First-order Nonlinear Ordinary Differential Equations Research Articles

Hamiltonian formalism for Bose excitations in a plasma with a non-Abelian interaction I: Plasmon – hard particle scattering

Hamiltonian theory for collective longitudinally polarized gluon excitations (plasmons) interacting with classical high-energy test color-charged particle propagating through a high-temperature gluon plasma is developed. A generalization of the Lie-Poisson bracket to the case of a continuous medium involving bosonic normal field variable ak⁎a and a non-Abelian color charge Qa is performed and the corresponding Hamilton equations are presented. The canonical transformations including simultaneously both bosonic degrees of freedom of the soft collective excitations and degree of freedom of hard test particle connecting with its color charge in the hot gluon plasma are written out. A complete system of the canonicity conditions for these transformations is derived. The notion of the plasmon number density Nkaa1′, which is a nontrivial matrix in the color space, is introduced. An explicit form of the effective fourth-order Hamiltonian describing the elastic scattering of a plasmon off a hard color particle is found and the self-consistent system of Boltzmann-type kinetic equations taking into account the time evolution of the mean value of the color charge of the hard particle is obtained. On the basis of these equations, a model problem of the interaction of two infinitely narrow wave packets is considered. A system of nonlinear first-order ordinary differential equations defining the dynamics of the interaction of the colorless Nkl and color Wkl components of the plasmon number density is derived. The problem of determining the third- and fourth-order coefficient functions entering into the canonical transformations of the original bosonic variable ak⁎a and color charge Qa is discussed. With the help of the coefficient functions obtained, a complete effective amplitude of the elastic scattering of plasmon off hard test particle is written out.

Influence of Raman gain on dynamics of spatiotemporal chaos in optical ring microresonators

We theoretically investigate on the evolution of spatiotemporal chaos under the influence of Raman gain, in an optical ring microresonator. The dynamics of the system is shown to be governed by the Lugiator–Lefever equation, with an additional real and imaginary Raman gain term. In the absence of pump field intensity and Raman gain, we observe an increase in the amplitude and frequency of the traveling optical field as the pump detuning frequency increases. However this field gradually vanishes as time increases, owing to the intra-cavity loss. By exploring an appropriate ansatz which incorporate a chirping parameter, the modified Lugiator–Lefever equation is transformed to a set of four coupled first-order nonlinear ordinary differential equations. Fixed point stability analysis strongly suggest that variation in Raman gain has the propensity of inducing the co-existence of bright and dark solitons in the anomalous dispersion regime. Results of numerical simulations underscore that the structural profiles of the optical field amplitude are greatly modified by the Raman effects. Variation in Raman gain equally leads to a bifurcation analysis of the system, in which the Lyapunov exponent diagrams reveal regions of period doubling and chaos. We finally observe that for a fixed value of the real part of Raman gain coefficient, an increase in the effects of soliton self-frequency shift generally drives the system to chaotic regime.

Theoretical investigation of the free fall of a spherical particle in a viscous fluid using continuous piecewise linearization method

This article presents a theoretical investigation of the problem of free fall of a spherical particle in a viscous fluid. The classic Boussinesq-Basset-Oseen (BBO) model for particle motion in laminar flow was modified for generalized flow by using a drag law that is applicable for 0<Re≤2.0×105. By assuming that the acceleration in the Basset force integral is constant, the Basset force effect was approximated to form an integrated added mass coefficient. Consequently, the integro-differential equation of the BBO model was transformed to a first-order nonlinear ordinary differential equation that accounts for the Basset force effect and was solved using the continuous piecewise linearization method (CPLM). The CPLM algorithm was developed based on the jerk-velocity relationship and is applicable to zero and non-zero initial conditions, steady motion, increasing or decreasing velocities and the corresponding acceleration and jerk responses. The CPLM algorithm was shown to predict published experimental results accurately and compared very well with numerical solutions and existing analytical solutions. Examination of the fall response under varying parameters showed that the fall distance, fall time and terminal velocity depend strongly on the sphere diameter, sphere density, and the density and viscosity of the fluid medium. Also, an analytical solution for the power dissipated in the fluid medium as the sphere falls to reach its terminal velocity was derived. The power dissipated was found to increase exponentially as the initial velocity deviates positively from the terminal velocity.

