Citation: Nicolas Nagahama, Bruno Gastaldi, Michael N. Clifford, María M. Manifesto, Renée H. Fortunato. The influence of environmental variations on the phenolic compound profiles and antioxidant activity of two medicinal Patagonian valerians (Valeriana carnosa Sm. and V. clarionifolia Phil.)[J]. AIMS Agriculture and Food, 2021, 6(1): 106-124. doi: 10.3934/agrfood.2021007
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Since the publication of the seminal paper [11] mathematical compartmental models are widely used to describe infectious diseases dymanics in large populations (see, for example [9], [4], [2] and [8]). It is well accepted that once an infected individual comes into contact with an unaffected population, the disease will spread by contact with the infectious individuals. Compartmental models divide the population into compartments characterizing the spread of the diseases and letters are used to denote the number of individuals in each compartment. Usually, the size of the population to be studied is
The basic reproduction number,
Many papers on optimal control applied to epidemiology propose
In this paper we focus on optimal control problems to control, via vaccination, the spread of a disease described by a SEIR model. We follow closely the approach in [16]: we consider
The normalized SEIR model differs from the usual SEIR model since the variables are fractions of the whole population instead of the number of individuals in each compartment. The theoretical and numerical treatment involving the latter model is usually done as if the variables are continuous and not integers; treating such variables as integers would demand the use of integer programming what is known to be very heavy computationally. When we turn to normalized models the variables are, by nature, continuous. In the literature, normalized models are common when the total population is assumed to remain constant during the time frame under study. This is not our case; here we normalize a SEIR model that incorporates exponential natural birth and death, as well as disease-caused death (similarly to what is done in [14]). As far as optimal control is concerned, normalizing such model brings out some new issues related to the choice of costs and the introduction of non standard constraints, questions we discuss here when comparing optimal control for normalized and not normalized SEIR models.
Herein, we refer to the SEIR model, where the variables
We emphasize that we do not concentrate on any particular disease. Rather, our aim is to illustrate how previously proposed optimal control formulations can be handled by this new model, when different scenarios are considered. Taking into account that the set of parameters for the population in [16], based on [17], are not to be found in today's world, we use different population's parameters closed related to some European countries.
Like other models in epidemiology, SEIR models represent only a rough approximation of reality. However, they provide new insights into the spreading of diseases and, when optimal control is applied, new insight on different vaccination policies.
This paper is organized in the following way. In Section 2 we introduce an optimal control problem with
The SEIR model is a compartmental model well accepted as modelling some infectious diseases. At each instant
Optimal control techniques for SEIR models allow the study of different vaccines policies; different policies are confronted in [17] and [1] where the minimizing cost is
˙S(t)=bN(t)−dS(t)−cS(t)I(t)N(t)−u(t)S(t),S(0)=S0, | (1) |
˙E(t)=cS(t)I(t)N(t)−(f+d)E(t),E(0)=E0, | (2) |
˙I(t)=fE(t)−(g+a+d)I(t),I(0)=I0, | (3) |
˙N(t)=(b−d)N(t)−aI(t),N(0)=N0, | (4) |
where
For some
0≤u(t)≤ˉua.e.t∈[0,T], | (5) |
where
˙R(t)=gI(t)−dR(t)+u(t)S(t),R(0)=R0. | (6) |
Here, the aim of applying optimal control to SEIR models is to control the spreading of the disease with some minimum financial cost. The cost should then be a weighted sum of the society financial costs of having, at each time,
JC(X,u)=∫T0(AI(t)+Bu(t)) dt, | (7) |
where
Throughout this paper we refer to the optimal control problem of minimizing
(P){Minimize∫T0(AI(t)+Bu(t)) dtsubject to˙S(t)=bN(t)−dS(t)−cS(t)I(t)N(t)−u(t)S(t),S(0)=S0,˙E(t)=cS(t)I(t)N(t)−(f+d)E(t),E(0)=E0,˙I(t)=fE(t)−(g+a+d)I(t),I(0)=I0,˙N(t)=(b−d)N(t)−aI(t),N(0)=N0,u(t)∈[0,ˉu] for a. e.t∈[0,T], with ˉu∈]0,1]. |
Next, we associate
s(t)=S(t)N(t),e(t)=E(t)N(t),i(t)=I(t)N(t),r(t)=R(t)N(t), | (8) |
we have
s(t)+e(t)+i(t)+r(t)=1 for all t. | (9) |
Notice that
˙s(t)=b−cs(t)i(t)−bs(t)+ai(t)s(t)−u(t)s(t), | (10) |
˙e(t)=cs(t)i(t)−(f+b)e(t)+ai(t)e(t), | (11) |
˙i(t)=fe(t)−(g+a+b)i(t)+ai2(t), | (12) |
˙r(t)=gi(t)−rb(t)+ai(t)r(t)+u(t)s(t). | (13) |
Remarkably, the dead rate parameters do not appear in this model (a feature we discuss in Remark 1 below). It is a simple matter to see that due to (9) we can discard equation (13), allowing us to reduce the number of differential equations from the normalized SEIR model (10)-(13).
