Global stability of models of humoral immunity against multiple viral strains

We analyse, from a mathematical point of view, the global stability of equilibria for models describing the interaction between infectious agents and humoral immunity. We consider the models that contain the variables of pathogens explicitly. The first model considers the situation where only a single strain exists. For the single strain model, the disease steady state is globally asymptotically stable if the basic reproductive ratio is greater than one. The other models consider the situations where multiple strains exist. For the multi-strain models, the disease steady state is globally asymptotically stable. In the model that does not explicitly contain an immune variable, only one strain with the maximum basic reproductive ratio can survive at the steady state. However, in our models explicitly involving the immune system, multiple strains coexist at the steady state.     

Multi-objective optimization in Aspergillus niger fermentation for selective product enhancement

A multi-objective optimization formulation that reflects the multi-substrate optimization in a multi-product fermentation is proposed in this work. This formulation includes the application of ε-constraint to generate the trade-off solution for the enhancement of one selective product in a multi-product fermentation, with simultaneous minimization of the other product within a threshold limit. The formulation has been applied to the fed-batch fermentation of Aspergillus niger that produces a number of enzymes during the course of fermentation, and of these, catalase and protease enzyme expression have been chosen as the enzymes of interest. Also, this proposed formulation has been applied in the environment of three control variables, i.e. the feed rates of sucrose, nitrogen source and oxygen and a set of trade-off solutions have been generated to develop the pareto-optimal curve. We have developed and experimentally evaluated the optimal control profiles for multiple substrate feed additions in the fed-batch fermentation of A. niger to maximize catalase expression along with protease expression within a threshold limit and vice versa. An increase of about 70% final catalase and 31% final protease compared to conventional fed-batch cultivation were obtained. Novel methods of oxygen supply through liquid-phase H₂O₂ addition have been used with a view to overcome limitations of aeration due to high gas–liquid transport resistance. The multi-objective optimization problem involved linearly appearing control variables and the decision space is constrained by state and end point constraints. The proposed multi-objective optimization is solved by differential evolution algorithm, a relatively superior population-based stochastic optimization strategy.     

An HBV model with diffusion and time delay

In this paper, a hepatitis B virus (HBV) model with spatial diffusion and saturation response of the infection rate is investigated, in which the intracellular incubation period is modelled by a discrete time delay. By analyzing the corresponding characteristic equations, the local stability of an infected steady state and an uninfected steady state is discussed. By comparison arguments, it is proved that if the basic reproductive number is less than unity, the uninfected steady state is globally asymptotically stable. If the basic reproductive number is greater than unity, by successively modifying the coupled lower-upper solution pairs, sufficient conditions are obtained for the global stability of the infected steady state. Numerical simulations are carried out to illustrate the main results.     

Mathematical analysis of an age-structured population model with space-limited recruitment

In this paper, we investigate structured population model of marine invertebrate whose life stage is composed of sessile adults and pelagic larvae, such as barnacles contained in a local habitat. First we formulate the basic model as an Cauchy problem on a Banach space to discuss the existence and uniqueness of non-negative solution. Next we define the basic reproduction number R0 to formulate the invasion condition under which the larvae can successfully settle down in the completely vacant habitat. Subsequently we examine existence and stability of steady states. We show that the trivial steady state is globally asymptotically stable if R0 < or = 1, whereas it is unstable if R0 > 1. Furthermore, we show that a positive (non-trivial) steady state uniquely exists if R0 > 1 and it is locally asymptotically stable as far as absolute value of R0 - 1 is small enough.     

Retrospective optimization of time-dependent fermentation control strategies using time-independent historical data