Assessment of Numerical Performance of Some Runge-Kutta Methods and New Iteration Method on First Order Differential Problems

This research focuses on the assessment of the numerical performance of some Runge-Kutta methods and New Iteration Method “NIM” for solving first-order differential problems. The assessment is conducted through extensive numerical experiments and comparative analyses. Accuracy, efficiency, and stability are among the key factors considered in evaluating the performance of the methods. A range of first-order differential problems with diverse characteristics and complexity levels is employed to thoroughly examine the methods' capabilities and limitations. The numerical investigation that is defined in the study as well as the results that are stated in the Tables, demonstrates that all the approaches produce extremely accurate results. However, the “NIM” was shown to be the most effective of the three methods used in this study. Conclusively, the “NIM” should be employed to solve first-order nonlinear and linear ordinary differential equations in place of Runge-Kutta Fourth order method (RK4M) and Butcher Runge-Kutta Fifth order method (BRK5M). In addition, BRK5M is more applicable and efficient than RK4M when solving first order ordinary differential problems.

New Numerical Iteration Schemes Based on Perturbation Iteration Algorithms

Perturbation–Iteration algorithms (PIA) have been developed recently to solve differential equations analytically. A continuous solution in terms of closed form functions as an approximation of the original equation can be found using the method. The method has been implemented to algebraic equations, ordinary and partial differential equations successfully. Based on the formalism developed previously, in this work, purely numerical versions of the perturbation–iteration algorithms are proposed for the first time for first-order nonlinear ordinary differential equations. The new algorithms are called as the Numerical Perturbation–Iteration Algorithms (NPIA) to distinguish them from the continuous analytical ones (PIA) and the method in general will be called the Numerical Perturbation–Iteration Method (NPIM). Iteration schemes based on one correction term in the perturbation solution and first-order derivatives in the Taylor series expansions (NPIA(1,1)), and one correction term in the perturbations and second-order derivatives in the series expansions (NPIA(1,2)) are derived. The numerical results are compared with the exact solutions and a very good match is observed. NPIA(1,2) produced slightly better results compared to NPIA(1,1) with total errors being at least [Formula: see text], [Formula: see text] being the step size for both methods. An improvement suggestion for NPIA(1,1) is proposed in the last section to eliminate unrealistic blow-up solutions which are encountered for some specific equations.

DC operating points of Mott neuristor circuits

In 1960 Hewitt Crane conceptualized neuristors as electronic logic networks that could mimic action potential generation by biological neurons. In 2012 Mott memristors were presented as nano-scale electronic devices for physically constructing neuristor networks. Both the original Mott memristor model and the simplified model presented in 2020 are stiff nonlinear first-order ordinary differential equations (ODEs) that describe the voltage controlled threshold switching and current controlled negative differential resistance, attributed to a Mott insulator-to-metal phase transition. By design, a Mott neuristor contains two identical Mott memristors plus linear resistors and capacitors (no inductors), which enables alternating threshold switching by the memristors to generate periodic (AC) output voltage spikes in response to a DC current or voltage input. This paper presents a steady-state analysis of dynamic neuristor models, including four-, five-, and six-state variables (each of which is a system of first-order ODEs involving the states) based on the original phenomenological model of the Mott memristor. Specifically, it reveals the unique stable operating points that occur for applied currents or voltages below the amounts needed to induce resistive switching. These DC operating points represent the physically meaningful “OFF” states for the circuit when the neuristor does not output voltage spikes. Jacques Hadamard's mathematical criteria for a well-posed physical model (existence, uniqueness, continuity, stability) motivated the search for these DC operating points (OFF states), which are needed for efficiently initializing neuristor models before conducting dynamic neuristor simulations. The key result of our analysis is the neuristor steady-state eq. (29) which is useful for understanding neuristor OFF states as well as the role of negative-differential resistance (NDR) in the operation of the Mott memristor model and neuristor circuits. In addition to the steady-state analysis, three numerical examples of dynamic neuristor circuit operation are also given and MATLAB code for the dynamic four-state model is available for interested readers.

