DDE Problems

Mathematical Specification of a DDE Problem

To define a DDE Problem, you simply need to give the function $f$, the initial condition $u_0$ at time point $t_0$, and the history function $h$ which together define a DDE:

\[\frac{du}{dt} = f(u,h,p,t) \qquad & (t \geq t_0)\]
\[u(t_0) = u_0,\]
\[u(t) = h(t) \qquad &(t < t_0).\]

$f$ should be specified as f(u, h, p, t) (or in-place as f(du, u, h, p, t)), $u_0$ should be an AbstractArray (or number) whose geometry matches the desired geometry of u, and $h$ should be specified as described below. The history function h is accessed for all delayed values. Note that we are not limited to numbers or vectors for $u_0$; one is allowed to provide $u_0$ as arbitrary matrices / higher dimension tensors as well.

Functional Forms of the History Function

The history function h can be called in the following ways:

  • h(p, t): out-of-place calculation
  • h(out, p, t): in-place calculation
  • h(p, t, deriv::Type{Val{i}}): out-of-place calculation of the ith derivative
  • h(out, p, t, deriv::Type{Val{i}}): in-place calculation of the ith derivative
  • h(args...; idxs): calculation of h(args...) for indices idxs

Note that a dispatch for the supplied history function of matching form is required for whichever function forms are used in the user derivative function f.

Declaring Lags

Lags are declared separately from their use. One can use any lag by simply using the interpolant of h at that point. However, one should use caution in order to achieve the best accuracy. When lags are declared, the solvers can more efficiently be more accurate and thus this is recommended.

Problem Type

Constructors

DDEProblem(f[, u0], h, tspan[, p]; <keyword arguments>)
DDEProblem{isinplace}(f[, u0], h, tspan[, p]; <keyword arguments>)

Parameter isinplace optionally sets whether the function is inplace or not. This is determined automatically, but not inferred.

Parameters are optional, and if not given then a NullParameters() singleton will be used which will throw nice errors if you try to index non-existent parameters. Any extra keyword arguments are passed on to the solvers. For example, if you set a callback in the problem, then that callback will be added in every solve call.

For specifying Jacobians and mass matrices, see the DiffEqFunctions page.

Arguments

  • f: The function in the DDE.
  • u0: The initial condition. Defaults to the value h(p, first(tspan)) of the history function evaluated at the initial time point.
  • h: The history function for the DDE before t0.
  • tspan: The timespan for the problem.
  • p: The parameters with which function f is called. Defaults to NullParameters.
  • constant_lags: A collection of constant lags used by the history function h. Defaults to ().
  • dependent_lags A tuple of functions (u, p, t) -> lag for the state-dependent lags used by the history function h. Defaults to ().
  • neutral: If the DDE is neutral, i.e., if delays appear in derivative terms.
  • order_discontinuity_t0: The order of the discontinuity at the initial time point. Defaults to 0 if an initial condition u0 is provided. Otherwise it is forced to be greater or equal than 1.
  • kwargs: The keyword arguments passed onto the solves.

Example Problems

Example problems can be found in DiffEqProblemLibrary.jl.

To use a sample problem, such as prob_ode_linear, you can do something like:

#] add DiffEqProblemLibrary
using DiffEqProblemLibrary.ODEProblemLibrary
# load problems
ODEProblemLibrary.importodeproblems()
prob = ODEProblemLibrary.prob_ode_linear
sol = solve(prob)

DDEs with 1 constant delay

DiffEqProblemLibrary.DDEProblemLibrary.prob_dde_constant_1delay_ipConstant
prob_dde_constant_1delay_ip

Delay differential equation

\[u'(t) = -u(t - 1)\]

for $t \in [0, 1]$ with history function $\phi(t) = 0$ if $t < 0$ and $\phi(0) = 1$.

Solution

The analytical solution for $t \in [0, 10]$ can be obtained by the method of steps and is provided in this implementation.

