American Institute of Mathematical Sciences

Advanced
doi: 10.3934/dcdsb.2021048

Persistence of mosquito vector and dengue: Impact of seasonal and diurnal temperature variations

Naveen K. Vaidya 1, and  Feng-Bin Wang 2,

1. 

Department of Mathematics and Statistics, San Diego State University, San Diego, CA 92182, USA, Computational Science Research Center, San Diego State University, San Diego, CA 92182, USA, Viral Information Institute, San Diego State University, San Diego, CA 92182, USA
2. 

Department of Natural Science in the Center for General Education, Chang Gung University, Guishan, Taoyuan 333, Taiwan, Community Medicine Research Center, Chang Gung Memorial Hospital, Keelung Branch, Keelung 204, Taiwan

Received  August 2020 Revised  January 2021 Published  February 2021

Dengue, a mosquito-borne disease, poses a tremendous burden to human health with about 390 million annual dengue infections worldwide. The environmental temperature plays a major role in the mosquito life-cycle as well as the mosquito-human-mosquito dengue transmission cycle. While previous studies have provided useful insights into the understanding of dengue diseases, there is little emphasis put on the role of environmental temperature variation, especially diurnal variation, in the mosquito vector and dengue dynamics. In this study, we develop a mathematical model to investigate the impact of seasonal and diurnal temperature variations on the persistence of mosquito vector and dengue. Importantly, using a threshold dynamical system approach to our model, we formulate the mosquito reproduction number and the infection invasion threshold, which completely determine the global threshold dynamics of mosquito population and dengue transmission, respectively. Our model predicts that both seasonal and diurnal variations of the environmental temperature can be determinant factors for the persistence of mosquito vector and dengue. In general, our numerical estimates of the mosquito reproduction number and the infection invasion threshold show that places with higher diurnal or seasonal temperature variations have a tendency to suffer less from the burden of mosquito population and dengue epidemics. Our results provide novel insights into the theoretical understanding of the role of diurnal temperature, which can be beneficial for the control of mosquito vector and dengue spread.

Keywords: Dengue epidemics, diurnal temperature, global stability, periodic models, seasonal temperature.
Mathematics Subject Classification: 34D20, 37N25, 92B05, 92D30, 92D40.
Citation: Naveen K. Vaidya, Feng-Bin Wang. Persistence of mosquito vector and dengue: Impact of seasonal and diurnal temperature variations. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021048
References:
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N. K. Vaidya, X. Li and F.-B. Wang, Impact of spatially heterogeneous temperature on the dynamics of dengue epidemics, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 321-349.  doi: 10.3934/dcdsb.2018099.  Google Scholar

[37]

N. K. Vaidya and L. M. Wahl, Avian influenza dynamics under periodic environmental conditions, SIAM J. Appl. Math., 75 (2015), 443-467.  doi: 10.1137/140966642.  Google Scholar

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F.-B. Wang, S.-B. Hsu and W. Wang, Dynamics of harmful algae with seasonal temperature variations in the cove-main lake, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 313-335.  doi: 10.3934/dcdsb.2016.21.313.  Google Scholar

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show all references

References:
[1]

G. Aronsson and R. B. Kellogg, On a differential equation arising from compartmental analysis, Math. Biosci., 38 (1978), 113-122.  doi: 10.1016/0025-5564(78)90021-4.  Google Scholar

[2]

N. Bacaër and S. Guernaoui, The epidemic threshold of vector-borne diseases with seasonality: the case of cutaneous leishmaniasis in Chichaoua, Morocco, J. Math. Biol., 53 (2006), 421-436.  doi: 10.1007/s00285-006-0015-0.  Google Scholar

[3]

S. Banu, W. Hu, C. Hurst and S. Tong, Dengue transmission in the asia-pacific region: Impact of climate change and socio-environmental factors, Tropical Medicine and International Health, 11 (2011), 598-607.  doi: 10.1111/j.1365-3156.2011.02734.x.  Google Scholar

[4]

S. Bhatt, P. W. Gething, O. J. Brady, J. P. Messina and A. W. Farlow, The global distribution and burden of dengue, Nature, 496 (2013), 504-507.  doi: 10.1038/nature12060.  Google Scholar