We have previously shown the usefulness of historical data for fermentation process optimization. The methodology developed includes identification of important process inputs, training of an artificial neural network (ANN) process model, and ultimately use of the ANN model with a genetic algorithm to find the optimal values of each critical process input. However, this approach ignores the time-dependent nature of the system, and therefore, does not fully utilize the available information within a database. In this work, we propose a method for incorporating time-dependent optimization into our previously developed three-step optimization routine. This is achieved by an additional step that uses a fermentation model (consisting of coupled ordinary differential equations (ODE)) to interpret important time-course features of the collected data through adjustments in model parameters. Important process variables not explicitly included in the model were then identified for each model parameter using automatic relevance determination (ARD) with Gaussian process (GP) models. The developed GP models were then combined with the fermentation model to form a hybrid neural network model that predicted the time-course activity of the cell and protein concentrations of novel fermentation conditions. A hybrid-genetic algorithm was then used in conjunction with the hybrid model to suggest optimal time-dependent control strategies. The presented method was implemented upon an E. coli fermentation database generated in our laboratory. Optimization of two different criteria (final protein yield and a simplified economic criteria) was attempted. While the overall protein yield was not increased using this methodology, we were successful in increasing a simplified economic criterion by 15% compared to what had been previously observed. These process conditions included using 35% less arabinose (the inducer) and 33% less typtone in the media and reducing the time required to reach the maximum protein concentration by 10% while producing approximately the same level of protein as the previous optimum.     

Global dynamics of an epidemiological model with age of infection and disease relapse

In this paper, an epidemiological model with age of infection and disease relapse is investigated. The basic reproduction number for the model is identified, and it is shown to be a sharp threshold to completely determine the global dynamics of the model. By analysing the corresponding characteristic equations, the local stability of a disease-free steady state and an endemic steady state of the model is established. By means of suitable Lyapunov functionals and LaSalle's invariance principle, it is verified that if the basic reproduction number is less than unity, the disease-free steady state is globally asymptotically stable, and hence the disease dies out; if the basic reproduction number is greater than unity, the endemic steady state is globally asymptotically stable and the disease becomes endemic.     

Oscillatory viral dynamics in a delayed HIV pathogenesis model

We consider an HIV pathogenesis model incorporating antiretroviral therapy and HIV replication time. We investigate the existence and stability of equilibria, as well as Hopf bifurcations to sustained oscillations when drug efficacy is less than 100%. We derive sufficient conditions for the global asymptotic stability of the uninfected steady state. We show that time delay has no effect on the local asymptotic stability of the uninfected steady state, but can destabilize the infected steady state, leading to a Hopf bifurcation to periodic solutions in the realistic parameter ranges.     

An Algorithm for operating a fed-batch fermentation using quasi-steady state

The determination of an optimum feeding profile of a fed-batch fermentation requires the solution of a singular optimum control problem, which is often complicated by changes in the process kinetics during the fermentation. The procedure of optimization may be sufficiently simple, if the feeding part of fermentation is carried out in the quasi-steady state. In this work an algorithm for operating a fed-batch fermentation using mentioned regime is offered. The algorithm supposes a periodical correction of the feeding strategy. Applying to fed-batch lysine fermentation demonstrate efficacy of this algorithm over frequently used strategies.     

Fed-batch optimization of alpha-amylase and protease-producing Bacillus subtilis using Markov chain methods

A stoichiometry-based model for the fed-batch culture of the recombinant bacterium Bacillus subtilis ATCC 6051a, producing extracellular alpha-amylase as a desirable product and proteases as undesirable products, was developed and verified. The model was then used for optimizing the feeding schedule in fed-batch culture. To handle higher-order model equations (14 state variables), an optimization methodology for the dual-enzyme system is proposed by integrating Pontryagin's optimum principle with fermentation measurements. Markov chain Monte Carlo (MCMC) procedures were appropriate for model parameter and decision variable estimation by using a priori parameter distributions reflecting the experimental results. Using a simplified Metropolis-Hastings algorithm, the specific productivity of alpha-amylase was maximized and the optimum path was confirmed by experimentation. The optimization process predicted a further 14% improvement of alpha-amylase productivity that could not be realized because of the onset of sporulation. Among the decision variables, the switching time from batch to fed-batch operation (t(s)) was the most sensitive decision variable.     

Travelling waves of a delayed SIR epidemic model with nonlinear incidence rate and spatial diffusion

This paper is concerned with the existence of travelling waves to a SIR epidemic model with nonlinear incidence rate, spatial diffusion and time delay. By analyzing the corresponding characteristic equations, the local stability of a disease-free steady state and an endemic steady state to this system under homogeneous Neumann boundary conditions is discussed. By using the cross iteration method and the Schauder's fixed point theorem, we reduce the existence of travelling waves to the existence of a pair of upper-lower solutions. By constructing a pair of upper-lower solutions, we derive the existence of a travelling wave connecting the disease-free steady state and the endemic steady state. Numerical simulations are carried out to illustrate the main results.     