DDEs with 2 constant delays

DiffEqProblemLibrary.DDEProblemLibrary.prob_dde_constant_2delays_ipConstant
prob_dde_constant_2delays_ip

Delay differential equation

\[u'(t) = -u(t - 1/3) - u(t - 1/5)\]

for $t \in [0, 1]$ with history function $\phi(t) = 0$ if $t < 0$ and $\phi(0) = 1$.

Solution

The analytical solution for $t \in [0, 10]$ can be obtained by the method of steps and is provided in this implementation.

DDETest Problems

Some details:

# DDEs with time dependent delays
prob_dde_DDETST_A1, prob_dde_DDETST_A2,
# DDEs with vanishing time dependent delays
prob_dde_DDETST_B1, prob_dde_DDETST_B2,
# DDEs with state dependent delays
prob_dde_DDETST_C1, prob_dde_DDETST_C2, prob_dde_DDETST_C3, prob_dde_DDETST_C4,
# DDEs with vanishing state dependent delays
prob_dde_DDETST_D1, prob_dde_DDETST_D2,
# neutral DDEs with time dependent delays
prob_dde_DDETST_E1, prob_dde_DDETST_E2,
# neutral DDEs with vanishing time dependent delays
prob_dde_DDETST_F1, prob_dde_DDETST_F2, prob_dde_DDETST_F3, prob_dde_DDETST_F4, prob_dde_DDETST_F5,
# neutral DDEs with state dependent delays
prob_dde_DDETST_G1, prob_dde_DDETST_G2,
# neutral DDEs with vanishing state dependent delays
prob_dde_DDETST_H1, prob_dde_DDETST_H2, prob_dde_DDETST_H3, prob_dde_DDETST_H4
DiffEqProblemLibrary.DDEProblemLibrary.prob_dde_DDETST_A1Constant
prob_dde_DDETST_A1

Delay differential equation model of blood production, given by

\[u'(t) = \frac{0.2 u(t - 14)}{1 + u(t - 14)^{10}} - 0.1 u(t)\]

for $t \in [0, 500]$ and history function $\phi(t) = 0.5$ for $t \leq 0$.

References

Mackey, M. C. and Glass, L. (1977). Oscillation and chaos in physiological control systems, Science (197), pp. 287-289.

DiffEqProblemLibrary.DDEProblemLibrary.prob_dde_DDETST_A2Constant
prob_dde_DDETST_A2

Delay differential equation model of chronic granulocytic leukemia, given by

\[u_1'(t) = \frac{1.1}{1 + \sqrt{10} u_1(t - 20)^{5/4}} - \frac{10 u_1(t)}{1 + 40 u_2(t)},\]
\[u_2'(t) = \frac{100 u_1(t)}{1 + 40 u_2(t)} - 2.43 u_2(t),\]

for $t \in [0, 100]$ and history function

\[\phi_1(t) = 1.05767027/3,\]
\[\phi_2(t) = 1.030713491/3,\]

for $t \leq 0$.

References

Wheldon, T., Kirk, J. and Finlay, H. (1974). Cyclical granulopoiesis in chronic granulocytic leukemia: A simulation study., Blood (43), pp. 379-387.

DiffEqProblemLibrary.DDEProblemLibrary.prob_dde_DDETST_B1Constant
prob_dde_DDETST_B1

Delay differential equation

\[u'(t) = 1 - u(\exp(1 - 1/t))\]

for $t \in [0.1, 10]$ with history function $\phi(t) = \log t$ for $t \in (0, 0.1]$.

Solution

The analytical solution for $t \in [0.1, 10]$ is

\[u(t) = \log t.\]

References

Neves, K. W. (1975). Automatic integration of functional differential equations: An approach, ACM Trans. Math. Soft. (1), pp. 357-368.

