Abstract
The maximum likelihood inference framework for delay differential equation models in the multivariate settings is developed. The number of delay parameters is assumed to be one or more. This study does not make any restrictive assumptions on the form of the underlying delay differential equations which was one of the limitations of some of the previous work. Thus, the maximum likelihood inference framework can be applied to general delay differential equation models with multiple delay parameters. To obtain the maximum likelihood estimator and estimate of the information matrix, two numerical algorithms are developed: (i) the adaptive grid and (ii) the gradient descent algorithms. Two examples of multivariate delay differential equation models related to the epidemic and pharmacokinetic models, respectively, are presented in this paper. For the unknown parameters, standard errors and confidence intervals are constructed, and formulas and techniques for producing the information matrix are developed. The code and computations are developed with the help of the mathematical software MATLAB.
Keywords: Delay differential equation (DDE), Delay differential equation models (DDEMs), Delayed pharmacokinetic models, Delayed susceptible-infective-recovered (SIR) model, Maximum likelihood estimation (MLE)
Subject terms: Applied mathematics, Software, Statistics
Introduction
Many researchers have incorporated various types of time delays into biological models to represent the regeneration times of resources, periods of maturation, feeding times, response times, and more; see, for examples,1–6. Also, delays are widespread in pharmacokinetic and pharmacodynamics studies in the form of absorption delay, delayed drug response, etc. see7–14. Numerous delay differential equation models (NDDEMs) are proposed by investigators according to their understanding of the underlying dynamical system. In such systems, the unknown parameters must be estimated based on actual observations from the experiment. Some research works on various parameter estimation techniques for dynamical system is available in the literature; see15–19. However, only a few methods in the statistical literature have been proposed to estimate and infer parameters in delay differential equation models (DDEMs), see20–26, albeit in a univariate setting. In the statistical literature, there aren’t a lot of methods for parameter estimation and inference for DDEMs in multivariate setting.
We present a review of parameter inference methods for DDEMs that have been reported in the literature for the multivariate case. Murphy27 developed parameter estimation algorithm to estimate unknown time delays and other parameters (e.g., initial condition) appearing within a nonlinear nonautonomous functional differential equation. The unknown parameters are estimated within the approximating systems by minimizing a least squares fit-to-data criterion. Murphy assumed that the model equation is a functional differential equation of the general form:
 |
1 |
where 
, 
denotes the function 
where 
varies over 
and 
represents a vector of parameters occurring in the model equation. Any of the parameters , 
or 
might be unknowns. If the delay is explicitly time independent, it is function 
that will be unknown, see27. Murphy introduced a full discretization approach where he uses linear splines to describe solution and delay functions. The problem then is treated as a very large minimization problem.
The two-step method relies on the estimation of derivatives of the underlying ODEs from the data. Many researchers suggested the two-step approach: In the first step, the first order derivative of the ODE is estimated from the observations in28–34. It is then treated as a response variable in the second step where estimation of the parameters is carried out. In the second step, the ODE parameters are estimated using a standard nonlinear least squares method, where the estimation of 
is now obtained from the first step.
Bünner et al.19 studied the special case of DDEMs whose underlying delay differential equation (DDE) is of the form
 |
2 |
where 
is the dynamic process on 
and is a constant delay parameter. Bünner et al. introduced a method to estimate from time series of the observation process (1.2) in the special case of 
. They take advantage of the fact that 
at time point 
as 
. Next, they infer the time delay using the value that gives the smoothest graph of 
versus 
. Their method amounts to the two-step method with an approximation of the derivative by 
.
Ellner et al.25 unified the approaches which have been introduced in20,21 and extended them to more realistic cases, in which the data have measurement errors. They used nonparametric smoothing methods to estimate the derivative of a univariate DDEM from noisy data that is presumed to be additive. see35. From the fitting of a generalized additive model, they then infer the constant delay parameter, τ. Even though Ellner’s method combines earlier research, it can only be used with DDEMs that satisfy the presumptive additive form.
A version of two-step method has been utilized to estimate certain types of DDE models with time varying coefficient whereas Siamak et al.36,37 presented an estimation method according to Least Squares Support Vector Machines (LS-SVMs), which, in deterministic parameter-affine DDEMs, transforms the solution of the differential equations (DEs) into an optimization of a set of algebraic equations for each of approximating constant and time-varying parameters. However, neither confidence intervals nor standard errors of estimations are given.
