Plane Thermoelastic Waves in Infinite Half-Space Caused Operational Research in Engineering Sciences: Theory and Applications Vol. 3, Issue 1, 2020, pp. 1-15 ISSN: 2620-1607 eISSN: 2620-1747 DOI: https:// doi: 10.31181/oresta200101t * Corresponding author. suncicat@uns.ac.rs (S. Kocić-Tanackov), ilijat@uns.ac.rs (I. Tanackov), lmojovic@tmf.bg.ac.rs (Lj. Mojović), jpejin@uns.ac.rs (J. Pejin), feta.sinani@unite.edu.mk (F. Sinani) SYNERGY EFFECTS OF NATURAL FUNGAL INHIBITORS CALCULATED BY QUEUING MODEL Sunčica Kocić-Tanackov 1, Ilija Tanackov 2*, Ljiljana Mojović 3, Jelena Pejin 1, Feta Sinani 4 1 University of Novi Sad, Faculty of Technology, Serbia, 2 University of Novi Sad, Faculty of Technical Sciences, Serbia, 3 University of Belgrade, Faculty of Technology and Metallurgy, Serbia, 4 Faculty of Applied Sciences, State University of Tetovo, Republic of North Macedonia Received: 24 October 2019 Accepted: 28 January 2020 First online: 06 February 2020 Original scientific paper Abstract. Model is based on the fungal birth and death processes. Model is suited for Petri dish. Growth of fungal colony diameter in Petri dish is described with exponential function. The value of diameter is declared as integer variable. Integer variable with 1 mm increment is a discrete state of the system. Time in the system is continuously. Discrete states, continuous time and exponential growth are basis for the application of queuing systems in the Petri dish. Queuing system clearly separated the intensity of birth and death. Difference between the birth intensity and death intensity is declared as the fungal life cycle. Fungal life cycle variable is extra sensitive to the inhibitors effects. The procedures for parameters calculation are mathematically explained, as well as the significance of the obtained parameters. Application of the model is performed for F. verticilloides in control conditions and at 16% concentration of basil and clove essential oils. Life cycle minimum is the synergetic inhibition maximum. For F. verticilloides, synergetic inhibition maximum is at 42% of basil and 58% of clove in 16% essential oil concentration. Key words: fungi, synergy, inhibition, essential oil, natural extract 1. Introduction Exponential probability distribution has exceptional constitutive characteristics such as maximum entropy, constant hazard function and it is memoryless. If the random evolution of a system is exponentially distributed, then this system is memoryless. In memoryless system, future state depends only on its present state, and not on any past states. The exponential distribution is the only distribution that Kocić-Tanackov et al./Oper. Res. Eng. Sci. Theor. Appl. 3 (1) (2020) 1-15 2 has memoryless property. Also, exponential function with the base e = 2.718281, f(x)=ex has one unique feature. Function f(x)=ex and her arbitrary derivation are identical, f(x)= f(x)  =…= f(x)(n)= ex. This feature is the basis of memoryless. Mycelium growth (Kang et al., 2003, Vargas-Arispuro et al., 2005; Boyraz and Özcan, 2006, Judith et al., 2008, Villa et al., 2009), hifae growth (Larralde-Corona et al., 1997 ; Kampichler et al., 2004; Diéguez-Uribeondo et al., 2004), spore count (Wagner et al., 2001) and fungal germination under extreme conditions (Onofri et al., 2007) have exponential properties. Alteration of the fungal colony diameter (Roller and Covill, 1999; Tzortzakis and Economakis, 2007; Taniwaki et al., 2009, Tang et al., 2009), fungal colonies in the presence of bacteria (Brule et al., 2001), the development of fungal biomass (Damar et al., 2006), the influence of various inhibitors (Collopy-Junior et al., 2006), essential nutritienats (Ramirez et al., 2004) and the impact of radiation on the fungal growth (Maity et al., 2008) can be described with exponential distribution. Indirect evidence about exponential fungal dynamics we can find in the fungal degradation process (Kim et al., 2000; Schober and Trösch, 2000; Mal-Nam et al., 2000; Ruiz-Aguilar et al., 2002; Ishii et al., 2007, 2008; Wakaizumi et al., 2009; Tanaka et al., 2009, Elsherbiny et al., 2017). Fungal growth occurs in the system of self-replicator species (Scheuring and Szathmáry, 2001; Chertov et al., 2004; Milne, 2008; Boswell, 2008). Self-replicator growth system is the basis of analogy between fungal growth and exponential function. Indirectly, through an exponential distribution, fungal systems are memoryless. Analogously, the fungal growth is the Markovian process. Management of microbiological systems has significant economic, environmental and health aspects. The microbiological control of foods is particularly