DiffEqProblemLibrary.DDEProblemLibrary.prob_dde_DDETST_B2Constant
prob_dde_DDETST_B2

Delay differential equation

\[u'(t) = - 1 - u(t) + 2 [u(t / 2) < 0]\]

for $t \in [0, 2 \log 66]$ with history function $\phi(0) = 1$.

Solution

The analytical solution for $t \in [0, 2 \log 66]$ is

\[u(t) = \begin{cases} 2 \exp(-t) - 1 & \text{if } t \in [0, 2 \log 2], \\ 1 - 6 \exp(-t) & \text{if } t \in (2 \log 2, 2 \log 6], \\ 66 \exp(-t) - 1 & \text{if } t \in (2 \log 6, 2 \log 66]. \end{cases}\]

References

Neves, K. W. and Thompson, S. (1992). Solution of systems of functional differential equations with state dependent delays, Technical Report TR-92-009, Computer Science, Radford University.

DiffEqProblemLibrary.DDEProblemLibrary.prob_dde_DDETST_C1Constant
prob_dde_DDETST_C1

Delay differential equation

\[u'(t) = - 2 u(t - 1 - |u(t)|) (1 - u(t)^2)\]

for $t \in [0, 30]$ with history function $\phi(t) = 0.5$ for $t \leq 0$.

References

Paul, C. A. H. (1994). A test set of functional differential equations, Technical Report 249, The Department of Mathematics, The University of Manchester, Manchester, England.

DiffEqProblemLibrary.DDEProblemLibrary.prob_dde_DDETST_C2Constant
prob_dde_DDETST_C2

Delay differential equation

\[u_1'(t) = - 2 u_1(t - u_2(t)),\]
\[u_₂'(t) = \frac{|u_1(t - u_2(t))| - |u_1(t)|}{1 + |u_1(t - u_2(t))|},\]

for $t \in [0, 40]$ with history function

\[\phi_1(t) = 1,\]
\[\phi_2(t) = 0.5,\]

for $t \leq 0$.

References

Paul, C. A. H. (1994). A test set of functional differential equations, Technical Report 249, The Department of Mathematics, The University of Manchester, Manchester, England.

DiffEqProblemLibrary.DDEProblemLibrary.prob_dde_DDETST_C3Constant
prob_dde_DDETST_C3

Delay differential equation model of hematopoiesis, given by

\[u_1'(t) = \hat{s}_0 u_2(t - T_1) - \gamma u_1(t) - Q,\]
\[u_2'(t) = f(u_1(t)) - k u_2(t),\]
\[u_3'(t) = 1 - \frac{Q \exp(\gamma u_3(t))}{\hat{s}_0 u_2(t - T_1 - u_3(t))},\]

for $t \in [0, 300]$ with history function $\phi_1(0) = 3.325$, $\phi_3(0) = 120$, and

\[\phi_2(t) = \begin{cases} 10 & \text{if } t \in [- T_1, 0],\\ 9.5 & \text{if } t < - T_1, \end{cases}\]

where $f(y) = a / (1 + K y^r)$, $\hat{s}_0 = 0.0031$, $T_1 = 6$, $\gamma = 0.001$, $Q = 0.0275$, $k = 2.8$, $a = 6570$, $K = 0.0382$, and $r = 6.96$.

References

Mahaffy, J. M., Belair, J. and Mackey, M. C. (1996). Hematopoietic model with moving boundary condition and state dependent delay, Private communication.

DiffEqProblemLibrary.DDEProblemLibrary.prob_dde_DDETST_C4Constant
prob_dde_DDETST_C4

Delay differential equation model of hematopoiesis, given by the same delay differential equation as prob_dde_DDETST_C3

\[u_1'(t) = \hat{s}_0 u_2(t - T_1) - \gamma u_1(t) - Q,\]
\[u_2'(t) = f(u_1(t)) - k u_2(t),\]
\[u_3'(t) = 1 - \frac{Q \exp(\gamma u_3(t))}{\hat{s}_0 u_2(t - T_1 - u_3(t))},\]

for $t \in [0, 100]$ with history function $\phi_1(0) = 3.5$, $\phi_3(0) = 50$, and $\phi_2(t) = 10$ for $t \leq 0$, where $f(y) = a / (1 + K y^r)$, $\hat{s}_0 = 0.00372$, $T_1 = 3$, $\gamma = 0.1$, $Q = 0.00178$, $k = 6.65$, $a = 15600$, $K = 0.0382$, and $r = 6.96$.