Wood38 devised spline-based model fitting methodologies for scenarios where the DDEMs are partially defined with noisy data for modeling ecological population dynamics. Wood considered a partially specified predator–prey model with 
being an unknown function, is given by the following set of equations:
 |
3 |
where 
and 
denote the predator and prey population size respectively, 
and 
are the derivatives of and with respect to the time 
respectively, 
, 
, 
are parameters, 
is the delayed prey population size and is the time delay. The model parameters are estimated by minimizing the distance between the observed data and the numerical solution of the DDE given a nonparametric spline estimate of and other parameters , and . This is a very difficult optimization problem since the DDE numeric solutions not only depend on the parameter values, but also on the history of the dynamic process, which is essentially an infinite-dimensional optimization problem. Because the cross validation and the estimations of the unknown parameters are needed to choose the smoothing coefficients linked to the penalty term, the spline-based approach has significant computing costs.
Cangelosi and Hooten39 utilized a fourth-order Runge–Kutta (RK4) approximation to the system of differential equations in conjunction with Markov chain Monte Carlo (MCMC) methods to implement the formal statistical model where RK4 is a useful approximation, well known for its accuracy, efficiency, robustness, and applicability to dynamical processes of any finite dimension. They proposed MCMC to estimate the parameters in the Lotka-Volterra (LV) model but assuming the data comes from a truncated normal distribution with the mean function being the solution of the DDE.
Also, Calderhead et al.40 proposed a Bayesian approach based on Gaussian processes (GP) to estimate DDE parameters, see41, which also requires repeatedly solving DDEs numerically when sampling for values of DDE parameters. They used GP to predict the state variables of the model as well as their derivatives, thus avoiding the need to solve the system explicitly.
Based on the maximum likelihood (ML) inference framework, we examine the estimation and inference of parameters for multivariate DDEMs that incorporate multiple delays. ML estimators are efficient estimator which is used to estimate the unknown parameters (fixed as well as random) of parameterized probability density functions.
Properties of MLEs are well understood and exhibit good behavior. Fisher placed maximization of the likelihood function on a sound footing under the name of the method of maximum likelihood which achieved widespread popularity because of its properties in the theory of estimation, Its mathematical basis has been extensively studied, see42,43. Berk44,45 takes a Bayesian approach and mentions in passing the information theoretic interpretation of maximum likelihood estimation (MLE). For large sample sizes, Huber46 presents common conditions under which the MLE is guaranteed to give the accurate values for the parameters that are unknown and he provides very general conditions, building on those of Wald47 under which the ML estimator converges to a well-defined limit, even when the probability model is not correctly specified. Huber’s limit is identical to that of Berk but, Huber does not explicitly discuss the information theoretic interpretation of this limit. This interpretation has been emphasized by Akaike48, who has observed that the ML estimator is a natural estimator for the parameters when the true distribution is unknown. The ML estimator is a natural estimator for the parameters which minimize the Kullback–Leibler49 Information Criterion (KLIC).
Generally, the parameters are often unknown in a DDE, and it is important to estimate these parameters based on observed data. A delay differential equation model is a statistical model that incorporates the observational process over the underlying DDE. As was previously mentioned, there were not many methods offered for parameter estimation and inference from DDEMs in the statistical literature.
In this paper, we use MLE technique to infer unknown parameters in DDEMs. To obtain the MLE, we develop the adaptive grid and gradient descent methods in a multivariate setting. The remainder of the paper is organized as follows. Sect. “The mathematical model” defines the mathematical model for DDEMs in a multivariate setting and presents two examples. The MLE technique for DDEMs and the associated numerical procedures are described in Sect. “Proposed methodology” Based on the outcomes of the simulation, Sect. “Numerical solution and results” resents the results and discussion based on the suggested methodology.
The mathematical model
A model of the DDE with multiple delays equals, 
as noisy realizations from an underlying DDE:
 |
4 |
where 
and 
is a vector of error variables where it is presumed that every component comes from a normal distribution with mean equals to zero and 
where 
is unknown standard deviation, i.e., 
. In (4), is the solution, 
, of the DDE
 |
5 |
calculated at the 
time points, 
; in (5), 
and 
where 
is the -th delay term with 
, where 
is the delay parameter, 
and 
are other parameters of concern that control the trajectories of the underlying DDE in (5). Equations (4) and (5) construct a multivariate DDEM in the most comprehensive form. In a DDEM, the parameters 
and are unknown and should be estimated according to observations 
The observation 
at the 
-th sampled time point, , with 
is obtained. The function on 
has a complete trajectory which is determined by (5) based on an initial condition function 
where 
, 
and 
is chosen to be
 |
6 |
which is also unknown besides 
and 
.
For prescribed values of , and 
, where 
, we notice that the solution is a function of , and ; thus, the solution is represented as an explicit function of the unknown quantities , and , that is 
. According to the observational model (4), the 
are gathered at the 
sampled time points 
.Our objective is to estimate the parameters , and 
(the variance of noise) based on the observations 
Note that here is an unknown vector of parameters 
, but it is actually a nuisance parameter that affects the estimation of the parameters of interest, namely and . The dynamical system properties are determined by and and not .