significant. In the case of fungi, control of growth by using inhibitors is based on compromise. Inhibition need to meet the requirements of microbiological quality and in the same time, to preserve the nutritional, health and organoleptic properties of food. The intensity of fungal inhibition is commonly investigated with the agar plate method, based on the measurement of the colony diameter, in the presence of essential oil or herbal extract during the time. (Nielsen and Rios, 2000; Guynot et al., 2003; Suhr and Nielsen, 2003; Benkeblia, 2004; Pereira, et al., 2006; Sheng-Hsien et al., 2007; Lopez-Malo et al., 2007; Fung and Zheng, 2007; Tullio et al., 2007; Soylu et al., 2007; Viuda-Martos et al., 2007, 2008; Tzortzakis, 2009; Reddy et al., 2009; Tatsadjieu et al., 2009, Tanackov S. et al., 2014; Badea et al., 2016; Llana-Ruiz-Cabello et al., 2016 Tancinová et al., 2018, 2019). Inhibitors can be synthetic and natural. Use of synthetic inhibitors is not always desirable, especially in food. Essential oils and plant extract are the main natural inhibitors. A special analytical challenge is the potential synergic effects in the application of inhibitors. Synergy inhibitors may improve the composition of the combinatorial selection of inhibitors with greater antifungal effect and more acceptable organoleptic characteristics. The inhibition intensity is determined by the comparative method a posteriori. This method is based on determining the initial birth rate without inhibition. In the presence of inhibitors, a reduced birth rate is obtained. The comparison (difference) Synergy effects of natural fungal inhibitors calculated by queuing model 3 of these two intensities represents the difference in the birth intensity, again. The intensity of dying due to inhibitory effects remains unknown. Individual inhibitor intensities can be estimated by standard procedure, but the synergistic effect of two or more inhibitors is difficult to describe by existing models. Considering the growth of colonies as a stochastic system opens up the possibility of applying a queuing system (QS). The birth and death intensities in microbiology analysis are analogous to the intensities of clients arrivals and clients servicing from queueing systems. The capacity of QS models has been proven in many fields of research (Fazlollahtabar anf Gholizadeh, 2019a; Fazlollahtabar and Gholizadeh, 2019b; Tanackov I. et al, 2019a, Tanackov I. et al, 2019b) 2. Materials and methods 2.1. Experimental setup For the antifungal activity testing, commercially available, food grade clove and basil extract was provided from ETOL “Tovarna arom in eteričnih olj” d.d., Celje, Slovenia. As test microorganisms, the following fungal strain from the genus Fusarium was used: F. verticillioides (Sacc.) Nirenberg (syn. F. moniliforme Sheld.). The fungal culture were isolated from cakes and maintained on Potato Dextrose Agar (PDA) at 4C as a part of the collection of the Laboratory for Food Microbiology at the Faculty of Technology, University of Novi Sad, Serbia. The agar plate method was applied in the testing of the antifungal activity of extracts. The basic medium for the antifungal tests was PDA. The medium was divided into equal volumes (150 ml), poured into Erlenmeyer (250 ml) flasks and autoclaved at 121ºC for 15 min. Concentrations 0 i 0.16% (v/v) were tested self-contained extracts, and basil-clove combinations: 50%:50%; 75%:25% i 25%:75%. The extracts were added to medium after cooling to 45C. The culture medium was then poured into sterile Petri dishes (9 cm), 12 ml into each plate. To prepare the conidial suspension dispute we used the seven-day culture F. verticillioides grown on PDA. Suspension of fungi prepared in a medium which contained 0.5% Tween 80 and 0.2% agar dissolved in distilled water and were adjusted to provide initial spore count of 106 spores/mL by using a haemocytometer. For each extract dose and fungi species, including the controls were centrally inoculated by spreading 1 µl of spore suspension (103 spores/ml) using an inoculation needle. After inoculation, the Petri plates were closed with parafilm. The efficacy of the treatment was evaluated by daily measurement of the diameter of radial colony growth during 14 days of incubation at 25 2ºC (table 1.). 