References

Mahaffy, J. M., Belair, J. and Mackey, M. C. (1996). Hematopoietic model with moving boundary condition and state dependent delay, Private communication.

DiffEqProblemLibrary.DDEProblemLibrary.prob_dde_DDETST_D1Constant
prob_dde_DDETST_D1

Delay differential equation

\[u_1'(t) = u_2(t), \]
\[u_2'(t) = - u_2(\exp(1 - u_2(t))) u_2(t)^2 \exp(1 - u_2(t)),\]

for $t \in [0.1, 5]$ with history function

\[\phi_1(t) = \log t, \]
\[\phi_2(t) = 1 / t,\]

for $t \in (0, 0.1]$.

Solution

The analytical solution for $t \in [0.1, 5]$ is

\[u_1(t) = \log t, \]
\[u_2(t) = 1 / t.\]

References

Neves, K. W. (1975). Automatic integration of functional differential equations: An approach, ACM Trans. Math. Soft. (1), pp. 357-368.

DiffEqProblemLibrary.DDEProblemLibrary.prob_dde_DDETST_D2Constant
prob_dde_DDETST_D2

Delay differential equation model of antigen antibody dynamics with fading memory, given by

\[u_1'(t) = - r_1 u_1(t) u_2(t) + r_2 u_3(t), \]
\[u_2'(t) = - r_1 u_1(t) u_2(t) + \alpha r_1 u_1(t - u_4(t)) u_2(t - u_4(t)),\]
\[u_3'(t) = r_1 u_1(t) u_2(t) - r_2 u_3(t), \]
\[u_4'(t) = 1 + \frac{3 \delta - u_1(t) u_2(t) - u_3(t)}{u_1(t - u_4(t)) u_2(t - u_4(t)) + u_3(t - u_4(t))} \exp(\delta u_4(t)),\]

for $t \in [0, 40]$ with history function

\[\phi_1(t) = 5, \]
\[\phi_2(t) = 0.1, \]
\[\phi_3(t) = 0, \]
\[\phi_4(t) = 0,\]

for $t \leq 0$, where $r_1 = 0.02$, $r_2 = 0.005$, $\alpha = 3$, and $\delta = 0.01$.

References

Gatica, J. and Waltman, P. (1982). A threshold model of antigen antibody dynamics with fading memory, in Lakshmikantham (ed.), Nonlinear phenomena in mathematical science, Academic Press, New York, pp. 425-439.

DiffEqProblemLibrary.DDEProblemLibrary.prob_dde_DDETST_E1Constant
prob_dde_DDETST_E1

Delay differential equation model of a food-limited population, given by

\[u(t) = r u(t) (1 - u(t - 1) - c u'(t - 1))\]

for $t \in [0, 40]$ with history function $\phi(t) = 2 + t$ for $t \leq 0$, where $r = \pi / \sqrt{3} + 1/20$ and $c = \sqrt{3} / (2 \pi) - 1 / 25$.

References

Kuang, Y. and Feldstein, A. (1991). Boundedness of solutions of a nonlinear nonautonomous neutral delay equation, J. Math. Anal. Appl. (156), pp. 293-304.