[5]

O. J. Brady, P. W. Gething, S. Bhatt, J. P. Messina, J. S. Brownstein, et al., Refining the global spatial limits of dengue virus transmission by evidence-based consensus, PLoS Negl. Trop. Dis., 6 (2012), e1760. doi: 10.1371/journal.pntd.0001760.  Google Scholar

[6]

G. Chowell, P. Diaz-Dueñas, J. C. Miller, A. Alcazar-Velazco, J. M. Hyman, P. W. Fenimore and C. Castillo-Chavez, Estimation of the reproduction number of dengue fever from spatial epidemic data, Math. Biosci., 208 (2007), 571-589.  doi: 10.1016/j.mbs.2006.11.011.  Google Scholar

[7]

N. C. Dom, Z. A. Latif, A. H. Ahmad, R. Ismail and B. Pradhan, Manifestation of gis tools for spatial pattern distribution analysis of dengue fever epidemic in the city of Subang Jaya, Malaysia, Environment Asia, 5 (2012), 82-92.   Google Scholar

[8]

T. P. Endy, A. Nisalak, S. Chunsuttiwat, D. H. Libraty and S. Green, Spatial and temporal circulation of dengue virus serotypes: A prospective study of primary school children in Kamphaeng Phet, Thailand, Am. J. Epidemiol., 156 (2002), 52-59.  doi: 10.1093/aje/kwf006.  Google Scholar

[9]

D. A. Focks, E. Daniels, D. G. Haile and J. E. Keesling, A simulation model of the epidemiology of urban dengue fever: Literature analysis, model development, preliminary validation, and samples of simulation results, Am. J. Trop. Med. Hyg., 53 (1995), 489-506.  doi: 10.4269/ajtmh.1995.53.489.  Google Scholar

[10]

A. K. Githeko, S. W. Lindsay, U. E. Confalonieri and J. A. Patz, Climate change and vector-borne diseases: A regional analysis, Bulletin of the World Health Organization, 78 (2000), 1136-1147.   Google Scholar

[11]

D. J. Gubler, Epidemic dengue/dengue hemorrhagic fever as a public health, social and economic problem in the 21st century, Trends in Microbiology, 10 (2002), 100-103.  doi: 10.1016/S0966-842X(01)02288-0.  Google Scholar

[12]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society Providence, RI, 1988. doi: 10.1090/surv/025.  Google Scholar

[13]

M. W. Hirsch, Systems of differential equations that are competitive or cooperative Ⅱ: Convergence almost everywhere, SIAM J. Math. Anal., 16 (1985), 423-439.  doi: 10.1137/0516030.  Google Scholar

[14]

M. J. Hopp and J. A. Foley, Global-scale relationships between climate and the dengue fever vector, aedes aegypti, Climatic Change, 48 (2001), 441-463.   Google Scholar

[15]

S. Karl, N. Halder, J. K. Kelso, S. A. Ritchie and G. J. Milne, A spatial simulation model for dengue virus infection in urban areas, BMC Infec. Dis., 14 (2014), p447. doi: 10.1186/1471-2334-14-447.  Google Scholar

[16]

A. Khan, M. Hassan and M. Imran, Estimating the basic reproduction number for single-strain dengue fever epidemics, Infectious Diseases of Poverty, 3 (2014), p12. doi: 10.1186/2049-9957-3-12.  Google Scholar

[17]

L. Lambrechts, K. P. Paaijmans, T. Fansiri, L. B. Carrington, L. D. Kramer, M. B. Thomas and T. W. Scott, Impact of daily temperature fluctuations on dengue virus transmission by aedes aegypti, PNAS, 108 (2011), 7460-7465.  doi: 10.1073/pnas.1101377108.  Google Scholar

[18]

M.-T. Li, G.-Q. Sun, L. Yakob, H.-P. Zhu, Z. Jin and W.-Y. Zhang, The driving force for 2014 dengue outbreak in Guangdong, China, PLoS ONE, 11 (2016), e0166211. doi: 10.1371/journal.pone.0166211.  Google Scholar

