23] Du, M.; Wang, Z.; Hu, H., Measuring memory with the order of fractional derivative, Sci Rep, 3, 1-3, (2013)
[24] Dung, N.; Day, N.; Tam, D.; Loan, H.; Chau, H., Fluid replacement in dengue shock syndrome: a randomized, double-blind comparison of four intravenous-fluid regimens, Clin Infect Dis, 29, 4, 787-794, (1999)
[25] Esteva, L.; Vargas, C., Analysis of a dengue disease transmission model, Math Biosci, 150, 2, 131-151, (1998) · Zbl 0930.92020
[26] Esteva, L.; Vargas, C., A model for dengue disease with variable human population, J Math Biol, 38, 3, 220-240, (1999) · Zbl 0981.92016
[27] Esteva, L.; Vargas, C., Coexistence of different serotypes of dengue virus, J Math Biol, 46, 1, 31-47, (2003) · Zbl 1015.92023
[28] Esteva, L.; Yang, H., Mathematical model to assess the control of aedes aegypti mosquitoes by the sterile insect technique, Math Biosci, 198, 132-147, (2005) · Zbl 1090.92048
[29] Feng, Z.; Velasco-Hernández, J., Competitive exclusion in a vector-host model for the dengue fever, J Math Biol, 35, 5, 523-544, (1997) · Zbl 0878.92025
[30] Garba, S.; Gumel, A.; Bakar, A., Backward bifurcations in dengue transmission dynamics, Math Biosci, 215, 11-25, (2008) · Zbl 1156.92036
[31] Gopalsamy, K.; He, X.-Z., Delay-independent stability in bidirectional associative memory networks, IEEE Trans Neural Networks, 5, 6, 998-1002, (1994)
[32] Gratz, N., Emergency control of aedes aegypti as a disease vector in urban areas, J Am Mosq Control Assoc, 7, 3, 353-365, (1991)
[33] Gubler, D., Dengue and dengue hemorrhagic fever, Clin Microbiol Rev, 11, 3, 480-496, (1998)
[34] Hanert, E.; Schumacher, E.; Deleersnijder, E., Front dynamics in fractional-order epidemic models, J Theor Biol, 279, 1, 9-16, (2011) · Zbl 1397.92636
[35] Hii, J.; Chew, M.; Sang, V.; Munstermann, L.; Tan, S., Population genetic analysis of host seeking and resting behaviors in the malaria vector, anopheles balabacensis (diptera: culicidae), J Med Entomol, 28, 5, 675-684, (1991)
[36] Kelly, D., Why are some people bitten more than others?, Trends Parasitol, 17, 12, 578-581, (2001)
[37] Kelly, D.; Thompson, C., Epidemiology and optimal foraging: modeling the ideal free distribution of insect vectors, Parasitology, 120, 319-327, (2000)
[38] Lakshmikantham, V.; Leela, S.; Martynyuk, A., Stability analysis of nonlinear systems, (1989), Marcel Dekker Inc New York, Basel · Zbl 0676.34003
[39] Matignon, D., Stability results for fractional differential equations with applications to control processing, (Proceedings of the multiconference on computational engineering in systems and application IMICS, vol. 2, (1996), IEEE-SMC Lile, France), 963-968
[40] McCall, P.; Kelly, D., Learning and memory in disease vectors, Trends Parasitol, 18, 10, 429-433, (2002)
[41] Min X, Xing Z. Hopf-type bifurcation and synchronization of a fractional order Van der Pol oscillator. In: Proceedings of the 31st Chinese control conference, Hefei, China; 2012. p. 193-198.