2.2. Markovian process in Petri dish Exponential parameter (Whitt, 2018; Tanackov et al., 2019) of fungal growth  is a function of the intensity of birth  and death , =f(,), provided . In Petri dish queuing system, number of microorganisams determines the state of the system. Description with Markovian birth-death process is based on the exponential intensity of birth  and death . If the initial state of the system is defined with zero Kocić-Tanackov et al./Oper. Res. Eng. Sci. Theor. Appl. 3 (1) (2020) 1-15 4 microorganisms, then the initial intensity of death  is also equal to zero. The system goes into a state one microorganism with intensity . Simultaneously with the transition to a state one microorganism, death process with intensity  is started. In the same time with the transition to a state with one microorganism, process of death with intensity  start. From the state with one microorganism, system exceeds to the state with two microorganisms by the same intensity of the birth, . If the system implemented another birth with intensity , and the first micro-organism has not finished the process of dying, the system exceeds in to the state two microorganisms. The dying process of the first microorganism is not complited and the second microorganism begins the process of dying. Therefore, the intensity of death in a state two microorganisms is equal 2. System with the same birth intensity exceeds in the next state, and with multiplicity intensity of death exceeds to a previous state. If the number of microorganisms is equal to k in the system, then all k microorganisms are in the process of dying. Therefore, the intensity of dying in system with k microorganisms is equal to k. System with k microorganisms cannot realize (k +1) intensity of the death. This relationship between the intensity of birth  and intensity of death , can be described with exponential development of fungal colonies in Petri dish (Fig. 1). Figure 1. Fugal colony, exponential development At asymptote diameter, the intensity of birth and k multiplicated death are identical, =k. The value of the colony diameter is equal to asymptote value which is maximal colony diameter Dmax. This point have a crucial importance for solving the explicit form of the function =f(,). The solution to the intensity of dying is now Synergy effects of natural fungal inhibitors calculated by queuing model 5 available, initially in steady-state mode. If necessary, the intensity of dying can be considered as non-stationary in time, and additional possibilities are consideration of non-stationary intensity of dying in conditions of different temperatures, humidity, initial inoculation or other important microbiological parameters. 2.3. Petri dish Queuing system Marcovian processes in the system are determined with the time and state of the system. Time in the system can be discretely and continuously. State of the system can be discrete and continuous, also. Consecutive time intervals for fungal colony diameter measurements were determined by SI unit, time. These intervals are discrete, usually 1 day. Measurement of fungal colony diameters in Petri dish, are recorded in the SI unit of length, in millimetres or centimetres. Development of fungal colonies declared this dimension as a variable. Diameters values are represent in the time series. The time is independent variable, and the diameter is dependent variable. Due to the nature of the fungal colony development, the total number of fungi cannot be precisely determined. All elements of the fungal colony development are synthesized in the diameter. Diameter is a generalized variable of the system. The state of the fungal colony is continuous variable. Fungal colony diameter (d0, d1, d2, d3, …, dk), f(ti)=di time series in Petri dish, for discrete time intervals t, (t0, t1, t2, t3, …, tk), t(i+1) = ti+t , k0, 1, 2, … , n have a form of exponential function: )e1(Dd)t(f k t m axkk − −== k0, 1, 2, … , n (1) A high approval of empirical and theoretical data is necessary condition for regular description of the fungal colony diameter with the exponential function. This agreement can be expressed with the correlation coefficient. The linear regression of empirical and theoretical data must be described with fulfil values of parameters a1 and b0, in addition to the high value of correlation coefficient r1. A fulfilment of these conditions gives a representative description of the empirical time series with exponential theoretical function. Discrete values of colony diameter can be obtained with integer function values from representative function.    )e1(DINT)t(f)t(s tmax −−== (2) Approximation of f(t) with the function s(t) depends from the increment size. Smaller increment has a better approximation. For fungal colony diameter measuring in millimetres, integer increment 1 mm gives a satisfactory approximation. With discrete values of the colony diameter, are fulfilment conditions for the application of Markovian process with continuous time. Continuous time with discrete states of the system allows the Petri dish to formation Markovian queuing system (Fig. 2). Kocić-Tanackov et al./Oper. Res. Eng. Sci. Theor. Appl. 3 (1) (2020) 1-15 6 Figure 2. Petri dish Markovian queuing system Explicit form of the function =f(,) have two unknown variables,  and . For the calculation of their values, it is necessary to define a system of two equations. The first equation is obtained from initial conditions. At the beginning of growth, at t=0, the intensity of death is equal to zero, =0. Intensity of birth  is equal to: Dmax(1−e−t)t=0 = (Dmaxe−t) t=0 =    = Dmax (3) The second equation can be obtained from the development of state and calculation of the average number of clients in the Markovian system (Fig. 3). Figure 3. Development of Markovian’s system mold colonies in Petri dishes Differential equations of queuing states in the stationary mode, with constant values of birth and death intensity (t)= =const and (t)==const, are: 01100 pppp0)t(p   =+−== 0 2 220 2 21101 p 21 1 ppp0p2ppp0)t(p          =+   −=+−−== 0 3 3202 3 21112 p 321 1 pp3p 2 0p3pp2p0)t(p          =+   −=+−−== Synergy effects of natural fungal inhibitors calculated by queuing model 7 . . . . . 0 k k1kkk1kk p !k 1 ppppp0)t(p         =−−−== +− . . . . . 0 n nn1nn p !n 1 ppnp0)t(p         =−+== − (4) In the n previous equations we have (n +1) unknown variables, k0, 1, 2, … , n. Equation needed to solve this system of equations, we can obtaine from the basic requirements of probability states: 1p !n 1 ...p !k 1 ...pp1pp...ppp 0 n 0 k 00n1n210 =        ++        ++   +=+++++ − From these (n +1) equations, probability of state p0 is:          ==         =        ++        ++   + = = n 0k k0 n 0k k 0 nk 0 0 !k 1 1 p1 !k 1 p !n 1 ... !k 1 ...1(p (5) Recurrent equation for calculating the probabilities of the queuing system states is obtained from (4) and (5):                  =        = = n 0k k k k0 k k !k 1 !k 1 pp !k 1 p (6) The average number of clients in the system is obtained from the (6):                   −          =                  =                  =  = = − = = = = = n 0k k n 1s 1s n 0s n 0k k s n 0s n 0k k s n 0s s !k 1 )!1s( 1 !k 1 !s s !k 1 !s 1 sps (7) The exponential function   e may be written as a Taylor series: Kocić-Tanackov et al./Oper. Res. Eng. Sci. Theor. Appl. 3 (1) (2020) 1-15 8    = =+         +         +         +         =         e !3!2!1!0!n 3210 0n n  1 e e !k 1 )!1s( 1 lim 0k k 1s 1s n ==                   −      =  = − → For a large enough n, we can adopt 1 !k 1 )!1s( 1 n 0k k n 1s 1s                    − = = − Average number of clients in the system is equal:   =  = n 0s sps (8) In stationary mode of ergodic Marcovian queuing system, this value (8) is equal to the average diameter of the fungal colony dave. From birth intensity initial conditions and from average number of clients, the value of death intensity  is: ave aveave d DD dd  =   =   = maxmax (9) Values Dmax,  and dave we can calculated from experimental measurements of Petri dish. Dmax is the parameter of the fungal colony asymptote.  is the parameter of the exponential function.  is calculated by the heuristic search of the colony diameter time series d0, d1, d2, d3, … , dk, Parameter dave is equal to:  ++++ == = k i k iave k dddd d k d 0 210 ...1 (10) In a long time of measurement, average diameter dave converge to asymptote of fungal colony Dmax. max 210 ...limlim D k dddd d k k ave k = ++++ = →→ (11) Also, the intensity of death converges to the growth rate. =  = ++++  = →→→ max max 210 max lim ... limlim D D k dddd D kkkk (12) Synergy effects of natural fungal inhibitors calculated by queuing model 9 Therefore, entry in to the deep asymptotic region should be limited. The introduction of another diameter dk+1 from time series in to the dave calculation (11), should be stopped because of small differences between successive diameter, dkdk+1. Significance of this difference, p= (1−q) can be directly set and calculated from the integral equation (13): qeeD t t eDdteD kk k k tt k kt t t t −== − ++ + −−+−− )()1( 11 1 max 1 maxmax (13) Adding and subtracting the value of 1, we relate the diameter and significance (14): qeeDeeD kkkk tttt −−−−=−+−− ++ −−−− ))1(1()11( 11 maxmax (14) Successive diameters are )e1(Dd k t m axk  −= i )e1(Dd 1k t m ax1k + + −= , and limit of the diameter difference dend is equal to (15) : endkkkk d D q ddqddD =  −−− ++ max 11max )()( (15) Equation (14) provides reliable intensity of birth  and death , with the required significance p. 2.4. Results The calculation of all relevant parameters is given in table 2. Calculated