Examples
As a special case, we consider two examples: The delayed Susceptible-Infective-Recovered (SIR) differential equation model with unknown parameters related to biological systems and the two-compartment model with time delay related to the pharmacokinetic models.
Delayed SIR model
The epidemic delayed SIR model with two delays which is derived from the original Kermack and McKendrick’s classical SIR model reported in50. The original SIR model divided the total population into three groups of individuals, namely, the group of susceptible individuals denoted by S, I is the group of infective individuals and R is the group of recovered individuals, where the principle of mass action governs the movement from the S to I to R groups51, governs the diffusion of an epidemic.
Epidemic models using ordinary differential equations in the simplest forms were given in52,53 as the following:
The classic epidemic model is the SIR model given by the initial value problem
 |
7 |
where is the contact rate (i.e., the average number of adequate contacts of a person per unit time), 
is average infectious period, 
, 
, and 
are the numbers in these classes, so that 
, 
is the total population size.
Recently, the epidemic models have been studied and extended by many authors and they can be found in54–60, for example. One of the most basic epidemic models which is proposed to explain the rise and fall in the number of infected patients observed in an epidemic is the SIR model. If the immunity that is obtained upon recovery is permanent, then the model is a SIR model. If recovery does not give immunity, then the model is called a susceptible-infective-susceptible (SIS) model. Generally, SIR models are applicable for viral agent diseases such as mumps, measles, and smallpox, while SIS models are more applicable for protozoan agent diseases such as sleeping sickness and malaria, and for some bacterial agent diseases such as plague, meningitis, and venereal diseases.
A SIR model with time delay has been studied in many literature reviews61–64. A SIR model with discrete delay was analyzed as in59. The delay was used to model the fact that an individual may not be infectious for a period of time after infection. In62–64, A SIR model with distributed delay was studied. The SIR model with incubation time delay and logistic growth rate with carrying capacity was analyzed and the dynamic properties of the suggested system are presented in65. A new system was extended and proposed by66 instead of the system in65 because of considering the behavioral changes of susceptible individuals.
Now, let 
, 
and 
denote the numbers of susceptible, infective, and recovered individuals at time , respectively. A SIR epidemic model with time delays is as follows:
 |
8 |
where the initial conditions 
, and 
for 
and 
, represents the derivative of 
with respect to . The differential equation model in (8) is an extension of the Kermack and McKendrick’s model48 incorporating delay parameters 
and 
. The delay parameter relates to the time delay for a newly infected susceptible to exhibit symptoms of his/her infectiousness. Thus, represents the incubation period of an infection in the susceptible. Similarly, is the infectious period before the patient is fully recovered, i.e., stop being infectious. Thus, the delayed SIR model represents a more realistic infection experience for a population under study.
The other parameters in the model, namely; 
; where 
are as follows: 
is the daily contact rate, i.e., the average number of contacts per infective per unit of time, is the rate of return to susceptibility after a period of infection of duration and is the daily recovery rate of the infectives per unit of time. In this example, 
, 
and 
.
Pharmacokinetic models: the two-compartment model with time delay
Pharmacokinetic study of drugs subject to enterohepatic circulation has gained importance in recent years. A two-compartment model was developed by Harrison and Gibaldi67 representing the body and the GI, or digestive, tract then by Chen and Gross68 to describe the pharmacokinetics of drugs undergoing reabsorption. In many cases, a compartmental model must include delays to give a more isomorphic description of the biological system see69–71. Cobelli and Rescigno72 and Steimer et al.73 proposed a modification to the previous model by adding a time-lag on the transfer pathway from compartment I to compartment II, where compartment I represents the body including the liver while compartment II represents the GI tract, as shown in Fig. 1. A two-compartment model with time delay is suggested to characterize the pharmacokinetics of drugs that undergo enterohepatic circulation.
Fig. 1.
Time delay pharmacokinetic model for a drug subject to enterohepatic circulation67.
The model illustrated in Fig. 1 represents a two-compartment system varying from the fundamental model described in67, by adding a time delay in the transfer pathway from compartment I to compartment II.
It is assumed that a time interval exists following biliary excretion, prior to the occurrence of reabsorption. The transfer processes are assumed to be of first-order kinetics; elimination occurs both from compartment I () and compartment II (); the rate constant for biliary excretion and reabsorption are, respectively, and . The hypothesis that the reabsorption of a drug molecule is lagged after its biliary excretion is represented by the addition of a time delay in the transition from the first compartment to the second one.
Now, let 
be the amount of drug at time in compartment I. The corresponding mathematical formulation is:
 |
9 |
The mathematical formulation is described by (9) with the corresponding initial conditions.
