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This work is carried out to model jointly time delay and unknown disturbances in batch fermentation process of cell proliferation C. acetobutylicum P262. The proposed model is formulated as delay logistic model (DLM), stochastic logistic model (SLM) and stochastic delay logistic model (SDLM). The strong solution of the resulting models is numerically simulated by using numerical method of Euler-Maruyama and their respective root mean square errors (RMSE) were calculated to assess the prediction quality of the model.
Keywords: Stochastic Differential Equation (SDE), Delay Differential Equation (DDE), Stochastic Delay Differential Equations (SDDE) and Euler-Maruyama.
Modelling of physical phenomena and biological system by using ordinary differential equations (ODEs) and stochastic differential equations (SDEs) has become an intensive research over last few years. In both types of equations the unknown function and its derivatives are evaluated at the same instant time, t. However, many of the natural phenomena do not have an immediate effect from the moment of their occurrence. The growth of a microbe for example, is noninstantaneous but responds only after some time lag. In general, we can see many systems in almost any area of science, for which the principle of causality, i.e. the future state of a system is independent of the past states and is determined solely by the present, does not apply. A crucial point with delay equations is the dynamics of the systems differ dramatically if the corresponding characteristic equations involve time delay. Therefore, ODEs and SDEs which are simply dependent on the present state are insufficient to explain the process that need for the incorporation of time delay. This phenomenon which involves the incorporation of time delay and stochastic effect can be modelled by SDDEs.
In fermentation process we can see that, time delay occur when the biological self-regulatory reaction is not instantaneous, but responds only after some time lag . During this phase microbial are adapting themselves to the new environment, thus no growth occur. At the end of the lag phase, the population of microorganisms is well-adjusted to new environment, cells multiply rapidly and cell mass (number of living cells) doubles regularly with time. This period is called an exponential phase. Cell population grows in transient state. Each transient state may be described by different types of kinetics models. As time evolves, the system is subjected to an intrinsic variability of the competing within species and deviations from exponential growth arise. It happens as a result of the nutrient level and toxin concentration achieves a value which can no longer support the maximum growth rate. These disturbances may be regarded as stochastic effects. Bear in mind all the phases involve in batch fermentation there is a need to model this system by using SDDE. However, most of the SDDE do not have analytical solution and numerical method provides tool in handling this problem.
Thus, this study is carried out to propose a stochastic delay logistic model (SDLM) in order to describe the cell growth of C. acetobutylicum P262 in batch fermentation. The prediction quality of the model is assessed by comparing the numerical simulation with the experimental data. The outline of this study is; in Section 2, the description of delay logistic model (DLM) and stochastic logistic model (SLM) used to describe the cell growth of C. acetobutylicum P262 is given. We introduced SDLM for C. acetobutylicum P262 proliferation in Section 3, followed by the description of numerical method which is Euler Maruyama adopted to simulate the strong solution of underlying model is given in Section 4. Then in Section 5, we numerically simulate the strong solution of DLM, SLM and SDLM via EM in order to assess the validity of the model, whose prediction has been compared with the experimental data.
Mathematical Model of C. acetobutylicum P262 Proliferation.