limit difference dend for significance p=0.95 satisfies all the requirements of measuring up to 14 days. Empirical results of measuring diameter and theoretical exponential functions linear regression parameters a and b were a1 and b0, in control conditions and 16% concentration for all compositions of basil and clove essential oils. The correlation coefficient is high r2 0.99, for all conditions, also. Calculation of the parameters  i Dmax is valid. Parameter dave is obtained from (11) as the average diameter of fungal colonies. Birth intensity  is obtained form (3), and death intensity  is obtained from (9) 3. Discusion In this example, approximated diameter converges to value 120.154 mm for 50% basil and 50% clove in 16% essential oil concentration. This value is larger than the approximated diameter for control conditions, 117.176 mm. From diameter comparation, synergetic stimulation off fungal growth is deduced by classic approach. However, the diameter does not express explicit morphological changes. Morphological changes are contained in the parameter of the life cycle. The life cycle of fungi represent the subtraction between birth intensity and death intensity. Subtraction between birth intensity in control conditions control=9,608 mm/day and the death intensity in control conditions control=2,058 mm/day is the life cycle of fungi F. verticilloides in control conditions control=7,594 mm/day. Life cycle control value is constant for all range of concentration and arbitrary inhibitors Kocić-Tanackov et al./Oper. Res. Eng. Sci. Theor. Appl. 3 (1) (2020) 1-15 10 relation. Intensity of birth and death values, basil clove i basil clove respectively, at 16% concentration for different relations basil and clove, do not have a constant value. These values are functions of relations between basil and clove. The calculations of the F. verticilloides life cycles in control condition and basil-clove synergetic conditions are given in table 3. Graphic presentation of the results in table 3 is given in Fig. 4. Figure 4. F. verticilloides life cycle dynamic From Fig. 4 is the apparent dynamics of birth and death process for the F. verticilloides. Birth process intensity basil clove on the 16% concentration has pronounced variations of value. Birth intensity at 100% basil in 16% essential oil concentration (8.850 mm/day) have higher birth intensity from 100% clove in 16% essential oil concentration (6,197 mm/day). With the increasing participation of clove in 16% essential oil concentration, come to a sudden fall of the birth intensity. Minimum intensity birth is about 60% basil and 40% clove participation. With further increase participation of clove, comes an increase in the intensity of birth. Stabilization of about 75% of clove participation, remains constant until the end of the domain. The intensity of the death basil clove at 16% concentration has not pronounced variations. Death intensity have basil clove are less than the death intensity in control conditions. Lacking the expected death intensity increase. However, reducing the intensity of growth directly reduces the quantity of the system, and thus the intensity of dying. Birth intensity minimum is at 50% basil and 50% clove. Death intensity minimum is at 25% basil and 70% clove. Approximate function of the life cycle basil clove = basil clove − basil clove , has a minimum at 42% basil and 58% clove. F. verticilloides life cycle minimum is the maximum basil-clove synergetic inhibition (Fig. 4.). Synergy effects of natural fungal inhibitors calculated by queuing model 11 Equilibrium line Eq gets points through 100% of the selected concentrations of two inhibitors. Equilibrium line is a set of values that is linearly proportional to the inhibitor participation in a 16% essential oil concentration. The synergetic stimulation zone (SS) is above from Equilibrium line The synergetic inhibitona zone (SI) is below from Equilibrium line. Area from Equilibrium line to the life cycle control level line is the synergic stimulation area. The values of the life cycle can vary about Equilibrium line. In such variations, values above the Equilibrium line are synergistic stimulation, even though they are less from the control level value. Synergetic inhibition values are below the Equilibrium line. In the shown case, for 16% concentration of essential oil relationships Basil and Clove, all values of the life cycle are under Equilibrium line. Selection of essential oils have a distinct inhibiting effect of synergy in the whole area, with a pronounced minimum of 42% basil and 58% clove. Standard models based on the difference in growth intensity between non-inhibited and inhibited sample, cannot directly express the maximum synergistic effect of the two inhibitors. The formation of the two-dimensional function of the action of two inhibitors using standard models requires an incomparably larger number of experiments with different concentrations, and one post-process application of some heuristic model. The results show that the QS model is more accurate, reliable and less expensive for research. 