For intravenous administration:
 |
10 |
and for oral administration:
 |
11 |
Equations (9), (10) and (11) represent two models: model I coincides with an intravenous single dose and is represented by (9) and (10); model II coincides with an oral single dose and is represented by (9) and (11).
In this research, model I and model II are studied and the goal is to estimate the parameters related to these models.
Proposed methodology
The maximum likelihood inference approach for DDEMs
Let 
denote the probability density function (pdf) parameterized in terms of the unknown parameters , and for independent (vector) observations 
. Based on the independence and normality assumptions on 
’s in (4), the likelihood function is given by
 |
For now, we consider that is known and fixed; the situation where is unknown is addressed later. So, for now, it is assumed that the aforementioned likelihood is a function of 
. Using the log-likelihood function is the standard procedure for statistical inference, which is provided by
 |
12 |
The log-likelihood function 
expressions are frequently easier to work with than the likelihood function, 
, since the expressions of the log-likelihood function involve summations and not products. Hence, the differentiation of the log-likelihood function is easier and the results are more reliable. Here the MLE of is denoted as 
. The MLE is a point estimate where
 |
13 |
which is depending on 
. Now, let us study the case in which is unknown. 
denotes to the MLE of and the in (12) is maximized as a function of after finding as in (13). In closed form, can be obtained and is provided by
 |
14 |
Over the grid, the function global maximum can be found by the grid algorithms. First, evaluate the function value on the grid, then find the grid value corresponding to the maximum value. The space of the grid is quite refined, and the grid value is close to the domain value. Therefore, we are close to the global maximum value. The algorithm of the adaptive grid improves the original algorithm of the grid, thereby bringing us increasingly closer to the global maximum. Nevertheless, the primary disadvantage of the grid algorithm or any grid is its slow convergence.
To find the MLE numerically, the log-likelihood in Eq. (12) is the function to maximize. Therefore, it is best to choose the value with largest log-likelihood to get 
and 
as in (13). This maximization can be done by a two-step procedure: First, an adaptive grid procedure is used which is then followed by a quasi-Newton algorithm. The first step ensures we are close to the MLE, whereas the second procedure guarantees a swift convergence towards the MLE.
The grid procedure
Utilizing an adaptive grid procedure can help us to get and , where the value with largest log-likelihood is chosen. The gridding is executed for and . A numerical Newton–Raphson method is utilized to determine the maximum value of in the grid space 
, is defined as
 |
15 |
where the grid space 
, 
which covers 
values of 
is used. By maximizing the log-likelihood above for each fixed value of and in , we find the MLE of , 
. The MLE 
is obtained by solving
 |
16 |
Denote 
to be the 
column vector consisting of the entries 
where
 |
By using Newton–Raphson method, the numerical problem is solved as follows:
 |
17 |
where
 |
18 |
And
 |
19 |
As noted from (18) and (19), 
and 
values are needed at each 
where 
and 
are the first and second partial derivative of 
regarding , respectively. This is performed repetitively as in the following:
is got by differentiating (5) regarding , where and are independent of each other, where 
, 
, 
, which gives:
 |
20 |
which suggests that 
gives a DDE that is obtained by (20). Note that 
if (
and 
if (
.
Since the initial condition of the -process is 
1, for all 
. Based on the initial condition using Euler method, the above DDE can be solved numerically.
Also, 
is got by differentiating (5) regarding to get
 |
21 |
where 
is the derivative of 
regarding , and 
is the delayed version of , that is 
The initial condition for the DDE in (
) is 
since the derivative of the initial values regarding 
is 
.
In the same way and by differentiating (5) regarding , we get
 |
22 |
where 
is the derivative of 
, and 
is the delayed version of 
, that is.
, 
. The initial condition for the DDE in (
) is 
since the derivative of the initial values regarding 
is 0.
To find 
at each 
by using the Euler method, the numerical solution is calculated using step size of 
, the number of equal segments of the interval 
, 
, and by using 
, 
where 
, 
. Here, is a natural number chosen to be large. To get the initial conditions of the process of the derivatives, it is observed that
 |
23 |
Thus, 
and 
for all , 
.
In the same way, the second derivative process of 
is got by differentiating (20) regarding , to get
 |
24 |
which implies that satisfies another DDE depending on 
which has been given earlier.
can be obtained by differentiating (21) with respect to as given below
 |
25 |
In the same way,
 |
26 |
where 
as 
, 
and 
as 
.
Thus, the second derivative process of 
can be obtained by differentiating (22) with respect to as given above.
The log-likelihood 
is computed depending on (17) using 
. Then, by finding the maximum 
as a function of 
, the point based maximum is found. The MLE got from the gridding algorithm is defined as
 |
27 |
The adaptive grid procedure
Applying the generic grid procedure repeatedly at increasingly finer intervals for 
) is the adaptive grid (AG) algorithm. These are the steps in the AG algorithm:
-
i.