In a typical batch process the number of living cells varies with time. After a lag phase, where no increase in cell numbers evident, a period of rapid growth ensues during which cell numbers increase exponentially with time. The logistic equations which had been used over decades to model the cell grow have several weaknesses. They show no lag phase and give us no insight into the variables which influence the growth (Bailey, 1986). Besides, this biological system in fact is subjected to randomness and it is necessitate considering stochastic modelling instead of deterministic counterpart. Therefore, the mathematical models which explain this process should take into account both of the uncertainty and gestation period to model this process. In this study we shall consider both of these paths and take into account stochastic delay differential equation, SDDE to model the growth of C. acetobutylicum P262 of batch process. The differential equation governing is
where is the population at time t of an isolated colony of microbial and is the growth rate. We immediately recognize this as growth rate characteristic of exponential growth. Recall that the general solution of differential equation (3.74) is of the form
where is the microbial population at time that we first consider it. The sign of k represents either population will grow without limit or it will become extinct. It easy to see that if , the population will become extinct and for , the population will grow unboundedly. It is obvious that no population grows without bounds, so there is a need to modify differential equation (3.75) somewhere to predict the fact that many population in particular microbial populations have a so-called limiting population determined by the carrying capacity of their environment. More realistic model is obtained if we admit that the growth rate coefficient will not be constant but will diminish as grows, because of overcrowding of population and limitation of essential nutrient. It was highlighted in Bailey (1986) that; to model limiting population, Verlhust (1844) and Pearl & Red (1920) contributed to a theory which included an inhibiting factor to population growth. They introduce new term into population model (3.75) assuming that inhibition is proportional to . This lead to the ordinary differential equations
where contribute to the coefficient of overcrowding. The negative sign in second term of equation (3.76) decreases the population of microbial. Differential equation (3.76) is called a logistic model. If the differential equation (3.76) has a solution does not change over time, let say for some constant, then we said that the system reach an equilibrium population level. To find equilibria solution to differential equations we need to find the values of the population which . Let introduce the equilibrium population level P and , hence we have (3.76) in the form of
Now suppose that the biological self-regulatory reaction represented by the factor in (3.77) is not instantaneous but responds only after some time lag . Then instead of (3.77), we have delay differential equation of
where the end point of . In ODEs we must have some sort of experimental observations to tell us the initial value . This contradict with DDEs, that is an initial function, define on interval with length is indispensible. The equation (3.78) had been studied extensively by Hutchinson (1948). Wright (1945) considered (3.78) in a modification form. In deterministic setting it is well defined that the original equation is transformed to a simplified version of DDE without loss it generality before the behaviour of the model being analyzed (Hairer et al. (1992), Kolmanovskii & Myshkis. (1992) and Kuang (1993)). This method required us to identify a suitable transformation function so that for any delay can be reduced to by a coordinate change. Simplified form of DDE being obtained by transforming the state variable of
Introducing new function, (Hairer et al. (1992)). This gives the equation
Introducing new function, (Wright (1961), Kuang (1993), Kolmanovskii & Myshkis (1992), Driver, (1977)), obtaining the equation
Madihah (2002) considered a deterministic equation (3.77) to describe the cell growth of batch fermentation. To have physical meaning, we let and where is defined as maximum cell concentration and is a maximum specific growth. Let us consider an equation (3.77) that subject to random perturbations caused by the random changes in the system. By allowed stochastic perturbations to equation (3.77) which is
where , the stochastic differential equation with multiplicative noise takes place and has the form of
The term is frequently taken as an increment of Brownian motion which had been treated former in the previous section.
Stochastic Delay Logistic Model
A more sophisticated insight into physical phenomena may be achieved if we consider problems with both time-lag and assume that the observed biological system operate in noisy environment. The following figure depicted the flow of the process and it showed that two types of SDDE will be obtained at the end of the process. The transformation of original DDE to a simplified DDE due to Hairer et al. (1992) will be contributed to an SDDE without noise in diffusion.
An autonomous SDDE which explain the behaviour of the so-called system therefore will take the form of (4.40)
However, the difficulty arise when delay term appear in both drift and diffusion coefficients. The presence of delay in diffusion may contribute to the complexity in approximating of multiple stochastic integral with time delay. A random perturbation through DDE after transformation is necessary in order to attain an SDDE without time delays in diffusion. The presence of delay in diffusion may contribute to the complexity of approximation multiple stochastic integral with time delay and subsequently higher order numerical method turn out to be complicated to develop. Thus, in this research we attempt to model SDDE without delay in diffusion function as well as to show that an original SDDE can be solved and analyzed by using SDDE with transformation.
Let we have delay logistic equation (4.38) in the following form
Introducing new function, (Hairer et. al (1992)), we have
where the perturbation through parameter , we obtained the following SDDE (Remark )
Estimation Parameter of SDLM
Figure 4.2: Modelling using SDDE
A simplified batch fermentation kinetic model for cell growth of C. acetobutylicum P262, substrate consumption and production of solvents (ABE) are given below