4. Conclisuions Queuing model has a high sensitivity. The basis of sensitivity is in intensity differentiation. These intensities are obtained from growth rate. At the same time, it is necessary to percieve a clear distinction between growth rate and life cycle. Growth rate is a feature of the of birth and death process. The life cycle is a feature of the birth and death intensitys. The queuing model has limitations. In the case of higher concetrations, application of inhibitors may delay the start of fungal growth. During the asymptotic inhibition, birth intensity is equal to zero. Changes in diameter does not correspond to exponential function. This period lasts until the beginning of delayed growth. Start of growth changes the value of the birth intensity. Existence of two values for the same intensity is a feature of nonstationary queing system. Solving the nonstationary queuing system can not be done by the proposed mathematical procedure. Therefore, the model gives solutions only in the case of small concentrations of inhibitors. These concentrations must be less then MIC (minimal inhibitory concentracion). The model can be applied to the analysis of bacterial inhibition, as well as the simultaneous application of three or more inhibitors. Kocić-Tanackov et al./Oper. Res. Eng. Sci. Theor. Appl. 3 (1) (2020) 1-15 12 References Badea, G., Bors, A.G., Lacatusu, I., Oprea, O., Ungureanu, C., Stan, R.Email Author, Meghea, A. (2015) Influence of basil oil extract on the antioxidant and antifungal activities of nanostructured carriers loaded with nystatin, Comptes Rendus Chimie, 18, 668-677 Benkeblia, N. (2004). Antimicrobial activity of essential oil extracts of various onions (Allium cepa) and garlic (Allium sativum). Lebensmittel-Wissenschaft und Technologie, 37, 263-268. Boswell G., P., 2008. Modelling mycelial networks in structured environments, Mycological research, 112, 1015 – 25. Boyraz, N., Özcan M. (2006). Inhibition of phytopathogenic fungi by essential oil, hydrosol, ground material and extract of summer savory (Satureja hortensis L.) growing wild in Turkey, International Journal of Food Microbiology, Volume 107, Issue 3, 238-242. Brulé, C., Frey-Klett, P., Pierrat, J., C., Courrier, S., Gérard, F., Lemoine, M., C., Rousselet, J., L., Sommer, G., Garbaye, J. (2001). Survival in the soil of the ectomycorrhizal fungus Laccaria bicolor and the effects of a mycorrhiza helper Pseudomonas fluorescens, Soil Biology and Biochemistry, Volume 33, Issues 12-13, 1683-1694. Chertov, O., Gorbushina, A., Deventer, B. (2004). A model for microcolonial fungi growth on rock surfaces, Ecological Modelling, Volume 177, Issues 3-4, 415-426. Collopy-Junior, I., Esteves, F., F., Nimrichter, L., Rodrigues, M., L., Alviano, C., S., Meyer-Fernandes, J., R. (2006). An ectophosphatase activity in Cryptococcus neoformans, FEMS Yeast Research, Volume 6, Issue 7, 1010-1017. Damare, S., R., Nagarajan, M., Raghukumar, C. (2008). Spore germination of fungi belonging to Aspergillus species under deep-sea conditions, Deep Sea Research Part I: Oceanographic Research Papers, 55, 670-678. Damare, S., Raghukumar, C., Muraleedharan, U., D., Raghukumar, S. (2006). Deep-sea fungi as a source of alkaline and cold-tolerant proteases, Enzyme and Microbial Technology, Volume 39, Issue 2, 172-181. Diéguez-Uribeondo, J., Gierz, G., Bartnicki-Garcıá, S. (2004). Image analysis of hyphal morphogenesis in Saprolegniaceae (Oomycetes), Fungal Genetics and Biology, 41, 293-307. Elsherbiny, E.A., Safwat, N.A, Elaasser, M.M. (2017). Fungitoxicity of organic extracts of Ocimum basilicum on growth and morphogenesis of Bipolaris species (teleomorph Cochliobolus), Journal of Applied Microbiology, 123, 841-852. Fazlollahtabar, H., Gholizadeh, H. (2019a). Application of queuing theory in quality control of multi-stage flexible flow shop. Yugoslav Journal of Operations Research. Fazlollahtabar, H., Gholizadeh, H. (2019b). Economic Analysis of the M/M/1/N Queuing System Cost Model in a Vague Environment. International Journal of Fuzzy Logic and Intelligent Systems, 19(3), 192-203. Fung, W., Zheng, X. (2007). Essential oils to control Alternaria alternata in vitro