Determine an initial grid space
containing the grid points 
, 
and . -
ii.
Let
to be maximized regarding 
and as given in Sect. “The grid procedure” -
iii.
Get
as in (27). -
iv.
Refine the Grid: Assume that
as in step iii. The upper and lower -grid points of new grid space 
is given by 
. The congruous upper and lower -grid points are 
. The original grid space is expanded such that the MLE result in the interior of if neither the upper nor lower bounds can be found. -
v.
Repeat steps ii—step iv to obtain
depending on the generic grid procedure. Continue repeating to form the sequence 
, 
. Stop at 
when 
, a pre-specified threshold. -
vi.
Depnding on the adaptive grid approach, the final MLE is
Remark: The first step selects a large domain 
as the initial grid space, which is expected to contain the MLE. The domain is chosen around the true values of 
in simulation experiments because these true values are known. Practically, an exhaustive search is carried out within the bounds of upper and lower of and . The lower bounds can be taken to be zero if the parameters are positive, which is typically the situation. Subsequently, take 
and 
are two large positive numbers, and build the grid in 
with 
equally spaced marginal grid points as its constituent parts. Given that the only goal is to discover the profile of log-likelihood, the value of needs not to be so large. To visualize the characteristics of the resultant surface, the log-likelihood is assessed and plotted at these grid points. It is preferable to either fix or increase and until the MLE is within the chosen domain, depending on this plot (see74).
When the adaptive grid procedure above yields the first step approximation to the MLE, we can use the MATLAB function “fminunc” as an alternative. The final MLE can also be obtained by minimizing the negative log-likelihood function using a quasi-Newton procedure, and this MATLAB function requires the gradient vector, which is provided by
 |
28 |
where 
is unknown parameter vector, 
and the last step MLE is expressed as
 |
29 |
where 
with defined in (15).
Statistical inference based on MLE
Let’s integrate into the estimation procedure. After getting 
, the MLE of is obtained analytically as follows:
 |
30 |
Let 
is the 
vector of all unknown parameters, including where 
. The information matrix 
can be expressed by:
 |
31 |
The information matrix in (31) can be written in the form
 |
32 |
Now, recall (12), the equation which is mentioned before
 |
By using log-likelihood function and substitution in (32), it follows that
 |
33 |
as well
 |
34 |
and
 |
35 |
where the derived DDEs for the quantities 
, and 
for 
, 
can be found based on the derived DDEs and Euler method is provided in Appendix A.
The observed information matrix, 
, is the information matrix assessed at the ML estimate, 
of 
The inverse of which is computed at the MLE, is as follows:
 |
36 |
The significance level 
for any element of 
, say, 
, for 
the margin of error estimate is needed. The estimated standard error of the MLE, 
, of is defined as:
 |
37 |
where the variance matrix, 
, is given in (36). The substituting from (33) into (36) gives the explicit terms of the covariance matrix. The confidence interval for is
 |
38 |
where 
norminv 
, 
is the required level of significance and 
is the margin of error. Similarly, confidence intervals for the components of 
can be found. In certain cases, we can see that some negative values are contained in the estimated confidence interval in (38). Here, the parameter is transformed logarithmically. After this log-transformation, the confidence interval for is
 |
39 |
as was explained for the parameter 
in the previous section.
Numerical solution and results
In this section, the simulation results for two examples of DDEMs in the multivariate setting with single and multiple delays are discussed and presented, respectively, based on the proposed developed methodology in Sect. “Proposed methodology” as follows:
Numerical experiment for delayed SIR model
We numerically solving the delayed SIR model in (8) by using the dde23 (MATLAB function) with parameters of fixed values at the positive constants 
which give a Siamak’s model as in34, the true values of the delays 
, the initial values 
and the standard deviation 
. At discrete time intervals, the sample observations 
from the DDE as in (4) are obtained. The starting point of the time intervals is 
whereas the end point is 
and each time interval has width 
. The total number of time points considered is 
equally spaced values. The aim is to estimate 
and based on observations 
. Figure 2 illustrates the numerical solution of the delayed SIR model (8), using fixed values parameters in 
at with and for the initial conditions 
, and 
. Also, Fig. 2 shows the different behavior of based on different parameter specifications 
, 
, 
and . Figure 3 gives the underlying trajectories of the solution from the delayed SIR model in (8) and sample observations. The stars are simulated noisy data by adding noise to the delayed SIR differential equations solutions at time points that are equally distributed in 
at with , and 
.
Fig. 2.
Periodic outbreak of disease and numerical solution of the delayed SIR model (8) in 
with and .
Fig. 3.
Numerical solution of the delayed SIR model (8), using estimated parameter values.