Sensitivity Analysis-Based Validation of the Modified NERA Model for Improved Performance

Marijuana is an illicit narcotic with multiple negative health effects as it continues to pose a severe danger to the health of people in emerging regions. Marijuana is spread through interactions between users and non-users. This study utilizes first-order non-linear ordinary differential equations to modify the Non-user, Experimental, Recreational, and Addicted (NERA) users’ model by adding a new class known as the hospitalized class of marijuana smokers. Moreover, this paper focuses on validating the NERA model using the sensitivity analysis concept. The NERA model presented in a previous study for marijuana usage among the target population consists of four distinct stages: the first stage is known as the non-user stage and is represented by the letter N; the second stage is known as the experimental stage of marijuana smoking and is represented by the letter E; the third stage is known as the recreational stage of marijuana smoking and is represented by the letter R; the fourth stage is represented by the letter A; and this last stage is known as the addicted stage of marijuana smoking. The suggested mathematical method achieved the reproduction number () for marijuana use. The results of the sensitivity tests indicate how crucial a variety of variables are to the spread of marijuana. The parameters that have a key impact on marijuana consumption were discovered based on sensitivity analysis. All these procedures, including the reproductive number and invariant region, using the sensitivity analysis concept were utilized to validate the updated model. In this study, the NERA model is modified by utilizing first-order non-linear ordinary differential equations. Furthermore, as demonstrated by the graphical and technique sections, the proposed modified models were successfully validated for improved performance.

APPLICATION OF NON-STANDARD FINITE DIFFERENCE METHOD ON COVID-19 MATHEMATICAL MODEL WITH FEAR OF INFECTION

This study presents a novel application of Non -Standard Finite Difference (NSFD) Method to solve a COVID-19 epidemic mathematical model with the impact of fear due to infection. The mathematical model is governed by a system of first-order non-linear ordinary differential equations and is shown to possess a unique positive solution that is bounded. The proposed numerical scheme is used to obtain an approximate solution for the COVID-19 model. Graphical results were displayed to show that the solution obtained by NSFD agrees well with those obtained by the Runge-Kutta-Fehlberg method built-in Maple 18.

A Method of Qualitative Analysis for Determining Monotonic Stability Regions of Particular Solutions of Differential Equations of Dynamic Systems

Developing stability analysis methods for modern dynamical system solutions has been a significant challenge in the field. This study aims to formulate a qualitative analysis approach for the monotone stability region of a specific solution to a single differential equation within a dynamical system. The system in question comprises two first-order nonlinear ordinary differential equations of a particular kind. The method proposed hinges on applying elements of combinatorics to the traditional mathematical investigation of a function with a single independent variable. This approach enables the exact determination of the different qualitative scenarios in which the desired solution changes, under the assumption that the function values monotonically diminish from a specified value down to a finite zero. This paper outlines the creation and decomposition of the monotone stability region associated with the solution under consideration.

Real-time adaptive sparse-identification-based predictive control of nonlinear processes

This study introduces a sparse identification-based model predictive control (MPC) framework that incorporates on-line updates of the sparse-identified model to account for nonlinear dynamics and model uncertainty in process systems. The methodology involves obtaining a nonlinear first-order ordinary differential equation model using sparse identification for nonlinear dynamics (SINDy), which is integrated into two control schemes: Lyapunov-based MPC (LMPC) for achieving steady-state operation and Lyapunov-based economic MPC (LEMPC) for achieving both closed-loop stability and optimal economic performance. To improve prediction accuracy, an on-line model update scheme is proposed for the SINDy models. Specifically, an error-trigger mechanism that utilizes prediction errors and then uses the most recent process data to update the parameters of the SINDy model in real-time is designed. By incorporating the error-triggered on-line model updates in the SINDy-based LMPC and LEMPC, the dynamic performance of the process is enhanced, ensuring closed-loop stability, optimality, and smooth control actions. Following theoretical results on the boundedness of the closed-loop states and detailed discussions on the selection criteria for parameters of the error-triggered SINDy update scheme, the effectiveness of the proposed methodology is demonstrated through a chemical process example with time-varying disturbances under the LEMPC framework.