Now we are faced with the choice of the cost for the normalized model. Taking into account that the main aim is to control or to eliminate the disease from the population under study, different costs is may be considered, reflecting different concerns.
The choice of the cost for
An almost straightforward translation of this reasoning to our normalized model yields
We postpone this discussion of the introduction of different costs to future research and we proceed now with the cost
(Pn){Minimize∫T0(ρi(t)+u(t)) dtsubject to˙s(t)=b−cs(t)i(t)−bs(t)+ai(t)s(t)−u(t)s(t),s(0)=s0,˙e(t)=cs(t)i(t)−(f+b)e(t)+ai(t)e(t),e(0)=e0,˙i(t)=fe(t)−(g+a+b)i(t)+ai2(t),i(0)=i0,u(t)∈[0,ˉu] for a. e.t∈[0,T], with ˉu∈]0,1]. |
Note that the dynamics is of the form
Remark 1. A word of caution regarding the way the system (10)-(13) is viewed. We cannot interpret the dynamics between these new compartments in the same way as with the classical model. Indeed, in equation (10) the term
We will discuss pros and cons of
Optimal control problems can be solved numerically by direct or indirect methods. Here, we opt to use the direct method (for a description these two methods see, for example, [19]): first the problem is discretized and the subsequent optimization problem is then solved using software packages with large scale nonlinear continuous optimization solvers. In this work all the simulations were made with the Applied Modelling Programming Language (AMPL), developed by [7], and interfaced to the Interior-Point optimization solver IPOPT, developed by [21]. Alternatively, the optimization solver WORHP (see [3]) can also be interfaced with AMPL. We refer the reader to [16] and references within for more information on software for optimal control problems.
The application of the Maximum Principle to problems in the form of
In all the computations we consider the time horizon to be 20 years: thus
Parameter | Description | Value |
b | Natural birth rate | 0.01 |
d | Death rate | 0.0099 |
c | Incidence coefficient | 1.1 |
f | Exposed to infectious rate | 0.5 |
g | Recovery rate | 0.1 |
a | Disease induced death rate | 0.2 |
T | Number of years | 20 |
Parameter | Description | Value |
A | weight parameter | 1 |
B | weight parameter | 2 |
S0 | Initial susceptible population | 1000 |
E0 | Initial exposed population | 100 |
I0 | Initial infected population | 50 |
R0 | Initial recovered population | 15 |
N0 | Initial population | 1165 |
Parameter | Description | Value |
s0 | Percentage of initial susceptible population | 0.858 |
e0 | Percentage of initial exposed population | 0.086 |
i0 | Percentage of initial infected population | 0.043 |
The problem
Although the two problem
Clearly, the reason why
When considering problem
u(t)S(t)≤V0, | (14) |
can be mathematically translated to normalized models but they loose their meaning. However, this drawback may be overcome by considering
We now focus on the Maximum Principle for the problem
H(x,p,u)=pf(x)+pg(x)u−λ(ρi+u), |
for appropriated
(ⅰ)
(ⅱ)
(ⅲ)
(ⅳ)
Since
It is a simple matter to see that condition (ⅲ) is equivalent to
ϕ(t)u∗(t)=maxu{ϕ(t)u(t):0≤u≤ˉu}. | (15) |
It follows that
u∗(t)={ˉu, if ϕ(t)>0,0, if ϕ(t)<0,singular, if ϕ(t)=0. | (16) |
In terms of the data of
−˙ps(t)=(ai(t)−ci(t)−b−u(t))ps(t)+ci(t)pe(t), | (17) |
−˙pe(t)=(ai(t)−b−f)pe(t)+fpi(t), | (18) |
−˙pi(t)=(as(t)−cs(t))ps(t)+(cs(t)+ae(t))pe(t)+(2ai(t)−a−b−g)pi(t)−ρ. | (19) |
Also, we have
ϕ(t)=−1−ps(t)s(t). | (20) |
Since our computations show that a singular arc appear, let us assume that
1Since the initial condition belong to the interior of