For the adaptive grid procedure, the initial grid space is taken to be 
with 
and 
; for this example as well as for the rest, see the remark that is given in Sect. “The adaptive grid procedure” belongs to the selection of . We choose 
(stopping criteria threshold) to be 0.0001. The adaptive grid results are presented in Table 1 for 
and time points that are equally distributed in at and 
, whereas the quasi-Newton results are given in Table 2. Recall that which is used for calculating the quantities 
and ; see Sect. “The grid procedure” As gets larger, the MLE accuracy increases but at the cost of longer computation times. It has been observed that when is set to 
, the results obtained are already satisfactory regarding their closeness to the actual parameter values of 
, 
and . Then, is taken into account in order to find the information matrix, calculate the parameter confidence intervals, and find the maximum of the log-likelihood function.
Table 1.
The values of 
, 
, 
and 
in delayed SIR model by using an adaptive grid for 
and time points that are equally distributed in at .
|
|
|
|
|
|---|---|---|---|---|
| 10 | − 0.0169 | (4.9994,0.0999,0.9994) | (0.9840,0.9640,0.9960,0.9880,9.9880) | 0.00011047 |
Table 2.
The values of , 
and 
for delayed SIR model with .
|
|
|
 |
 |
|---|---|---|---|---|
| 10 | 0.0169 |  |
(1,1,1,1,10) | (0.9840,0.9640,0.9960,0.9880,9.9880) |
where 
.
The information matrix as for delayed SIR model at is
 |
and variance of 
by using MLE is given by
 |
The 
confidence intervals for parameters 
in delayed SIR model as shown in Table 3.
Table 3.
The confidence intervals for parameters 
and in delayed SIR model.
|
=10 | |
|---|---|
 |
(0.9632,1.0048) |
 |
(0.9210,1.0070) |
 |
(0.9895,1.0025) |
 |
(0.9565,1.0195) |
 |
(4.4384,22.4766) |
 |
(4.9954,5.0138) |
 |
(0.0998,0.1038) |
 |
(0.9847,1.0050) |
 |
10−3×(0.0477,0.1732) |
Comparison
The purpose of this section is to compare the proposed model estimation (8) with that of a previously reported estimation procedure by Siamak et al.36.
First, recall the delayed SIR epidemic model with time delays (8) as in the following:
 |
 |
 |
where , and 
represent the numbers of individuals of susceptible, infective, and recovered at time .
Now, recall the mathematical formulation which be introduced as Siamak’s model36:
 |
 |
 |
40 |
where the initial conditions
, and for 
. It is noted that there are no parameters modeled in Siamak’s model. So, no estimation of parameters is possible by his method (i.e. Siamak’s model doesn’t estimate the parameters , and but estimates the values of and because Siamak’s model implicitly assumes 
, 
and 
). Due to this property of Siamak’s model, it cannot estimate and correctly for general values of , and .
By using generated data from the delayed SIR model (8) where 
, , 
, 
and 
, the delayed SIR model in (8) is numerically solved using dde23 with values of fixed parameters at the positive constants 
, the true values of the delays , the initial values and the standard deviation . Sample observations 
from the DDE as in (4) were obtained at discrete time intervals of width which start at point and end at point corresponding to sampled points 
the aim is to estimate and based on observations . Figure 4 illustrate the numerical solution of the delayed SIR model (8), using values of fixed parameters in at with and for the initial conditions
, and and the different behaviour of according to various parameter configurations , 
, and . Figure 5 shows the underlying trajectories of the solution from the delayed SIR model in (8) and the sample observations. The stars are simulated noisy data by including noise in the delayed SIR differential equations solutions at time points that are equally distributed in at with , and .
Fig. 4.
Periodic outbreak of disease and numerical solution of the delayed SIR model (8) in 
with and 
Fig. 5.
Numerical solution of the delayed SIR model (8), using estimated parameter values.
For the adaptive grid procedure, the initial grid space is considered to be 
with 
, 
and . We set the standard for to be 0.0001. The adaptive grid results are given in Table 4 for , and time points that are equally distributed in at and . Recall that which is needed for calculating the quantities 
and 
. As gets larger, the MLE accuracy increases but at the cost of longer computation times. It is noted that for , sufficiently good results are carried out in terms of closeness to the true parameter values of , , and .
Table 4.
The values of , and in delayed SIR model by using an adaptive grid for 
and time points that are equally distributed in at .
|
|
|
|
|---|---|---|---|
| 10 | − 0.0169 | (4.9991,0.1002,0.9997) | (0.4960,1.0000,1.4880,0.9800,9.9880) |
Now, let give illustration of the poor estimation by Siamak. By using same generated data from the delayed SIR model (8) where , , , and , the Siamak’s model above is numerically solved using dde23 with the initial values and the standard deviation . Sample observations 
from the DDE as in (4) were obtained at discrete time intervals of width which start at point and end at point corresponding to sampled points the aim is to estimate 
and 
based on observations . The initial grid space for the adaptive grid procedure is taken to be 
with 
and 
. The standard for δ was set to be 
. As , the adaptive grid results are given in Table 5.