Nonlinear dynamics of Kerr optical microresonators with spatially fluctuating loss

Dissipative soliton crystals (the so-called soliton combs) form in Kerr microresonators as a result of the competition between the group-velocity dispersion and the Kerr nonlinearity on one hand, and the balance of cavity loss by an external pump on the other hand. In some physical contexts, the loss can fluctuate within the microresonator cavity, inducing a saturable-absorption process which impacts the dynamics of the optical field. In this study, dissipative soliton crystals are investigated in a Kerr optical microresonator with spatially fluctuating loss. The underlying mathematical model consists of a modified Lugiato–Lefever equation with a space-dependent loss, coupled to a rate equation for the fluctuating loss. Adopting an ansatz that describes the optical-field envelope as a complex function of real amplitude and real phase with a characteristic modulation frequency, the mathematical model is reduced to a set of first-order nonlinear ordinary differential equations which are solved numerically. Simulations suggest that when the homogeneous cavity loss is small enough, the impact of loss fluctuation on the soliton-comb profile is rather moderate. The effect of loss fluctuations becomes noticeable when the homogeneous loss is sizable, with the recovery time of the induced saturable-absorption process being reasonably long to promote a slow saturable absorption. An analysis of the influence of the detuning on the amplitude and phase of the dissipative soliton crystal, as well as on the spatial variation of the loss for a fixed value of the characteristic frequency, is taken into consideration in the study.

Internal ring waves in a three-layer fluid on a current with a constant vertical shear

Oceanic internal waves often have curvilinear fronts and propagate over vertically sheared currents. We present the first study of long weakly-nonlinear internal ring waves in a three-layer fluid in the presence of a horizontally uniform background current with a constant vertical shear. The leading order of this theory leads to the angular adjustment equation—a nonlinear first-order ordinary differential equation describing the dependence of the linear long-wave speed on its angle to the direction of the current. The compact ring waves, well studied in the absence of a current, correspond to the singular solution (envelope of the general solution) of this equation, and they can exist only under certain conditions. The constructed solutions reveal qualitative differences in the shapes of the wavefronts of the two baroclinic modes: the wavefront of the faster mode is elongated in the direction of the current, while the wavefront of the slower mode is squeezed. Moreover, depending on the vorticity strength, several different regimes have been identified. When the vertical shear is weak, part of the wavefront is able to propagate upstream, while when the shear is strong enough, the whole wavefront propagates downstream. A richer pattern of behaviour is observed for the slower mode. As the shear increases, singularities of the swallowtail-type may arise and, eventually, solutions with compact wavefronts crossing the downstream axis cease to exist. We show that the latter is related to the long-wave instability of the base flow. We obtain the cKdV-type amplitude equation and examine analytical expressions for its coefficients. Using this cKdV-type equation we numerically model the evolution of the waves for both modes. The initial stage of the evolution is in agreement with the leading-order predictions for the deformations of the wavefronts. Then, as the wavefronts expand, strong dispersive effects occur in the upstream direction. Moreover, when nonlinearity is enhanced, fission of waves is observed in the upstream part of the ring waves.

Soliton trains induced by femtosecond laser filamentations in transparent materials with saturable nonlinearity

Femtosecond laser inscriptions in optical media current offer the most reliable optical technology for processing of transparent materials, among which is the laser micromachining technology. In this process, the nonlinearity of the transparent medium can be either intrinsic or induced by multiphoton ionization processes. In this work, a generic model is proposed to describe the dynamics of femtosecond laser inscription in transparent materials characterized by a saturable nonlinearity. The model takes into account multiphoton ionization processes that can induce an electron plasma of inhomogeneous density and electron diffusions. The mathematical model is represented by a one-dimensional complex Ginzburg–Landau equation with a generalized saturable nonlinearity term in addition to the residual nonlinearity related to multiphoton ionization processes, coupled to a rate equation for time evolution of the electron plasma density. Dynamical properties of the model are investigated focusing on the nonlinear regime, where the model equations are transformed into a set of coupled first-order nonlinear ordinary differential equations, which are solved numerically with the help of a sixth-order Runge–Kutta algorithm with a fixed time step. Simulations reveal that upon propagation, spatiotemporal profiles of the optical field and of the plasma density are periodic pulse trains, the repetition rates and amplitudes of which are increased with an increase of both the multiphoton ionization order and the saturable nonlinearity. When electron diffusions are taken into account, the system dynamics remains qualitatively unchanged; however, the electron plasma density gets strongly depleted, leaving almost unchanged the amplitude of pulses composing the femtosecond laser soliton crystals.

Mechanistic Modelling of Biomass Growth, Glucose Consumption and Ethanol Production by Kluyveromyces marxianus in Batch Fermentation.