[19]

L. Liu, X.-Q. Zhao and Y. Zhou, A tuberculousis model with seasonality, Bull. Math. Biol., 72 (2010), 931-952.  doi: 10.1007/s11538-009-9477-8.  Google Scholar

[20]

A. Nisalak, T. P. Endy, S. Nimmannitya, S. Kalayanarooj and U. Thisayakorn, Serotype-specific dengue virus circulation and dengue disease in Bangkok, Thailand from 1973 to 1999, Am. J. Trop. Med. Hyg., 68 (2003), 191-202.  doi: 10.4269/ajtmh.2003.68.191.  Google Scholar

[21]

M. Oki and T. Yamamoto, Climate change, population immunity, and hyperendemicity in the transmission threshold of dengue, PLoS ONE, 7 (2010), e48258. doi: 10.1371/journal.pone.0048258.  Google Scholar

[22]

K. P. Paaijmans, A. F. Read and M. B. Thomas, Understanding the link between malaria risk and climate, PNAS, 106 (2009), 13844-13849.  doi: 10.1073/pnas.0903423106.  Google Scholar

[23]

A. Pakhare, Y. Sabde, A. Joshi, R. Jain, A. Kokane and R. Joshi, A study of spatial and meteorological determinants of dengue outbreak in bhopal city in 2014, PLoS Negl. Trop. Dis., 53 (2014), 225-233.   Google Scholar

[24]

W. G. Panhuisa, M. Choisyb, X. Xionga, N. S. Choka and P. Akarasewid, Region-wide synchrony and traveling waves of dengue across eight countries in southeast asia, Proc. Nat. Acad. Sci., 112 (2015), 13069-13074.   Google Scholar

[25]

J. A. Patz, D. Campbell-Lendrum, T. Holloway and J. A. Foley, Impact of regional climate change on human health, Nature, 438 (2005), 310-317.  doi: 10.1038/nature04188.  Google Scholar

[26]

S. T. R. Pinho, C. P. Ferreira, L. Esteva, F. R. Barreto, V. C. Morato e Silva and M. G. L. Teixeira, Modelling the dynamics of dengue real epidemics, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 368 (2010), 5679-5693.  doi: 10.1098/rsta.2010.0278.  Google Scholar

[27]

V. Racloz, R. Ramsey, S. Tong and W. Hu, Surveillance of dengue fever virus: A review of epidemiological models and early warning systems, PLoS Negl. Trop. Dis., 6 (2012), e1648. doi: 10.1371/journal.pntd.0001648.  Google Scholar

[28]

D. J. Rogers and S. E. Randolph, Climate change and vector-borne diseases, Adv. Parasitol., 62 (2006), 345-381.  doi: 10.1016/S0065-308X(05)62010-6.  Google Scholar

[29]

T. W. Scott, A. C. Morrison, L. H. Lorenz, G. G. Clark and D. Strickman, Longitudinal studies of aedes aegypti (diptera: Culicidae) in Thailand and puerto rico: Population dynamics, J. Med. Entomol., 37 (2000), 77-88.   Google Scholar

[30]

P. M. Sheppard, W. W. Macdonald, R. J. Tonnand and B. Grab, The dynamics of an adult population of aedes aegypti in relation to dengue haemorrhagic fever in bangkok, J. Anim. Ecol., 38 (1969), 661-702.  doi: 10.2307/3042.  Google Scholar

[31]

C. P. Simmons, J. J. Farrar, N. van Vinh Chau and B. Wills, Dengue, J. Vector Borne Dis, 6 (2012), e1648. doi: 10.1056/NEJMra1110265.  Google Scholar

[32]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, American Mathematical Society Providence, RI, 1995.  Google Scholar

[33] H. L. Smith and P. Waltman, The Theory of the Chemostat, CCambridge University Press, Cambridge, 1995.  doi: 10.1017/CBO9780511530043.  Google Scholar
[34]

R. W. Sutherst, Global change and human vulnerability to vector-borne diseases, N. Engl. J. Med., 366 (2012), 1423-1432.  doi: 10.1128/CMR.17.1.136-173.2004.  Google Scholar

[35]