Table 5.
The values of , 
and 
in Siamak’s model (40) by using an adaptive grid.
|
|
|
|
|---|---|---|---|
| 10 | − 4.4608 | (5.0626,0.0889,1.1050) | (4.4800,6.4800) |
Table 5 showing values of maximum value of log-likelihood (
, 
and in Siamak’s model (40) by using an adaptive grid for generated data from the delayed SIR model (8) and time points that are equally distributed in at .
From the previous discussion, it is noted that the estimated values of 
, 
, are very different from 
(the original values used to simulate the data). This means the general SIR model (8) with , and cannot be estimated using Siamak’s model (40) and method, in which case the estimation is incorrect. However, note that the model and estimation proposed in this paper are able to obtain an accurate estimate of as well as 
.
Numerical experiment for pharmacokinetic models
Model I for intravenous administration
Recall the mathematical formulation in (9) which be introduced as pharmacokinetic model:
 |
 |
with the corresponding initial conditions:
, and 
for for intravenous administration and
, and 
for for oral administration.
The equations above represent two models: model I corresponds to an intravenous single dose and is represented by (9) and (10); model II corresponds to an oral single dose and is represented by (9) and (11).
By utilizing dde23 with fixed values of parameters 
in at with 
and 
or 
or 
, the trajectories of the solutions 
and 
are obtained. As in (9) and (10) for the initial conditions 
and 
, the effect of time delay on pharmacokinetic after intravenous injection of the drug and the various traits of the solution based on the parameter specifications is presented in Fig. 6. Therefore, the parameters are fixed at 
. Figure 7 shows the underlying trajectories of the solution and from the DDE model (9) for intravenous administration, (Model I), and the sampled observations.SECT
Fig. 6.
Variations of and versus time and numerical solution of (9) for intravenous administration (Model I) using fixed parameter values as a) and b) .
Fig. 7.
Numerical solution of (9) for intravenous administration (Model I) using estimate of the parameter.
The stars and squares are simulated noisy data by adding noise to the solutions of two-compartment model with time delay at time points that are equally distributed in at with and or or and initial conditions: a) and b) 
respectively.
The initial grid space for the adaptive grid procedure is considered as 
with 
and 
. The results of adaptive grid are given in Table 6 for 
and time points that are equally distributed in at and and Newton–Raphson procedures are given in Table 7. As in previous example, it is noted that for , acceptable results are accomplished in terms of closeness to the true parameter values of 
, 
and 
. Thus, is considered for finding the MLE function.
Table 6.
The values of , 
, 
and in (9) Model I for intravenous administration by using an adaptive grid for and time points that are equally distributed in at .
|
|
|
|
|
|---|---|---|---|---|
| 10 | − 1.0685 | (100.1167,-0.1216) | (0.5040,0.9880,0.1960,0.9920,0.4960) | 0.0134 |
Table 7.
The values of , and 
for model I with .
|
|
|
 |
|
|---|---|---|---|---|
| 10 | 1.0128 |  |
(0.5,1.0,0.2,1.0,0.5) | (0.4991,0.9905,0.1984,0.9935,0.4950) |
The information matrix as for Model I at is
 |
and variance of by using MLE is given by
 |
The confidence intervals for parameters 
in Model II as shown in Table 8.
Table 8.
The confidence intervals for parameters 
and in model I.
|
=10 | |
|---|---|
|
(0.4789, 0.5211) |
|
(0.9875, 1.0125) |
|
(0.1827, 0.2173) |
 |
(0.9862, 1.0138) |
 |
(0.4877, 0.5123) |
|
(99.7482, 100.4852) |
|
(-0.4820, 0.2389) |
|
(0.0081, 0.0189) |
Model II for oral administration
In the same mannerand by using dde23 with values of fixed parameters in at with 
and or 
or 
, the trajectories of the solutions and are obtained. As in (9) and (11) for the initial conditions 
and 
, the effect of time delay on pharmacokinetic after oral intake of the drug and the various traits of the solution according to the parameter specifications is shown in Fig. 8. The underlying trajectories of the solution and from the DDE model (9) for oral administration, (Model II), and the sampled observations are displayed in Fig. 9.
Fig. 8.
Variations of and versus time and numerical solution of (9) for intravenous administration (Model II) using fixed parameter values as a) and b) 
Fig. 9.
Numerical solution of (9) for oral administration (Model II) using estimated parameter values.
The stars and squares are simulated noisy data by adding noise to the solutions of two-compartment model with time delay at time points that are equally distributed in at with and or or and initial conditions: a) and b) 
respectively.