R:={(s,e,i)∈R3: s≥0, e≥0, i≥0} |
and so we deduce that
ϕ(t)=0 implies that ps(t)=−1s(t)<0. |
In the interior of the singular interval we have
dϕdt=cs(t)i(t)pe(t)−bps(t)=0 |
implies that that
d2ϕdt=aci(t)2pe(t)s(t)−aci(t)pe(t)s(t)−bci(t)pe(t)s(t)+cfe(t)pe(t)s(t)−cgi(t)pe(t)s(t)−c2i(t)2pe(t)s(t)−ci(t)pe(t)s(t)u(t)+2bci(t)pe(t)+cfi(t)pe(t)s(t)−cfi(t)s(t)pi(t)+abi(t)ps(t)−bci(t)ps(t)−b2ps(t)−bps(t)u(t), | (21) |
depends on the control variable
ddu(d2ϕdt)=−ci(t)pe(t)s(t)−bps(t)>0. |
Thus the strict Generalized Legendre-Clebsch Condition (GLC) holds and we can solve
using(x,p)=c(−c+a)s(t)i(t)2pe(t)ci(t)pe(t)s(t)+bps(t)+cefs(t)pe(t)−b2ps(t)ci(t)pe(t)s(t)+bps(t)+((((−a−b+f−g)s(t)+2b)pe(t)−fs(t)pi(t)−bps(t))c+abps(t))i(t)ci(t)pe(t)s(t)+bps(t) | (22) |
It is important to observe that the above expression for singular controls depends on the multipliers. Since we do not establish that the multipliers are unique, we can only expect to use (22) to validate numerical findings but not to prove to optimality. In fact, to prove optimality of computed solution we need to check numerically sufficient conditions. Unfortunately, there are no numerically verifiable sufficient conditions for problems with singular arcs.
We now present and discuss the results of our simulations for
● Case 1:
● Case 2:
● Case 3:
In the first two cases the computed optimal control exhibits a bang-singular-bang structure while in the last one the optimal control is bang-bang. For all the three cases we present graphs with the computed controls and trajectories. As in [16] and to keep the exposition short, we do not present the graphs of the multipliers but we give their computed initial values, and we also present the final states, the costs and the switching times
Case 1. Taking
Numerical results for Case 1:
s(T)=0.095341,e(T)=0.00051104,i(T)=0.0020380,ps(0)=−126.5,pe(0)=−2253,pi(0)=−3219. |
Case 2. The results of the simulations are shown in figures 6 and 7. In figure 6 the optimal control,
Numerical results for case 2:
s(T)=0.16598,e(T)=0.0033060,i(T)=0.0079433,ps(0)=−377.0,pe(0)=−5137,pi(0)=−7081. |
When we go from case 1 to case 2, the optimal control goes from singular to bang-bang. This is because when we decrease the value of
Case 3. While keeping
Numerical results for case 3:
s(T)=0.071623,e(T)=0.0031999,i(T)=0.015275,ps(0)=−1025,pe(0)=−4692,pi(0)=−6872. |
If the control is bang-bang as in case 2 and 3 second order sufficient conditions may be checked numerically as described in [15] and [16]. Here we refrain from engaging in such discussion to keep the exposition short. Here we compute numerically the switching times using the so called induced optimization problem as in [15]. Recall that the switching times are the points
Implementing the induced optimization problem with AMPL for case 2, with
For case 3, with
We now turn to case 1 where the computed control
We studied the optimal control of an epidemiological normalized SEIR model using a
Moreover, we confronted this problem with the one previously studied in [16] where the so called classical SEIR model is used. The normalized model may cover in one single problem populations of different size and it is defined with what may be seen as a more realistic cost. Because of the use of normalized model, the solution of
The authors would like to thank Prof. Helmut Maurer for numerous and enlightening discussions on this topic as well as his help in writing up AMPL codes for the problems reported here. Thanks are due to the anonymous referees whose many comments and suggestions greatly improved this paper.