M. Teurlai, C. E. Menkés, V. Cavarero, N. Degallier, E. Descloux, J.-P. Grangeon, et al., Socio-economic and climate factors associated with dengue fever spatial heterogeneity: A worked example in new caledonia, PLoS Negl. Trop. Dis., 9 (2015), e0004211. doi: 10.1371/journal.pntd.0004211.  Google Scholar

[36]

N. K. Vaidya, X. Li and F.-B. Wang, Impact of spatially heterogeneous temperature on the dynamics of dengue epidemics, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 321-349.  doi: 10.3934/dcdsb.2018099.  Google Scholar

[37]

N. K. Vaidya and L. M. Wahl, Avian influenza dynamics under periodic environmental conditions, SIAM J. Appl. Math., 75 (2015), 443-467.  doi: 10.1137/140966642.  Google Scholar

[38]

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.  Google Scholar

[39]

F.-B. Wang, S.-B. Hsu and W. Wang, Dynamics of harmful algae with seasonal temperature variations in the cove-main lake, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 313-335.  doi: 10.3934/dcdsb.2016.21.313.  Google Scholar

[40]

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.  Google Scholar

[41]

WHO, Dengue Guidelines for Diagnosis, Treatment, Prevention, and Control, TDR: World Health Organization, (2009). Google Scholar

[42]

R. E. Woodruff and T. McMichael, Climate change and human health: All affected bit somevvmore than others, Social Alternatives, 23 (2004), 17-22.   Google Scholar

[43]

H. M. Yang, M. de L. da G. Macoris, K. C. Galvani and M. T. M. Andrighetti, Follow up estimation of aedes aegypti entomological parameters and mathematical modellings, Biosystems, 103 (2011), 360-371.  doi: 10.1016/j.biosystems.2010.11.002.  Google Scholar

[44]

H. M. Yang, M. L. G. Macoris, K. C. Galvani, M. T. M. Andrighetti and D. M. V. Wanderley, Assessing the effects of temperature on dengue transmission, Epidemiol. Infect., 137 (2009), 1179-1187.  doi: 10.1017/S0950268809002052.  Google Scholar

[45]

H. M. Yang, M. L. G. Macoris, K. C. Galvani, M. T. M. Andrighetti and D. M. V. Wanderley, Assessing the effects of temperature on the population of aedes aegypti, the vector of dengue, Epidemiol. Infect., 137 (2009), 1188-1202.  doi: 10.1017/S0950268809002040.  Google Scholar