This paper presents results concerning mechanistic modeling to describe the dynamics and interactions between biomass growth, glucose consumption and ethanol production in batch culture fermentation by Kluyveromyces marxianus (K. marxianus). The mathematical model was formulated based on the biological assumptions underlying each variable and is given by a set of three coupled nonlinear first-order Ordinary Differential Equations. The model has ten parameters, and their values were fitted from the experimental data of 17 K. marxianus strains by means of a computational algorithm design in Matlab. The latter allowed us to determine that seven of these parameters share the same value among all the strains, while three parameters concerning biomass maximum growth rate, and ethanol production due to biomass and glucose had specific values for each strain. These values are presented with their corresponding standard error and 95% confidence interval. The goodness of fit of our system was evaluated both qualitatively by in silico experimentation and quantitative by means of the coefficient of determination and the Akaike Information Criterion. Results regarding the fitting capabilities were compared with the classic model given by the logistic, Pirt, and Luedeking-Piret Equations. Further, nonlinear theories were applied to investigate local and global dynamics of the system, the Localization of Compact Invariant Sets Method was applied to determine the so-called localizing domain, i.e., lower and upper bounds for each variable; whilst Lyapunov's stability theories allowed to establish sufficient conditions to ensure asymptotic stability in the nonnegative octant, i.e., R+,03. Finally, the predictive ability of our mechanistic model was explored through several numerical simulations with expected results according to microbiology literature on batch fermentation.

Analytical approaches for growth models in economics

This study deals with the group-theoretical analysis of nonlinear optimal control problems called the optimal growth model with the environmental asset and capitalist decision model of endogenous growth, which are expressed in terms of the current and present value Hamiltonian functions. The first-order conditions for optimal control problems based on Pontryagin’s maximum principle are considered. In addition, Lie point symmetries and their relation with the Prelle-Singer, λ-symmetries, adjoint symmetries, Darboux polynomials, and Jacobi’s last multiplier approaches are studied by analyzing the coupled nonlinear first-order ordinary differential equations corresponding to the first-order conditions. For both current and present Hamiltonian cases, the first integrals and the corresponding analytical (closed-form) solutions and graphical representations for the optimal control problems are represented.

Investigation of the self-propulsion of a wetting/nonwetting ganglion in tapered capillaries with arbitrary viscosity and density contrasts

The movement of a meniscus inside a capillary tube has been extensively studied in the context of displacing one fluid with another immiscible one. This phenomenon exists in many applications including pharmaceutical, oil production, filtration and separation processes, and others. When one of the phases is entrapped inside a capillary tube, it forms what is called a ganglion with two menisci between the two fluids. In a straight uniform capillary tube, a stagnant entrapped ganglion is symmetric. The situation is different if the capillary tube is tapered in which case the two menisci assume different curvatures. Such inhomogeneity of the capillary pressure self-propels the ganglion to move. The fate of the ganglion inside the tapered tube depends on whether it is wetting or nonwetting to the tube wall. That is, after the initial movement, a wetting ganglion accelerates towards the tapered end of the tube while a nonwetting one decelerates towards the wider end before reaching a terminal configuration. Such fates are linked to the variations of the capillary pressure, which continuously increases for a wetting ganglion and decreases for the nonwetting one. In this work, a generalized model is developed that not only describes capillary-driven dynamics over a wide range of viscosity and density contrasts but also pressure-driven scenarios with/without gravity. The model, however, neglects the inertial effect of the two fluids on account of the fact that it is confined to the very early time of the movement process. A first-order nonlinear ordinary differential equation is developed that describes the dynamic behavior of both the wetting and nonwetting ganglions. A fourth-order Runge-Kutta algorithm is developed to solve the model equations. Furthermore, a computational fluid dynamics (CFD) analysis was used to provide a comparison and verification framework.

ON SOLUTIONS OF NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS WITH VARIABLE COEFFICIENTS BASED ON ELASTIC TRANSFORMATION METHODS

We consider difficult problems in solving nonlinear ordinary differential equations with variable coefficients. We use elastic transformation methods (elastic upgrading transformation and elastic reduced transformation) to transform the first and third order equations into the associated Chebyshev equation. Then, according to the general solution of the associated Chebyshev equation, we obtain the general solutions of the first-order and third-order nonlinear ordinary differential equations with variable coefficients, giving curves of general solution. The elastic transformation method provides a new idea and expands the solvable classes for solving nonlinear ordinary differential equation with variable coefficients.