The financial support of FEDER funds through COMPETE and Portuguese funds through the Portuguese Foundation for Science and Technology (FCT), within the FCT project PTDC/EEI-AUT/1450/2012—FCOMP-01-0124-FEDER-028894, PTDC/EEI-AUT/2933/2014, TOCCATTA -funded by FEDER funds through COMPETE2020 -POCI and FCT as well as POCI-01-0145-FEDER-006933 -SYSTEC -funded by FEDER funds through COMPETE2020 – Programa Operacional Competitividade e Internacionalização (POCI) – and by national funds through FCT -Fundação para a Ciência e a Tecnologia, are gratefully acknowledged.
[1] | Borsini OE (1999) Valerianaceae. In: Correa MN (Ed.) Flora Patagónica. Col Cient Inst Nac Tec Agropec 8: 448-471. |
[2] | Kutschker A (2011) Revisión del género Valeriana (Valerianaceae) en Sudamérica austral. Gayana Bot 68: 244-296. |
[3] | Nagahama N, Bach H, Manifesto MM, et al. (2016) Valeriana gaimanensis (Valerianaceae nom. conserv.) a new species from the Patagonian semi-arid desert, Argentina. Syst Bot 41: 245-251. |
[4] | Molares S, Ladio AH, Nagahama N (2018) Recent reports on ethnopharmacological and ethnobotanical studies of Valeriana carnosa Sm. (Valerianaceae). In: Martinez JL, Munoz-Acevedo A, et al. (Eds.). Ethnobotany: Local Knowledge and Traditions. CRC Press, Boca Raton, Florida, US, 90-102. |
[5] | Nagahama N, Bonino MF (2020) Modelling the potential distribution of Valeriana carnosa in Argentinean Patagonia: a proposal for conservation and in situ cultivation considering climate change projections. J Appl Res Med Aromat Plants 16: 100240. |
[6] | Kutschker A, Morrone JJ (2012) Distributional patterns of the species of Valeriana (Valerianaceae) in southern South America. Plant Syst Evol 298: 535-547. |
[7] | Villalba R, Lara A, Boninsegna JA, et al. (2003) Large-scale temperature changes across the southern Andes: 20th-century variations in the context of the past 400 years. Clim Change 59: 177-232. |
[8] | Bianchi E, Villalba R, Viale M, et al. (2016) New precipitation and temperature grids for northern Patagonia: Advances in relation to global climate grids. J Meteorol Res 30: 38-52. |
[9] | Schultz J (2002) Biochemical ecology: how plants fight dirty. Nature 416: 267-267. |
[10] | Ramakrishna A, Ravishankar GA (2011) Influence of abiotic stress signals on secondary metabolites in plants. Plant Signal Behav 6: 1720-1731. |
[11] | Andola HC, Gaira KS, Rawal RS, et al. (2010) Habitat dependent variation in berberine content of Berberis asiática Roxb. Ex. DC. in Kumaon, western Himalaya. Chem Biodivers 7: 415-420. |
[12] | Jugran AK, Bahukhandi A, Dhyani P, et al. (2016) Impact of altitudes and habitats on valerenic acid, total phenolics, flavonoids, tannins, and antioxidant activity of Valeriana jatamansi. Appl Biochem Biotech 179: 911-926. |
[13] | Fonseca JM, Rushing JW, Rajapakse NC, et al. (2006) Potential implications of medicinal plant production in controlled environments: the case of feverfew (Tanacetum parthenium). HortScience 41: 531-535. |
[14] | Pavarini DP, Pavarini SP, Niehues M, et al. (2012) Exogenous influences on plant secondary metabolite levels. Anim Feed Sci Tech 176: 5-16. |
[15] | García D, Furlan MR, Diamante MS, et al. (2019) Promising phytochemical responses of Achyrocline satureioides (Lam.) DC. under various farming conditions. Ind Crop Prod 129: 440-447. |
[16] | Binns SE, Arnason JT, Baum BR (2002) Phytochemical variation within populations of Echinacea angustifolia (Asteraceae). Biochem Syst Ecol 30: 837-854. |
[17] | Figueiredo AC, Barroso JG, Pedro LG, et al. (2008) Factors affecting secondary metabolite production in plants: volatile components and essential oils. Flavour Frag J 23: 213-226. |