DiffEqProblemLibrary.DDEProblemLibrary.prob_dde_DDETST_E2Constant
prob_dde_DDETST_E2

Delay differential equation model of a logistic Gauss-type predator-prey system, given by

\[u_1'(t) = u_1(t) (1 - u_1(t - \tau) - \rho u_1'(t - \tau)) - \frac{u_2(t) u_1(t)^2}{u_1(t)^2 + 1}, \]
\[u_2'(t) = u_2(t) \left(\frac{u_1(t)^2}{u_1(t)^2 + 1} - \alpha\right),\]

for $t \in [0, 2]$ with history function

\[\phi_1(t) = 0.33 - t / 10, \]
\[\phi_2(t) = 2.22 + t / 10,\]

for $t \leq 0$, where $\alpha = 0.1$, $\rho = 2.9$, and $\tau = 0.42$.

References

Kuang, Y. (1991). On neutral delay logistics Gauss-type predator-prey systems, Dyn. Stab. Systems (6), pp. 173-189.

DiffEqProblemLibrary.DDEProblemLibrary.prob_dde_DDETST_F1Constant
prob_dde_DDETST_F1

Delay differential equation

\[u'(t) = 2 \cos(2t) u(t / 2)^{2 \cos t} + \log(u'(t / 2)) - \log(2 \cos t) - \sin t\]

for $t \in [0, 1]$ with history function $\phi(0) = 1$ and $\phi'(0) = 2$.

Solution

The analytical solution for $t \in [0, 1]$ is

\[u(t) = \exp(\sin(2t)).\]

References

Jackiewicz, Z. (1981). One step methods for the numerical solution of Volterra functional differential equations of neutral type, Applicable Anal. (12), pp. 1-11.

DiffEqProblemLibrary.DDEProblemLibrary.prob_dde_DDETST_F2Constant
prob_dde_DDETST_F2

Delay differential equation

\[u'(t) = u'(2t - 0.5)\]

for $t \in [0.25, 0.499]$ with history function $\phi(t) = \exp(-t^2)$ and $\phi'(t) = -2t \exp(-t^2)$ for $t \leq 0.25$.

Solution

The analytical solution for $t \in [0.25, 0.499]$ is

\[u(t) = u_i(t) = \exp(-4^i t^2 + B_i t + C_i) / 2^i + K_i\]

if $t \in [x_i, x_{i + 1}]$, where

\[x_i = (1 - 2^{-i}) / 2, \]
\[B_i = 2 (4^{i-1} + B_{i-1}), \]
\[C_i = - 4^{i-2} - B_{i-1} / 2 + C_{i-1}, \]
\[K_i = - \exp(-4^i x_i^2 + B_i x_i + C_i) / 2^i + u_{i-1}(x_i),\]

and $B_0 = C_0 = K_0 = 0$.

References

Neves, K. W. and Thompson, S. (1992). Solution of systems of functional differential equations with state dependent delays, Technical Report TR-92-009, Computer Science, Radford University.

DiffEqProblemLibrary.DDEProblemLibrary.prob_dde_DDETST_F3Constant
prob_dde_DDETST_F3

Delay differential equation

\[u'(t) = \exp(-u(t)) + L_3 \left[\sin(u'(\alpha(t))) - \sin\left(\frac{1}{3 + \alpha(t)}\right)\right]\]

for $t \in [0, 10]$ with history function $\phi(0) = \log 3$ and $\phi'(0) = 1 / 3$, where $\alpha(t) = 0.5 t (1 - \cos(2 \pi t))$ and $L_3 = 0.2$.

Solution

The analytical solution for $t \in [0, 10]$ is

\[u(t) = \log(t + 3).\]
DiffEqProblemLibrary.DDEProblemLibrary.prob_dde_DDETST_G1Constant
prob_dde_DDETST_G1

Delay differential equation

\[u'(t) = - u'(t - u(t)^2 / 4)\]

for $t \in [0, 1]$ with history function $\phi(t) = 1 - t$ for $t \leq 0$ and $\phi'(t) = -1$ for $t < 0$.

Solution

The analytical solution for $t \in [0, 1]$ is

\[u(t) = t + 1.\]

References

El'sgol'ts, L. E. and Norkin, S. B. (1973). Introduction to the Theory and Application of Differential Equations with Deviating Arguments, Academic Press, New York, p. 44.