and vivo. Food Control, 18, 1126-1130. mailto:rl_stan2000@yahoo.com Synergy effects of natural fungal inhibitors calculated by queuing model 13 Guynot, E., M., Ramos, J. A., Seto, L., Purroy, V., Sanchis V., Marin, S. (2003). Antifungal activity of volatile compounds generated by essential oils against fungi commonly causing deterioration of bakery products. Journal of Applied Microbiology, 94, 893-899. Ishii, N., Inoue, Y., Shimada, K., Tezuka, Y., Mitomo, H., Kasuya, K. (2007). Fungal degradation of poly(ethylene succinate), Polymer Degradation and Stability, Volume 92, Issue 1, 44-52. Ishii, N., Inoue, Y., Tagaya, T., Mitomo, H., Nagai, D., Kasuya, K. (2008). Isolation and characterization of poly(butylene succinate)-degrading fungi, Polymer Degradation and Stability, 93, 883-888. Judith, K., Pollack, J., K., Li, Z., J., Marten, M., R. (2008). Fungal mycelia show lag time before re-growth on endogenous carbon, Biotechnology and Bioengineering, 100, 458-465. Kampichler, C., Rolschewski, J., Donnelly, D., P., Boddy, L. (2004). Collembolan grazing affects the growth strategy of the cord-forming fungus Hypholoma fasciculare. Soil Biology and Biochemistry, 36, 591-599. Kang, H-C., Park, Y-H., Go S-J. (2003). Growth inhibition of a phytopathogenic fungus, Colletotrichum species by acetic acid, Microbiological Research, 158, 321-326. Kim, M-N., Lee, A-R., Yoon, J-S., Chin, I-J. (2000). Biodegradation of poly(3-hydroxybutyrate), Sky-Green® and Mater-Bi® by fungi isolated from soils, European Polymer Journal, 36, 1677-1685. Larralde-Corona, C., P., López-Isunza, F., Viniegra-González, G. (1997). Morphometric evaluation of the specific growth rate of Aspergillus niger grown in agar plates at high glucose levels, Biotechnology and Bioengineering, 56, 287-294. Llana-Ruiz-Cabello, Pichardo, S., Bermúdez, J.M., Baños, A, Núñez, C, Guillamón, E., Aucejo, S, Cameán, A.M, (2016). Development of PLA films containing oregano essential oil (Origanum vulgare L. virens) intended for use in food packaging, Food Additives and Contaminants - Part A Chemistry, Analysis, Control, Exposure and Risk Assessment, 33, 1374-1386. Lopez-Malo, A., Barreto-Valdivieso, J., Palou, E.,  San-Martyn F. (2007). Aspergillus flavus growth response to cinnamon extract and sodium benzoate mixtures, Food Control, 18, 1358–62. Maity, J., P., Chakraborty, A., Chanda, S., Santra S., C. (2008). Effect of gamma radiation on growth and survival of common seed-borne fungi in India, Radiation Physics and Chemistry, 77, 907-912 Milne, E., M., G. (2008). The natural distribution of survival, Journal of Theoretical Biology, 255, 223-236 Onofri, S., Selbmann, L., de Hoog, G., S., Grube, M., Barreca, D., Ruisi, S., Zucconi, L. (2007). Evolution and adaptation of fungi at boundaries of life, Advances in Space Research, Volume 40, Issue 11, 1657-1664 Pereira, C.M., Chlfoun, M.S., Pimenta, J.C., Angelico, L. C., Maciel, P., W. (2006). Spices, fungi mycelial develompent and ochratoxin A production. Scientific Research and Essay, 1, 038-042. Kocić-Tanackov et al./Oper. Res. Eng. Sci. Theor. Appl. 3 (1) (2020) 1-15 14 Ramirez, M., L., Chulze, S., N., Magan., N. (2004). Impact of Osmotic and Matric Water Stress on Germination, Growth, Mycelial Water Potentials and Endogenous Accumulation of Sugars and Sugar Alcohols in Fusarium graminearum, Mycologia, 96, 470-478 Reddy, N., R., K., Reddy, S., C., Muralidharan, K. (2009). Potential of botanical and biocontrol agents on growth and aflatoxin production by Aspergillus flavus infecting rice grains. Food Control, 20, 173-178. Roller, S., Covill, N. (1999). The antifungal properties of chitosan in laboratory media and apple juice, International Journal of Food Microbiology, 47, 67-77. Ruiz-Aguilar, G., M., L., Fernández-Sánchez, J., M., Rodríguez-Vázquez, R., Poggi-Varaldo, H. (2002). Degradation by white-rot fungi of high concentrations of PCB extracted from a contaminated soil, Advances in Environmental Research, 6, 4, 559-568. Scheuring, I., Szathmáry, E. (2001). Survival of Replicators with Parabolic Growth Tendency and Exponential Decay, Journal of Theoretical Biology, 212, 99-105 Schober, G., Trösch, W. (2000). Degradation of digestion residues by lignolytic fungi, Water Research, 34, 3424-3430. Sheng-Hsien L., Ku-Shang C., Min-Sheng S., Yung-Sheng H., Hung-Der J. (2007). Effects of some Chinese medicinal plant extracts on five different fungi. Food Control, 18, 1547–1554. Soylu S., Yigitbas, H., Soylu, E., M., Kurt, Ş. (2007). Antifungal effects of essential oils from oregano and fennel on Sclerotinia sclerotiorum, Journal of Applied Microbiology, 103, 