Well-Posedness and a Finite Difference Approximation for a Mathematical Model of HPV-Induced Cervical Cancer

We present a first-order finite difference scheme for approximating solutions of a mathematical model of cervical cancer induced by the human papillomavirus (HPV), which consists of four nonlinear partial differential equations and a nonlinear first-order ordinary differential equation. The scheme is analyzed and used to provide an existence-uniqueness result. Numerical simulations are performed in order to demonstrate the first-order rate of convergence. A sensitivity analysis was done in order to compare the effects of two drug types, those that increase the death rate of HPV-infected cells, and those that increase the death rate of the precancerous cell population. The model predicts that treatments that affect the precancerous cell population by directly increasing the corresponding death rate are far more effective than those that increase the death rate of HPV-infected cells.

The role of supporting and disruptive mechanisms of FT3 homeostasis in regulating the hypothalamic-pituitary-thyroid axis.

Thyroid hormones are controlled by the hypothalamic-pituitary-thyroid (HPT) axis through a complex network of regulatory loops, involving the hormones TRH, TSH, FT4, and FT3. The relationship between TSH and FT4 is widely used for diagnosing thyroid diseases. However, mechanisms of FT3 homeostasis are not well understood. We used mathematical modelling to further examine mechanisms that exist in the HPT axis regulation for protecting circulating FT3 levels. A mathematical model consisting of a system of four coupled first-order parameterized non-linear ordinary differential equations (ODEs) was developed, accounting for the interdependencies between the hormones in the HPT axis regulation. While TRH and TSH feed forward to the pituitary and thyroid, respectively, FT4 and FT3 feed backward to both the pituitary and hypothalamus. Stable equilibrium solutions of the ODE system express homeostasis for a particular variable, such as FT3, if this variable stays in a narrow range while certain other parameter(s) and system variable(s) may vary substantially. The model predicts that (1) TSH-feedforward protects FT3 levels if the FT4 production rate declines and (2) combined negative feedback by FT4 and FT3 on both TSH and TRH production rates keeps FT3 levels insensitive to moderate changes in FT4 production rates and FT4 levels. The optimum FT4 and FT3 feedback and TRH and TSH-feedforward ranges that preserve FT3 homeostasis were found by numerical continuation analysis. Model predictions were in close agreement with clinical studies and individual patient examples of hypothyroidism and hyperthyroidism. These findings further extend the concept of HPT axis regulation beyond TSH and FT4 to integrate the more active sister hormone FT3 and mechanisms of FT3 homeostasis. Disruption of homeostatic mechanisms leads to disease. This provides a perspective for novel testable concepts in clinical studies to therapeutically target the disruptive mechanisms.

The Transmission Dynamics of a Compartmental Epidemic Model for COVID-19 with the Asymptomatic Population via Closed-Form Solutions.

Unlike previous viral diseases, COVID-19 has an "asymptomatic" group that has no symptoms but can still spread the disease to others at the same rate as symptomatic patients who are infected. In the literature, the mass action or standard incidence rates are considered for compartmental models with asymptomatic compartment for studying the transmission dynamics of COVID-19, but the quarantined adjusted incidence rate is not. To bridge this gap, we developed a Susceptible Asymptomatic Infectious Quarantined (SAIQ) model with a Quarantine-Adjusted (QA) incidence to investigate the emergence and containment of COVID-19. COVID-19 models are investigated using various methods, but only a few studies take into account closed-form solutions. The knowledge of closed-form solutions simplifies the construction of the various epidemic indicators that describe the epidemic phenomenon and makes the sensitivity analysis to variations in the data under consideration possible. The closed-form solutions of the systems of four nonlinear first-order ordinary differential equations (ODEs) are established. The Epidemic Peak (EP), Force of Infection (FOI) and Rate of Infection (ROI) are the important indicators for the control and prevention of disease. We examined these indicators using closed-form solutions and particular parameter values. Different disease control scenarios are thoroughly examined. The four scenarios to analyze COVID-19 propagation and containment are (i) lockdown, (ii) quarantine and other preventative measures, (iii) stabilizing the basic reproduction rate to a level where the pandemic can be contained and (iv) containing the epidemic through an appropriate combination of lockdown, quarantine and other preventative measures.