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Figure 2.1.  A schematic diagram of the dengue transmission model. $ A, M_s, M_e, M_i $: aquatic, susceptible, exposed, and infected mosquitos. $ H_s, H_e, H_i, H_r $: susceptible, exposed, infected, and recovered humans. Solid arrows represent birth, maturation, infection, transfer, death, while dashed arrows indicate the effects of time dependent periodic environmental temperature, $ T(t) $
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44] for $ \delta (T) $ (oviposition rate), $ \mu_a(T) $ (aquatic phase mortality rate), $ \theta (T) $ (mosquito emergence rate from acuatic phase), and $ \mu_m(T) $ (mosquito mortality rate)">Figure 2.2.  Best-fit curves provided by the experimental data [44] for $ \delta (T) $ (oviposition rate), $ \mu_a(T) $ (aquatic phase mortality rate), $ \theta (T) $ (mosquito emergence rate from acuatic phase), and $ \mu_m(T) $ (mosquito mortality rate)
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17] for the transmission probability from human to mosquito, $ \beta_m (T) $">Figure 2.3.  Best-fit curve provided by the data generated from the previous estimates [17] for the transmission probability from human to mosquito, $ \beta_m (T) $
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Figure 2.4.  Temperature profile of $ T_m(t) $ [Left], $ \Psi_d(t) $ [Middle], and $ T(t) $ [Right]. Parameters used are $ T_0 = 25 ^oC, \epsilon_m = 5 ^oC, \tau_m $ = 365 day, $ \phi_m = 0 $, $ \epsilon_d = 5 ^oC $, $ \tau_d = 1 $ day, and $ \phi_d = 0 $
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Figure 4.1.  Mosquito reproduction number ($ \mathcal{R}^M $) [Left] and infection invasion threshold ($ \mathcal{R}^0 $) [right] for different values of the mean temperature ($ T_0 $) with amplitudes of seasonal temperature and diurnal temperature fixed at $ \epsilon_m = 5 $ $ ^oC $ and $ \epsilon_d = 5 $ $ ^oC $, respectively. For comparison purposes, $ \mathcal{R}^M $ and $ \mathcal{R}^0 $ for the constant temperature (i.e., $ \epsilon_m = \epsilon_d = 0 $ $ ^oC $) are also plotted
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Figure 4.2.  Mosquito reproduction number ($ \mathcal{R}^M $) [left column] and infection invasion threshold ($ \mathcal{R}^0 $) [right column] for different values of the amplitudes of seasonal temperature ($ \epsilon_m $) with the amplitude of diurnal temperature fixed at $ \epsilon_d = 5 $ $ ^oC $ and the mean temperature fixed at $ T_0 = 16 $ $ ^oC $ [top row], $ T_0 = 28 $ $ ^oC $ [middle row], and $ T_0 = 38 $ $ ^oC $ [bottom row]
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Figure 4.3.  Mosquito reproduction number ($ \mathcal{R}^M $) [left column] and infection invasion threshold ($ \mathcal{R}^0 $) [right column] for different values of the amplitudes of diurnal temperature ($ \epsilon_d $) with the amplitude of seasonal temperature fixed at $ \epsilon_m = 5 $ $ ^oC $ and the mean temperature fixed at $ T_0 = 16 $ $ ^oC $ [top row], $ T_0 = 28 $ $ ^oC $ [middle row], and $ T_0 = 38 $ $ ^oC $ [bottom row]
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Table 2.1.  Model parameters
Parameter Description Value Reference
$ k $ Fraction of female larvae from eggs 0.5 (0-1) [18,26]
$ b $ Per capita biting rate 0.1 [6,26]
$ \mu_h $ Natural death rate of humans 4.22$ \times 10^{-5} $ d$ ^{-1} $ Calculated, [16]
$ 1/\gamma_h $ Intrinsic period 10 days [6,16,18,26]
$ \alpha_h $ Human recovery rate 0.1 d$ ^{-1} $ [18,26]