The results of adaptive grid are given in Table 9 for 
and time points that are equally distributed in at and and Newton–Raphson procedures are given in Table 10. As in previous example, it is noted that for , acceptable results are actually carried out in terms of closeness to the true parameter values of 
, 
and . Thus, is considered for finding the MLE function.
Table 9.
The values of , , and in (9) Model II for oral administration by using an adaptive grid for and time points that are equally distributed in at .
|
|
|
|
|
|---|---|---|---|---|
| 10 | − 1.3643 | (− 0.0874,100.2061) | (0.2,0.2,0.2,0.5,4) | 0.0134 |
Table 10.
The values of , and for Model II with .
|
|
|
|
|
|---|---|---|---|---|
| 10 | 1.1293 |  |
(0.2,0.2,0.2,0.5,4) | (0.1992,0.1993,0.2019,0.4965,4.0007) |
The information matrix as for Model II at is
 |
and variance of by using MLE is given by
 |
The confidence intervals for parameters in Model II as shown in Table 11.
Table 11.
The confidence intervals for parameters and in model II.
|
=10 | |
|---|---|
|
(0.1964, 0.2036) |
|
(0.1980, 0.2020) |
|
(0.1946, 0.2054) |
|
(0.4962, 0.5038) |
|
(3.9710, 4.0290) |
|
(− 0.3231, 0.1484) |
|
(99.9714, 100.4409) |
|
(0.0080, 0.0187) |
Conclusion
The method of maximum likelihood inference was presented for unknown delay parameters as well as other parameters of interest in multivariate setting. As examples, we considered the delayed SIR model with multiple delays and the two-compartment model with time delay (Model I for intravenous administration and Model II for oral administration) and after that, we derived the unknown parameters in these models. A two-step approach that uses a gradient descent technique after an adaptive grid is suggested. Based on the proposed methodology and by using simulation results, the numerical solution and statistical inference on unknown parameters of two examples for DDEMs in the multivariate settings with single and multiple delays are solved and illustrated in graphs and tables. The comparison between obtained results and Siamak et al.36 results has been added and applied to delayed SIR model, example of multivariate DDEMs. Moreover, the likelihood inferential framework was used to find the gradient vector, information matrix, and confidence intervals for each unknown parameter.
Acknowledgements
Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2025R734), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.
Appendix
A. Quasi-Newton procedure
As input to the quasi-Newton procedure, it is required to find the gradient vector, 
, as in (28). Recall that the DDEM has multiple delays. Based on the log-likelihood, 
is defined as
 |
41 |
Denoting 
and 
, the first-order partial derivatives involve derivatives of 
with respect to the parameters are as follows:
 |
42 |
where
 |
43 |
and
 |
44 |
where
 |
45 |
The derivative expressions 
and 
for 
; 
; 
, can be numerically obtained from respective DDEs.
The DDEM was mentioned above can be numerically solved based on this initial conditions and the values of can be obtained from 
for each 
. The case of is similar and already discussed in Sect. “The grid procedure”.
Equations (42) and (44) also involve 
and 
. Recall (16), which is:
 |
46 |
By differentiating (16) regarding and , respectively, it fallows that:
 |
47 |
thus
 |
48 |
Similarly, from differentiating (16) with respect to , it follows that
 |
49 |
and hence,
 |
50 |
For the derivatives of l regarding its arguments, the explicit expressions are given as in the following:
For the DDE in (5) with 
and 
. The first-order partial derivatives of 
are as follows:
 |
51 |
 |
52 |
 |
53 |
These partial derivatives of can be numerically obtained from the derivative expression of 
and 
for 
since each of them form an additional DDE derived from (5) by differentiating it regarding the quantity of interest. Further it is obtained that
 |
54 |
Again, differentiating regarding the arguments of , the second-order partial derivatives will be on this form
 |
55 |
 |
56 |
 |
57 |
 |
58 |
 |
59 |
 |
60 |
 |
61 |
 |
62 |
 |
63 |
and
 |
64 |
Equations (58, (59), (63) are necessary for the quasi-Newton procedure.
B. Information matrix
The information matrix, 
can be obtained form
 |
65 |
Taking negative on the LHS of Eqs. (55)–(64), followed by expectation under the sampling distribution of each 
, it is noted that
 |
66 |
or
 |
67 |
on the RHS become zero since the expectation of equals 
Hence, it follows that
 |
68 |
as in (33). As well
 |
69 |
Finally, taking expectation in Eq. (64)
 |
70 |
Author contributions
Authors have worked equally to write and review the manuscript.
Data availability
The data that supports the findings of this study are available within the article.
Declarations
Competing interests
The authors declare no competing interests.
Footnotes
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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Data Availability Statement
The data that supports the findings of this study are available within the article.