An age-structured within-host HIV-1 infection model with virus-to-cell and cell-to-cell transmissions

In this paper, a within-host HIV-1 infection model with virus-to-cell and direct cell-to-cell transmission and explicit age-since-infection structure for infected cells is investigated. It is shown that the model demonstrates a global threshold dynamics, fully described by the basic reproduction number. By analysing the corresponding characteristic equations, the local stability of an infection-free steady state and a chronic-infection steady state of the model is established. By using the persistence theory in infinite dimensional system, the uniform persistence of the system is established when the basic reproduction number is greater than unity. By means of suitable Lyapunov functionals and LaSalle's invariance principle, it is shown that if the basic reproduction number is less than unity, the infection-free steady state is globally asymptotically stable; if the basic reproduction number is greater than unity, the chronic-infection steady state is globally asymptotically stable. Numerical simulations are carried out to illustrate the feasibility of the theoretical results.     

Senescence and antibiotic resistance in an age-structured population model

Different theories have been proposed to understand the growing problem of antibiotic resistance of microbial populations. Here we investigate a model that is based on the hypothesis that senescence is a possible explanation for the existence of so-called persister cells which are resistant to antibiotic treatment. We study a chemostat model with a microbial population which is age-structured and show that if the growth rates of cells in different age classes are sufficiently close to a scalar multiple of a common growth rate, then the population will globally stabilize at a coexistence steady state. This steady state persists under an antibiotic treatment if the level of antibiotics is below a certain threshold; if the level exceeds this threshold, the washout state becomes a globally attracting equilibrium.     

Dynamics of a delay differential equation model of hepatitis B virus infection

We formulate and systematically study the global dynamics of a simple model of hepatitis B virus in terms of delay differential equations. This model has two important and novel features compared to the well-known basic virus model in the literature. Specifically, it makes use of the more realistic standard incidence function and explicitly incorporates a time delay in virus production. As a result, the infection reproduction number is no longer dependent on the patient liver size (number of initial healthy liver cells). For this model, the existence and the component values of the endemic steady state are explicitly dependent on the time delay. In certain biologically interesting limiting scenarios, a globally attractive endemic equilibrium can exist regardless of the time delay length.     

Dynamic analysis of lactic acid fermentation in membrane bioreactor

In this paper, a mathematical model for the lactic acid fermentation in membrane bioreactor is investigated. This novel theoretical framework could result in an objective criterion on how to control the substrate concentration in order to keep a sustainable and steady output of lactic acid. Firstly, continuous input substrate is taken. The existence and local stability of two equilibria are studied. According to Poincaré-Bendixson Theorem, we obtain the conditions for the globally asymptotical stability of the equilibrium. Secondly, impulsive input substrate is also considered. Using Floquet's theorem and small-amplitude perturbation, we obtain the biomass-free periodic solution is locally stable if some conditions are satisfied. In a certain limiting case, it is shown that a nontrivial periodic solution emerges via a supercritical bifurcation. Finally, our findings are confirmed by means of numerical simulations.     

Productivity improvement in xanthan gum fermentation using multiple substrate optimization

A novel and more comprehensive formulation of the optimal control problem that reflects the operational requirements of a typical industrial fermentation has been proposed in this work. This formulation has been applied to a fed-batch bioreactor with three control variables, i.e., feed rates of carbon source, nitrogen source, and an oxygen source, to result in a 148.7% increase in product formation. Xanthan gum production using Xanthomonas campestris has been used as the model system for this optimization study, and the liquid-phase oxygen supply strategy has been used to supply oxygen to the fermentation. The formulated optimization problem has several constraints associated with it due to the nature of the system. A robust stochastic technique, differential evolution, has been used to solve this challenging optimization problem. The infinite dimensional optimization problem has been approximated to a finite dimensional one by control vector parametrization. The state constraints that are path constraints have been addressed by using penalty functions and by integrating them over the total duration to ensure a feasible solution. End point constraints on final working volume of the reactor and on the final residual concentrations of carbon and nitrogen sources have been included in the problem formulation. Further, the toxicity of the oxygen source, H(2)O(2), has been addressed by imposing a constraint on its maximum usable concentration. In addition, the initial volume of the bioreactor contents and feed concentrations have been handled as decision variables, which has enabled a well-grounded choice for their values from the optimization procedure; adhoc values are normally used in the industry. All results obtained by simulation have been validated experimentally with good agreements between experimental and simulated values.     