[42] Newton, E.; Reiter, P., A model of the transmission of dengue fever with an evolution of the impact of ultra-low volume (ulv) insecticide applications on dengue epidemics, Am J Trop Med Hyg, 47, 6, 709-720, (1992)
[43] Oldham, K.; Spanier, J., The fractional calculus: theory and applications of differentiation and integration to arbitrary order, (1974), Academic Press New York · Zbl 0292.26011
[44] Pinho, S.; Ferreira, C.; Esteva, L.; Barreto, F.; Morato e Silva, V., Modelling the dynamics of dengue real epidemics, Philos Trans R Soc A, 368, 5679-5693, (2010) · Zbl 1211.37116
[45] Podlubny, I., Fractional differential equations, (1999), Academic Press San Diego, California · Zbl 0918.34010
[46] Pooseh, S.; Rodrigues, H.; Torres, D., Fractional derivatives in dengue epidemics, (Simos, T.; Psihoyios, G.; Tsitouras, C.; Anastassi, Z., Numerical analysis and applied mathematics, ICNAAM, (2011), American Institute of Physics Melville), 739-742
[47] Qian, D.; Li, C.; Agarwal, R.; Wong, P., Stability analysis of fractional differential system with Riemann-Liouville derivative, Math Comput Model, 52, 862-874, (2010) · Zbl 1202.34020
[48] Rosen, L.; Roseboom, L.; Gubler, D.; Lien, J.; Chaniotis, B., Comparative susceptibility of mosquito species and strains to oral and parenteral infection with dengue and Japanese encephalitis viruses, Am J Trop Med Hyg, 34, 3, 603-615, (1985)
[49] Rosenbaum, J.; Nathan, M.; Ragoonanansingh, R.; Rawlins, S.; Gayle, C., Community participation in dengue prevention and control: a survey of knowledge, attitudes, and practice in trinidad and tobago, Am J Trop Med Hyg, 53, 2, 111-117, (1995)
[50] Schutz, G.; Trimper, S., Elephants can always remember: exact long-range memory effects in a non-Markovian random walk, Phys Rev E, 70, 045101, (2004)
[51] Sheppard, P.; MacDonald, W.; Tonn, R.; Grab, B., The dynamics of an adult population of aedes aegypti in relation to dengue haemorrhagic fever in Bangkok, J Anim Ecol, 38, 661-702, (1969)
[52] Sipahi, R.; Atay, F.; Niculescu, S.-L., Stability of traffic flow behavior with distributed delays modeling the memory effects of the drivers, SIAM J Appl Math, 68, 3, 738-759, (2007) · Zbl 1146.34058
[53] Southwood, T.; Murdie, G.; Yasuno, M.; Tonn, R.; Reader, P., Studies on the life budget of aedes aegypti in wat samphaya, Thailand Bull World Health Organ, 46, 2, 211-226, (1972)
[54] Stanislavsky, A., Memory effects and macroscopic manifestation of randomness, Phys Rev E, 61, 5, 4752-4759, (2000)
[55] Takken, W.; Verhulst, N., Host preferences of blood-feeding mosquitoes, Annu Rev Entomol, 58, 433-453, (2013)
[56] Tarasov, V., No violation of the Leibniz rule. no fractional derivative, Commun Nonlinear Sci Numer Simul, 18, 2945-2948, (2013) · Zbl 1329.26015
[57] Tavazoei, M., A note on fractional-order derivatives of periodic functions, Automatica, 46, 5, 945-948, (2010) · Zbl 1191.93062
[58] Tavazoei, M.; Haeri, M., Chaotic attractors in incommensurate fractional order systems, Physica D, 237, 20, 2628-2637, (2008) · Zbl 1157.26310
[59] Tavazoei, M.; Haeri, M.; Attari, M., A proof for non existence of periodic solutions in time invariant fractional order systems, Automatica, 45, 8, 1886-1890, (2009) · Zbl 1193.34006
[60] Tavazoei, M.; Haeri, M.; Attari, M.; Bolouki, S.; Siami, M., More details on analysis of fractional-order van der Pol oscillator, J Vib Control, 15, 6, 803-819, (2009) · Zbl 1273.70037
[61] Thome, R.; Yang, H.; Esteva, L., Optimal control of aedes aegypti mosquito by the sterile insect technique and insecticide, Math Biosci, 223, 1, 12-23, (2010) · Zbl 1180.92058
[62] van den Driessche, P.; Watmough, J., Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math Biosci, 180, 29-48, (2002) · Zbl 1015.92036
[63] Vinauger, C.; Buratti, L.; Lazzari, C., Learning the way to blood: first evidence of dual olfactory conditioning in a bloodsucking insect, rhodnius prolixus I. appetitive learning, J Exp Biol, 214, 3032-3038, (2011)
[64] Watts, D.; Burke, D.; Harrison, B.; Whitmire, R.; Nisalak, A., Effect of temperature on the vector efficiency of aedes aegypti for dengue 2 virus, Am J Trop Med Hyg, 36, 1, 143-152, (1987)
[65] Whitehorn, J.; Farrar, J., Dengue, Br Med Bull, 95, 161-173, (2010)
[66] World Health Organization. Dengue and severe dengue; 2013. <http://www.who.int/mediacentre/factsheets/fs117/en/index.html> [accessed 22.09.13].
[67] World Health Organization. Health topics (dengue); 2013. <http://www.who.int/topics/dengue/en/> [accessed 22.09.13].
[68] Zhang, F.; Changpin, L.; Chen, Y., Asymptotical stability of nonlinear fractional differential system with Caputo derivative, Int J Differ Equ, 2011, 1-12, (2011) · Zbl 1239.34008
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