[18] | Çırak C, Bertoli A, Pistelli L, et al. (2010) Essential oil composition and variability of Hypericum perforatum from wild populations of northern Turkey. Pharm Biol 48: 906-914. |
[19] | Guajardo JJ, Gastaldi B, González SB, et al. (2018) Variability of phenolic compounds at different phenological stages in two populations of Valeriana carnosa Sm. (Valerianoideae, Caprifoliaceae) in Patagonia. Bol Latinoam Caribe 17: 381-393. |
[20] | Zlatev ZS, Lidon FJ, Kaimakanova M (2012) Plant physiological responses to UV-B radiation. Emir J Food Agr 24: 481-501. |
[21] | Apel K, Hirt H (2004) Reactive oxygen species: metabolism, oxidative stress, and signal transduction. Annu Rev Plant Biol 55: 373-399. |
[22] | Laura A, Moreno-Escamilla JO, Rodrigo-García J, et al. (2019) Phenolic compounds, In: Postharvest physiology and biochemistry of fruits and vegetables, Woodhead Publishing, 253-271. |
[23] | Bravo L (1998) Polyphenols: chemistry, dietary sources, metabolism, and nutritional significance. Nut Rev 56: 317e333. |
[24] | Jackman RL, Smith JL (1996) Anthocyanins and betalains, In: Hendry GAF, Houghton JD, Natural Food Colorants, Eds, London, Blackie Academic & Professional, 249-250. |
[25] | Crozier A, Clifford MN, Ashihara H (2006) Plant Secondary Metabolites: Occurrence. Structure and Role in the Human Diet, Oxford, Blackwells 26: 1001-1013. |
[26] | Shahidi F, Janitha P, Wanasundara P (1992) Phenolic antioxidants. Crit Rev Food Sci 32: 67-103. |
[27] | Piccinelli A, Arana S, Caceres A, et al. (2004) New lignans from the roots of Valeriana prionophylla with antioxidative and vasorelaxant activities. J Nat Prod 67: 1135-1140. |
[28] | Russell W, Duthie G (2011) Plant secondary metabolites and gut health: the case for phenolic acids. Proc Nutr Soc 70: 389-396. |
[29] | Surveswaran S, Cai Y, Corke H, et al. (2007) Systematic evaluation of natural phenolic antioxidants from Indian medicinal plants. Food Chem 102: 938-953. |
[30] | Wojdylo A, Oszmiański J, Czemerys R (2007) Antioxidant activity and phenolic compounds in 32 selected herbs. Food Chem 105: 940-949. |
[31] | Bhatt ID, Dauthal P, Rawat S, et al. (2012) Characterization of essential oil composition, phenolic content, and antioxidant properties in wild and planted individuals of Valeriana jatamansi Jones. Sci Hortic 136: 61-68. |
[32] | Estomba D, Ladio A, Lozada M (2006) Medicinal wild plant knowledge and gathering patterns in a Mapuche community from North-western Patagonia. J Ethnopharmacol 103: 109-119. |
[33] | Molares S, Ladio AH (2012) Plantas aromáticas con órganos subterráneos de importancia cultural en la Patagonia argentina: una aproximación a sus usos desde la etnobotánica, la percepción sensorial y la anatomía. Darwiniana 2012: 7-24. |
[34] | Nagahama N, Manifesto MM, Fortunato RH (2019) Vegetative propagation and proposal for sustainable management of Valeriana carnosa Sm., a traditional medicinal plant from Patagonia. J Appl Res Med Aromat Plants 14: 100218. |
[35] | Sumner LW, Amberg A, Barrett D, et al. (2007) Proposed minimum reporting standards for chemical analysis. Metabolomics 3: 211-221. |
[36] | Chaisri P, Laoprom N (2017) Antioxidant properties and total phenolic content of selected traditional Thai medicinal plants. Thai Pharm Health Sci J 12: 10-18. |
[37] | Gastaldi B, Assef Y, van Baren C, et al. (2016) Actividad antioxidante en infusiones, tinturas y aceites esenciales de especies nativas de la Patagonia Argentina. Rev Cub Plant Med 21: 51-62. |
[38] | Simirgiotis MJ, Silva M, Becerra J, et al. (2012) Direct characterisation of phenolic antioxidants in infusions from four Mapuche medicinal plants by liquid chromatography with diode array detection (HPLC-DAD) and electrospray ionisation tandem mass spectrometry (HPLC-ESI-MS). Food Chem 131: 318-327. |