1021-1030. Suhr, K., I., Nielsen, P., V. (2003). Antifungal activity of essential oils evaluated by two different application techniques against rye bread spoilage fungi, Journal of Applied Microbiology, 94, 665-674. Tanackov, I., Dragić, D., Sremac, S., Bogdanović, V., Matić, B., Milojević, M. (2019a). New Analytic Solutions of Queueing System for Shared–Short Lanes at Unsignalized Intersections, Symmetry, MDPI, 11(1), 55. Tanackov, I., Prentskovskis, O., Jevtić, Ž., Stojić, G., Ercegovac, P. (2019a). A New Method for Markovian Adaptation of the Non-Markovian Queueing System Using the Hidden Markov Model, Algorithms, 12, 133. Tanackov, S., Dimić, G., Mojović, Lj., Pejin, J. (2014). Tanackov, I., Effect of caraway, basil, and oregano extracts and their binary mixtures on fungi in growth medium and on shredded cabbage, LWT Food Science and Technology, 59, 426-432 Tanaka, H., Koike, K., Itakura, S., Enoki, A. (2009). Degradation of wood and enzyme production by Ceriporiopsis subvermispora, Enzyme and Microbial Technology, 45, 5, 384-390 Tancinová, D., Mašková, Z., Foltinová, D., Štefániková, J., Árvay, J, (2018). Effect of essential oils of Lamiaceae Plants On The Rhizopus Spp, Potravinarstvo Slovak Journal of Food Sciences 12, 491-498. Tancinová, D., Medo, J., Mašková, Z., Foltinová, D., Árvay, J. (2019) Effect of essential oils of Lamiaceae plants on the Penicillium commune, Journal of Microbiology, Biotechnology and Food Sciences 8, 1111-1117. https://www.scopus.com/sourceid/21100823448?origin=recordpage https://www.scopus.com/sourceid/21100823448?origin=recordpage Synergy effects of natural fungal inhibitors calculated by queuing model 15 Tang, M., Sheng, M., Chen, H., Zhang, F., F., 2009, In vitro salinity resistance of three ectomycorrhizal fungi, Soil Biology and Biochemistry, 41, 5, 948-953 Taniwaki, M., H., Hocking, A., D., Pitt, J., I., Fleet G., H. (2009). Growth and mycotoxin production by food spoilage fungi under high carbon dioxide and low oxygen atmospheres, International Journal of Food Microbiology, 132, 100-108. Tatsadjieu, L., N., Dongmo, P., M., J., Ngassoum, B., M., Etoa, F., X., Mbofung, F., M., C. (2009). Investigations on the essential oil of Lippia rugosa from Cameroon for its potential use as antifungal agent against Aspergillus flavus Link ex. Fries. Food Control, 20, 161-166. Tullio, V., Nostro, A., Mandras, N., Dugo, P., Banche, G., Cannatelli, M., A., Cuffini, A., M., Alonzo, V., Carlone, N., A. (2007). Antifungal activity of essential oils against filamentous fungi determined by broth microdilution and vapour contact methods, Journal of Applied Microbiology, 102, 1544-1550. Tzortzakis, N., G. (2009). Impact of cinnamon oil-enrichment on microbial spoilage of fresh produce. Innovative Food Science & Emerging Technologies, 10, 7-102. Tzortzakis, N., G., Economakis, C., D. (2007). Antifungal activity of lemongrass (Cympopogon citratus L.) essential oil against key postharvest pathogens, Innovative Food Science & Emerging Technologies, 8, 253-258. Vargas-Arispuro, I., Reyes-Báez, R., Rivera-Castañeda, G., Martínez-Téllez, M., A., Rivero-Espejel, I. (2005). Antifungal lignans from the creosotebush (Larrea tridentata), Industrial Crops and Products, Volume 22, 101-107. Vilela, G., R., de Almeida, G., S., D'Arce, M., A., B., R., Moraes, M., H., D., Brito, J., O., da Silva, M., F., G., F., Silva, S., C., de Stefano Piedade, S., M., Calori -Domingues, M., A., da Gloria, E., M. (2009). Activity of essential oil and its major compound, 1,8-cineole, from Eucalyptus globulus Labill., against the storage fungi Aspergillus flavus Link and Aspergillus parasiticus Speare, Journal of Stored Products Research, 45, 108-111. Viuda-Martos, M., Ruiz-Navajas, Y., Fernández-López, J., Pérez-Álvarez, J. (2007). Antifungal activities of thyme, clove and origano essential oils. Journal of Food Safety 27, 91-101. Viuda-Martos, M., Ruiz-Navajas, Y., Fernández-López, J., Pérez-Álvarez, J. (2008). Antifungal activity of lemon (Citrus lemon L.), mandarin (Citrus reticulata L.), grapefruit (Citrus paradisi L.) and orange (Citrus sinensis L.) essential oils. Food Control, 19, 1130-8. Wagner, S., C., Skipper, H., D., Walley., F., Bridges, Jr. W., B. (2001). Long -Term Survival of Glomus claroideum Propagules from Soil Pot Cultures under Simulated Conditions, Mycologia, 93, 815-820 Wakaizumi, M,. Yamamoto, H., Fujimoto, N., Ozeki, K. (2009). Acrylamide degradation by filamentous fungi used in food and beverage industries, Journal of Bioscience and Bioengineering, 108, 391-393. © 2020 by the authors. Submitted for possible open access publication under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).