DiffEqProblemLibrary.DDEProblemLibrary.prob_dde_DDETST_G2Constant
prob_dde_DDETST_G2

Delay differential equation

\[u'(t) = - u'(u(t) - 2)\]

for $t \in [0, 1]$ with history function $\phi(t) = 1 - t$ for $t \leq 0$ and $\phi'(t) = -1$ for $t < 0$.

Solution

The analytical solution for $t \in [0, 1]$ is

\[u(t) = t + 1.\]

El'sgol'ts, L. E. and Norkin, S. B. (1973). Introduction to the Theory and Application of Differential Equations with Deviating Arguments, Academic Press, New York, pp. 44-45.

DiffEqProblemLibrary.DDEProblemLibrary.prob_dde_DDETST_H1Constant
prob_dde_DDETST_H1

Delay differential equation

\[u'(t) = - \frac{4 t u(t)^2}{4 + \log(\cos(2t))^2} + \tan(2t) + 0.5 \arctan\left(u'\left(\frac{t u(t)^2}{1 + u(t)^2}\right)\right)\]

for $t \in [0, 0.225 \pi]$ with history function $\phi(0) = 0$ and $\phi'(0) = 0$.

Solution

The analytical solution for $t \in [0, 0.225 \pi]$ is

\[u(t) = - \log(\cos(2t)) / 2.\]

References

Castleton, R. N. and Grimm, L. J. (1973). A first order method for differential equations of neutral type, Math. Comput. (27), pp. 571-577.

DiffEqProblemLibrary.DDEProblemLibrary.prob_dde_DDETST_H2Constant
prob_dde_DDETST_H2

Delay differential equation

\[u'(t) = \cos(t) (1 + u(t u(t)^2)) + L_3 u(t) u'(t u(t)^2) + (1 - L_3) \sin(t) \cos(t \sin(t)^2) - \sin(t + t \sin(t)^2)\]

for $t \in [0, \pi]$ with history function $\phi(0) = 0$ and $\phi'(0) = 1$, where $L_3 = 0.1$.

Solution

The analytical solution for $t \in [0, \pi]$ is

\[u(t) = \sin(t).\]

References

Hayashi, H. (1996). Numerical solution of retarded and neutral delay differential equations using continuous Runge-Kutta methods, PhD thesis, Department of Computer Science, University of Toronto, Toronto, Canada.

DiffEqProblemLibrary.DDEProblemLibrary.prob_dde_DDETST_H3Constant
prob_dde_DDETST_H3

Same delay differential equation as prob_dde_DDETST_H2 with $L_3 = 0.3$.

References

Hayashi, H. (1996). Numerical solution of retarded and neutral delay differential equations using continuous Runge-Kutta methods, PhD thesis, Department of Computer Science, University of Toronto, Toronto, Canada.

DiffEqProblemLibrary.DDEProblemLibrary.prob_dde_DDETST_H4Constant
prob_dde_DDETST_H4

Same delay differential equation as prob_dde_DDETST_H2 with $L_3 = 0.5$.

References

Hayashi, H. (1996). Numerical solution of retarded and neutral delay differential equations using continuous Runge-Kutta methods, PhD thesis, Department of Computer Science, University of Toronto, Toronto, Canada.

Radar5 Test Problems

DiffEqProblemLibrary.DDEProblemLibrary.prob_dde_RADAR5_oregonatorConstant
prob_dde_RADAR5_oregonator

Delay differential equation model from chemical kinetics, given by

\[ u_1'(t) = - k_1 A u_2(t) - k_2 u_1(t) u_2(t - \tau) + k_3 B u_1(t) - 2 k_4 u_1(t)^2, \]
\[ u_2'(t) = - k_1 A u_2(t) - k_2 u_1(t) u_2(t - \tau) + f k_3 B u_1(t),\]

for $t \in [0, 100.5]$ with history function

\[ \phi_1(t) = 1e-10, \]
\[ \phi_2(t) = 1e-5,\]

for $t \leq 0$, where $k_1 = 1.34$, $k_2 = 1.6e9$, $k_3 = 8000$, $k_4 = 4e7$, $k_5 = 1$, $f = 1$, $A = 0.06$, $B = 0.06$, and $\tau = 0.15$.