Sequential Decision Rules for Managing Nematodes with Crop Rotations

A dynamic model of nematode populations under a crop rotation that includes both host and nonhost crops is developed and used to conceptualize the problem of economic control. The steady state of the dynamic system is used to devise an approximately optimal decision policy, which is then applied to cyst nematode (Heterodera schachtii) control in a rotation of sugarbeet with nonhost crops. Long-run economic returns from this approximately optimal decision rule are compared with results from solution of the exact dynamic optimization model. The simple decision rule based on the steady state provides long-run average returns that are similar to the fully optimal solution. For sugarbeet and H. schachtii, the simplified rule can be calculated by maximizing a relatively simple algebraic expression with respect to the number of years in the sequence of nonhost crops. Maximization is easy because only integers are of interest and the number of years in nonhost crops is typically small. Solution of this problem indirectly yields an approximation to the optimal dynamic economic threshold density of nematodes in the soil. The decision rule requires knowledge of annual nematode population change under host and nonhost crops, and the relationship between crop yield and nematode population density.     

Fractional optimal control dynamics of coronavirus model with Mittag–Leffler law

Most countries around the world are battling to limit the spread of severe acute respiratory syndrome-coronavirus 2 (SARS-CoV-2). As the world strives to get an effective medication to control the disease, appropriate control measures for now remains one of the effective measures to reduce the spread of the disease. In this study, a fractional optimal control model is formulated in Atangana-Baleanu-Caputo derivative sense. The reproduction number and steady state of disease free of the Coronavirus model are examined and found to be globally stable. The existence and uniqueness of solution of the fractional Coronavirus model is established by using the Banach fixed point theorem approach. Three controls are considered in the model and Pontryagins Maximum Principle is used to establish the necessary conditions for optimal control solution. The numerical solution suggests that the best strategy is found to be the utilization of all three controls at the same time.     

Separable models in age-dependent population dynamics

A class of population models is considered in which the parameters such as fecundity, mortality and interaction coefficients are assumed to be age-dependent. Conditions for the existence, stability and global attractivity of steady-state and periodic solutions are derived. The dependence of these solutions on the maturation periods is analyzed. These results are applied to specific single and multiple population models. It is shown that periodic solutions cannot occur in a general class of single population age-dependent models. Conditions are derived that determine whether increasing the maturation period has a stabilizing effect. In specific cases, it is shown that any number of switches in stability can occur as the maturation period is increased. An example is given of predator-prey model where each one of these stability switches corresponds to a stable steady state losing its stability via a Hopf bifurcation to a periodic solution and regaining its stability upon further increase of the maturation period.     

Development of thermodynamic optimum searching (TOS) to improve the prediction accuracy of flux balance analysis

Flux balance analysis (FBA) has been widely used in calculating steady‐state flux distributions that provide important information for metabolic engineering. Several thermodynamics‐based methods, for example, quantitative assignment of reaction directionality and energy balance analysis have been developed to improve the prediction accuracy of FBA. However, these methods can only generate a thermodynamically feasible range, rather than the most thermodynamically favorable solution. We therefore developed a novel optimization method termed as thermodynamic optimum searching (TOS) to calculate the thermodynamically optimal solution, based on the second law of thermodynamics, the minimum magnitude of the Gibbs free energy change and the maximum entropy production principle (MEPP). Then, TOS was applied to five physiological conditions of Escherichia coli to evaluate its effectiveness. The resulting prediction accuracy was found significantly improved (10.7–48.5%) by comparing with the ¹³C‐fluxome data, indicating that TOS can be considered an advanced calculation and prediction tool in metabolic engineering. Biotechnol. Bioeng. 2013; 110: 914–923. © 2012 Wiley Periodicals, Inc.