[39] | Kaliora A, Kogiannou D, Kefalas P, et al. (2014) Phenolic profiles and antioxidant and anticarcinogenic activities of Greek herbal infusions; balancing delight and chemoprevention? Food Chem 142: 233-241. |
[40] | Simirgiotis M, Benites J, Areche C, et al. (2015) Antioxidant capacities and analysis of phenolic compounds in three endemic Nolana species by HPLC-PDA-ESI-MS. Molecules 20: 11490-11507. |
[41] | Navarrete A, Avula B, Choi YW, et al. (2006) Chemical fingerprinting of Valeriana species: simultaneous determination of valerenic acids, flavonoids, and phenylpropanoids using liquid chromatography with ultraviolet detection. J AOAC Int 89: 8-15. |
[42] | Meinhart AD, Damin FM, Caldeirão L, et al. (2017) Chlorogenic acid isomer contents in 100 plants commercialized in Brazil. Food Res Int 99: 522-530. |
[43] | Sen-Utsukarci B, Taskin T, Goger F, et al. (2019) Chemical composition and antioxidant, cytotoxic, and insecticidal potential of Valeriana alliariifolia in Turkey. Arch Ind Hyg Toxicol 70: 207-218. |
[44] | Sarikurkcu C, Jeszka-Skowron M, Ozer MS (2020) Valeriana dioscoridis aerial parts' extracts - A new source of phytochemicals with antioxidant and enzyme inhibitory activities. Ind Crop Prod 148: 112273. |
[45] | Mahibbur RM, Govindarajulu Z (1997) A modification of the test of Shapiro and Wilks for normality. J Appl Stat 24: 219-235. |
[46] | Conover WJ (1999) Practical nonparametric statistics. New York: John Wiley and Sons, Inc. |
[47] | Di Rienzo JA, Casanoves F, Balzarini MG, et al. (2015) InfoStat, v. 2015. Grupo InfoStat, Córdoba, Universidad Nacional de Córdoba. |
[48] | Lester G, Lewers K, Medina M, et al. (2012) Comparative analysis of strawberry total phenolics via Fast Blue BB vs. Folin-Ciocalteu: Assay interference by ascorbic acid. J Food Compos Anal 27: 102-107. |
[49] | Ludwig I, Bravo J, De Peña M, et al. (2013) Effect of sugar addition (torrefacto) during roasting process on antioxidant capacity and phenolics of coffee. LWT-Food Sci Technol 51: 553-559. |
[50] | Muñoz-Bernal O, Torres-Aguirre G, Núñez-Gastélum J, et al. (2017) Nuevo acercamiento a la interacción del reactivo de Folin-Ciocalteu con azúcares durante la cuantificación de polifenoles totales. Revista TIP 20: 23-28. |
[51] | Katsube T, Tabata H, Ohta Y, et al. (2004) Screening for antioxidant activity in edible plant products: Comparison of low-density lipoprotein oxidation assay, DPPH radical scavenging assay, and Folin-Ciocalteu assay. J Agric Food Chem 52: 2391-2396. |
[52] | de Sousa SHB, de Andrade Mattietto R, Chisté RC, et al. (2018) Phenolic compounds are highly correlated to the antioxidant capacity of genotypes of Oenocarpus distichus Mart. fruits. Food Res Int 108: 405-412. |
[53] | Djeridane A, Yousfi M, Nadjemi B, et al. (2006) Antioxidant activity of some Algerian medicinal plants extracts containing phenolic compounds. Food Chem 97: 654-660. |
[54] | Katalinic V, Milos M, Jukic M (2006) Screening of 70 medicinal plant extracts for antioxidant capacity and total phenols. Food Chem 94: 550-557. |
[55] | Ziani BE, Heleno SA, Bachari K, et al. (2019) Phenolic compounds characterization by LC-DAD-ESI/MSn and bioactive properties of Thymus algeriensis Boiss. & Reut. and Ephedra alata Decne. Food Res Int 116: 312-319. |
[56] | Akula R, Ravishankar GA (2011) Influence of abiotic stress signals on secondary metabolites in plants. Plant Signal Behav 6: 1720-1731. |
[57] | Selvam K, Rajinikanth R, Govarthanan M, et al. (2013) Antioxidant potential and secondary metabolites in Ocimum sanctum L. at various habitats. J Med Plants Res 7: 706-712. |