References

Epstein, I. and Luo, Y. (1991). Differential delay equations in chemical kinetics. Nonlinear models, Journal of Chemical Physics (95), pp. 244-254.

DiffEqProblemLibrary.DDEProblemLibrary.prob_dde_RADAR5_robertsonConstant
prob_dde_RADAR5_robertson

Delay differential equation model of a chemical reaction with steady state solution, given by

\[ u_1'(t) = - a u_1(t) + b u_2(t - \tau) u_3(t), \]
\[ u_2'(t) = a u_1(t) - b u_2(t - \tau) u_3(t) - c u_2(t)^2, \]
\[ u_3'(t) = c u_2(t)^2,\]

for $t \in [0, 10e10]$ with history function $\phi_1(0) = 1$, $\phi_2(t) = 0$ for $t \in [-\tau, 0]$, and $\phi_3(0) = 0$, where $a = 0.04$, $b = 10_000$, $c = 3e7$, and $\tau = 0.01$.

References

Guglielmi, N. and Hairer, E. (2001). Implementing Radau IIA methods for stiff delay differential equations, Computing (67), pp. 1-12.

DiffEqProblemLibrary.DDEProblemLibrary.prob_dde_RADAR5_waltmanConstant
prob_dde_RADAR5_waltman

Delay differential equation model of antibody production, given by

\[ u_1'(t) = - r u_1(t) u_2(t) - s u_1(t) u_4(t), \]
\[ u_2'(t) = - r u_1(t) u_2(t) + \alpha r u_1(u_5(t)) u_2(u_5(t)) [t \geq t_0], \]
\[ u_3'(t) = r u_1(t) u_2(t), \]
\[ u_4'(t) = - s u_1(t) u_4(t) - \gamma u_4(t) + \beta r u_1(u_6(t)) u_2(u_6(t)) [t > t_1], \]
\[ u_5'(t) = [t \geq t_0] \frac{u_1(t) u_2(t) + u_3(t)}{u_1(u_5(t)) u_2(u_5(t)) + u_3(u_5(t))}, \]
\[ u_6'(t) = [t \geq t_1] \frac{1e-12 + u_2(t) + u_3(t)}{1e-12 + u_2(u_6(t)) + u_3(u_6(t))},\]

for $t \in [0, 300]$ with history function

\[ \phi_1(t) = \phi_0, \]
\[ \phi_2(t) = 1e-15, \]
\[ \phi_3(t) = 0, \]
\[ \phi_4(t) = 0, \]
\[ \phi_6(t) = 0,\]

for $t \leq 0$, where $\alpha = 1.8$, $\beta = 20$, $\gamma = 0.002$, $r = 5e4$, $s = 1e5$, $t_0 = 32$, $t_1 = 119$, and $\phi_0 = 0.75e-4$.

References

Waltman, P. (1978). A threshold model of antigen-stimulated antibody production, Theoretical Immunology (8), pp. 437-453.

QS Example

DiffEqProblemLibrary.DDEProblemLibrary.prob_dde_qsConstant
prob_dde_qs

Delay differential equation model of Quorum Sensing (QS) of Pseudomonas putida IsoF in continuous cultures.

References

Buddrus-Schiemann et al. (2014). Analysis of N-Acylhomoserine Lactone Dynamics in Continuous Cultures of Pseudomonas Putida IsoF By Use of ELISA and UHPLC/qTOF-MS-derived Measurements and Mathematical Models, Analytical and Bioanalytical Chemistry.