$ \delta_m $ In $ \delta(t) $ 9.531 Data fitting
$ \delta_h $ In $ \delta(t) $ 22.55 Data fitting
$ N_{\delta} $ In $ \delta(t) $ 7.084 Data fitting
$ a_{\delta} $ In $ \delta(t) $ 0 Data fitting
$ \epsilon_{\delta} $ In $ \delta(t) $ $ 10^{-6} $ Data fitting
$ a_{0\mu_a} $ In $ \mu_a(t) $ 2.914 Data fitting
$ a_{1\mu_a} $ In $ \mu_a(t) $ -0.4986 Data fitting
$ a_{2\mu_a} $ In $ \mu_a(t) $ 0.03099 Data fitting
$ a_{3\mu_a} $ In $ \mu_a(t) $ -0.0008236 Data fitting
$ a_{4\mu_a} $ In $ \mu_a(t) $ 7.975$ \times 10^{-6} $ Data fitting
$ a_{0\theta} $ In $ \theta(t) $ 8.044$ \times 10^{-5} $ Data fitting
$ a_{1\theta} $ In $ \theta(t) $ 11.386 Data fitting
$ a_{2\theta} $ In $ \theta(t) $ 40.1461 Data fitting
$ \epsilon_{\theta} $ In $ \theta(t) $ $ 10^{-6} $ Data fitting
$ a_{0\mu_m} $ In $ \mu_m(t) $ 0.1901 Data fitting
$ a_{1\mu_m} $ In $ \mu_m(t) $ -0.0134 Data fitting
$ a_{2\mu_m} $ In $ \mu_m(t) $ 2.739$ \times 10^{-4} $ Data fitting
$ a_{0\gamma_m} $ In $ \gamma_m(t) $ 5$ \times 10^{4/3} $ Data fitting
$ a_{1\gamma_m} $ In $ \gamma_m(t) $ 0.0768 Data fitting
$ \beta_{mh} $ In $ \beta_m(t) $ 18.9871 Data fitting
$ N_{\beta_m} $ In $ \beta_m(t) $ 7 Data fitting
$ \epsilon_{\beta m} $ In $ \beta_m(t) $ $ 10^{-6} $ Data fitting
$ a_{\beta m} $ In $ \beta_m(t) $ 0 Data fitting
$ a_{0\beta_h} $ In $ \beta_h(t) $ 1.044$ \times 10^{-3} $ Data fitting
$ a_{1\beta_h} $ In $ \beta_h(t) $ 12.286 Data fitting
$ a_{2\beta_h} $ In $ \beta_h(t) $ 32.461 Data fitting
$ \epsilon_{\beta h} $ In $ \beta_h(t) $ $ 10^{-6} $ Data fitting
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Parameter Description Value Reference
$ k $ Fraction of female larvae from eggs 0.5 (0-1) [18,26]
$ b $ Per capita biting rate 0.1 [6,26]
$ \mu_h $ Natural death rate of humans 4.22$ \times 10^{-5} $ d$ ^{-1} $ Calculated, [16]
$ 1/\gamma_h $ Intrinsic period 10 days [6,16,18,26]
$ \alpha_h $ Human recovery rate 0.1 d$ ^{-1} $ [18,26]
$ \delta_m $ In $ \delta(t) $ 9.531 Data fitting
$ \delta_h $ In $ \delta(t) $ 22.55 Data fitting
$ N_{\delta} $ In $ \delta(t) $ 7.084 Data fitting
$ a_{\delta} $ In $ \delta(t) $ 0 Data fitting
$ \epsilon_{\delta} $ In $ \delta(t) $ $ 10^{-6} $ Data fitting
$ a_{0\mu_a} $ In $ \mu_a(t) $ 2.914 Data fitting
$ a_{1\mu_a} $ In $ \mu_a(t) $ -0.4986 Data fitting
$ a_{2\mu_a} $ In $ \mu_a(t) $ 0.03099 Data fitting
$ a_{3\mu_a} $ In $ \mu_a(t) $ -0.0008236 Data fitting
$ a_{4\mu_a} $ In $ \mu_a(t) $ 7.975$ \times 10^{-6} $ Data fitting
$ a_{0\theta} $ In $ \theta(t) $ 8.044$ \times 10^{-5} $ Data fitting
$ a_{1\theta} $ In $ \theta(t) $ 11.386 Data fitting
$ a_{2\theta} $ In $ \theta(t) $ 40.1461 Data fitting
$ \epsilon_{\theta} $ In $ \theta(t) $ $ 10^{-6} $ Data fitting
$ a_{0\mu_m} $ In $ \mu_m(t) $ 0.1901 Data fitting
$ a_{1\mu_m} $ In $ \mu_m(t) $ -0.0134 Data fitting
$ a_{2\mu_m} $ In $ \mu_m(t) $ 2.739$ \times 10^{-4} $ Data fitting
$ a_{0\gamma_m} $ In $ \gamma_m(t) $ 5$ \times 10^{4/3} $ Data fitting
$ a_{1\gamma_m} $ In $ \gamma_m(t) $ 0.0768 Data fitting
$ \beta_{mh} $ In $ \beta_m(t) $ 18.9871 Data fitting
$ N_{\beta_m} $ In $ \beta_m(t) $ 7 Data fitting
$ \epsilon_{\beta m} $ In $ \beta_m(t) $ $ 10^{-6} $ Data fitting
$ a_{\beta m} $ In $ \beta_m(t) $ 0 Data fitting
$ a_{0\beta_h} $ In $ \beta_h(t) $ 1.044$ \times 10^{-3} $ Data fitting
$ a_{1\beta_h} $ In $ \beta_h(t) $ 12.286 Data fitting
$ a_{2\beta_h} $ In $ \beta_h(t) $ 32.461 Data fitting
$ \epsilon_{\beta h} $ In $ \beta_h(t) $ $ 10^{-6} $ Data fitting
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