[58] | Rodríguez-Calzada T, Qian M, Strid Å, et al. (2019) Effect of UV-B radiation on morphology, phenolic compound production, gene expression, and subsequent drought stress responses in chili pepper (Capsicum annuum L.). Plant Physiol Biochem 134: 94-102. |
[59] | Devkota A, Dall Acqua S, Jha PK, et al. (2010) Variation in the active constituent contents in Centella asiatica grown in different habitats in Nepal. Botanica Orientalis: J Plant Sci 7: 43-47. |
[60] | Alonso-Amelot ME, Oliveros-Bastidas A, Calcagno-Pisarelli M (2007) Phenolics and condensed tannins of high altitude Pteridium arachnoideum in relation to sunlight exposure, elevation, and rain regime. Biochem Syst Ecol 35: 1-7. |
[61] | Oloumi H, Hassibi N (2011) Study the correlation between some climate parameters and the content of phenolic compounds in roots of Glycyrrhiza glabra. J Med Plants Res 5: 6011-6016. |
[62] | Nicolle C, Simon G, Rock E, et al. (2004) Genetic variability influences carotenoid, vitamin, phenolic, and mineral content in white, yellow, purple, orange, and dark-orange carrot cultivars. J Am Soc Hortic Sci 129: 523-529. |
[63] | Bell CD, Kutschker A, Arroyo MT (2012) Phylogeny and diversification of Valerianaceae (Dipsacales) in the southern Andes. Mol Phylogenet Evol 63: 724-737. |
[64] | Owen RW, Haubner R, Mier W, et al. (2003) Isolation, structure elucidation and antioxidant potential of the major phenolic and flavonoid compounds in brined olive drupes. Food Chem Toxicol 41: 703-717. |
[65] | Mendez J (2005) Dihydrocinnamic acids in Pteridium aquilinum. Food Chem 93: 251-252. |
[66] | Trejo-Machin A, Verge P, Puchot L, et al. (2017) Phloretic acid as an alternative to the phenolation of aliphatic hydroxyls for the elaboration of polybenzoxazine. Green Chem 19: 5065-5073. |
[67] | Kikuzaki H, Hisamoto M, Hirose K, et al. (2002) Antioxidant properties of ferulic acid and its related compounds. J Agric Food Chem 50: 2161-2168. |
[68] | Nićiforović N, Abramovič H (2014) Sinapic acid and its derivatives: natural sources and bioactivity. Comp Rev Food Sci F 13: 34-51. |
[69] | Nenadis N, Lazaridou O, Tsimidou MZ (2007) Use of reference compounds in antioxidant activity assessment. J Agric Food Chem 55: 5452-5460. |
[70] | Yun KJ, Koh DJ, Kim SH, et al. (2008) Anti-inflammatory effects of sinapic acid through the suppression of inducible nitric oxide synthase, cyclooxygase-2, and proinflammatory cytokines expressions via nuclear factor-κB inactivation. J Agric Food Chem 56: 10265-10272. |
[71] | Johnson ML, Dahiya JP, Olkowski AA, et al. (2008) The effect of dietary sinapic acid (4-hydroxy-3, 5-dimethoxy-cinnamic acid) on gastrointestinal tract microbial fermentation, nutrient utilization, and egg quality in laying hens. Poultry Sci 87: 958-963. |
[72] | Engels C, Schieber A, Gänzle MG (2012) Sinapic acid derivatives in defatted oriental mustard (Brassica juncea L.) seed meal extracts using UHPLC-DADESI-MSn and identification of compounds with antibacterial activity. Eur Food Res Technol 234: 535-542. |
[73] | Hudson EA, Dinh PA, Kokubun T, et al. (2000) Characterization of potentially chemopreventive phenols in extracts of brown rice that inhibit the growth of human breast and colon cancer cells. Cancer Epidemiol Biomarkers Prev 9: 1163-1170. |
[74] | Yoon BH, Jung JW, Lee JJ, et al. (2007) Anxiolytic-like effects of sinapic acid in mice. Life Sci 81: 234-240. |
[75] | Robbins RJ (2003) Phenolic acids in foods: An overview of analytical methodology. J Agric Food Chem 51: 2866-2887. |
[76] | Cuvelier ME, Richard H, Berset C (1992) Comparison of the antioxidative activity of some acid-phenols: structure-activity relationship. Biosci Biotech Bioch 56: 324-325. |
[77] | Kim DO, Lee CY (2004) Comprehensive study on vitamin C equivalent antioxidant capacity (VCEAC) of various polyphenolics in scavenging a free radical and its structural relationship. Crit Rev Food Sci 44: 253-273. |
[78] | Clifford MN, Jaganath IB, Ludwig IA, et al. (2017) Chlorogenic acids and the acyl-quinic acids: discovery, biosynthesis, bioavailability and bioactivity. Nat Prod Rep 34: 1391-1421. |
[79] | Clifford MN (2000) Chlorogenic acids and other cinnamates: nature, occurrence, dietary burden, absorption and metabolism. J Sci Food Agric 80: 1033-1042. |
[80] | Clifford MN, Zheng W, Kuhnert N (2006) Profiling the chlorogenic acids of Aster by HPLC-MSn. Phytochem Analysis 17: 384-393. |
[81] | Clifford MN, Kirkpatrick J, Kuhnert N, et al. (2008) LC-MSn analysis of the cis isomers of chlorogenic acids. Food Chem 106: 379-385. |
[82] | Makita C, Chimuka L, Cukrowska E, et al. (2017) UPLC-qTOF-MS profiling of pharmacologically important chlorogenic acids and associated glycosides in Moringa ovalifolia leaf extracts. S Afr J Bot 108: 193-199. |
[83] | Masike K, Khoza SB, Steenkamp AP, et al. (2017) A Metabolomics-guided exploration of the phytochemical constituents of Vernonia fastigiata with the aid of pressurized hot water extraction and liquid chromatography-mass spectrometry. Molecules 22: 1200. |
[84] | Jaiswal R, Kuhnert N (2011) How to identify and discriminate between the methyl quinates of chlorogenic acids by liquid chromatography-tandem mass spectrometry. J Mass Spectrom 46: 269-281. |
[85] | Clifford MN (2017) Some Notes on the Chlorogenic Acids. 3. LC and LC-MS. Available from: https://www.researchgate.net/publication/312590947_Some_Notes_on_the_Chlorogenic_Acids_3_LC_and_LC-MS_Version_3_January_2017. |
[86] | Naveed M, Hejazi V, Abbas M, et al. (2018) Chlorogenic acid (CGA): A pharmacological review and call for further research. Biomed Pharmacother 97: 67-74. |
[87] | Clifford MN, Kerimi A, Williamson G (2020) Bioavailability and metabolism of chlorogenic acids (acyl-quinic acids) in humans. Compr Rev Food Sci Food Saf 19: 1299-1352. |
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Parameter | Description | Value |
b | Natural birth rate | 0.01 |
d | Death rate | 0.0099 |
c | Incidence coefficient | 1.1 |
f | Exposed to infectious rate | 0.5 |
g | Recovery rate | 0.1 |
a | Disease induced death rate | 0.2 |
T | Number of years | 20 |
Parameter | Description | Value |
A | weight parameter | 1 |
B | weight parameter | 2 |
S0 | Initial susceptible population | 1000 |
E0 | Initial exposed population | 100 |
I0 | Initial infected population | 50 |
R0 | Initial recovered population | 15 |
N0 | Initial population | 1165 |
Parameter | Description | Value |
s0 | Percentage of initial susceptible population | 0.858 |
e0 | Percentage of initial exposed population | 0.086 |
i0 | Percentage of initial infected population | 0.043 |
Parameter | Description | Value |
b | Natural birth rate | 0.01 |
d | Death rate | 0.0099 |
c | Incidence coefficient | 1.1 |
f | Exposed to infectious rate | 0.5 |
g | Recovery rate | 0.1 |
a | Disease induced death rate | 0.2 |
T | Number of years | 20 |
Parameter | Description | Value |
A | weight parameter | 1 |
B | weight parameter | 2 |
S0 | Initial susceptible population | 1000 |
E0 | Initial exposed population | 100 |
I0 | Initial infected population | 50 |
R0 | Initial recovered population | 15 |
N0 | Initial population | 1165 |
Parameter | Description | Value |
s0 | Percentage of initial susceptible population | 0.858 |
e0 | Percentage of initial exposed population | 0.086 |
i0 | Percentage of initial infected population | 0.043 |