key: cord-0301536-pl17r7h2
authors: Bartley, D.
title: Comparison of Zeroth-Order Deterministic Local and Delay Pandemic Models
date: 2020-08-07
journal: nan
DOI: 10.1101/2020.08.05.20162164
sha: ff0b5ceaf11d04479450a66d580f8167edd5ba86
doc_id: 301536
cord_uid: pl17r7h2

Delay differential equations are set up for zeroth-order pandemic models in analogy with traditional SIR and SEIR models by specifying individual times of incubation and infectiousness prior to recovery. Independent linear delay relations in addition to a nonlinear delay differential equation are found for characterizing time-dependent compartmental populations. Asymptotic behavior allows a link between parameters of the delay and traditional models for their comparison. In analogy with transformation of the traditional equations into linear form giving populations and time in parametric form, approximation of the delay equations results in a simple accurate finite recursive solution. Otherwise, straightforward numerical solution is effected in terms of linearized boundary conditions specifying the distribution of instigators as to their initial infection progress--in contrast to traditional models specifying only initial average infectious and exposed populations. Examples contrasting asymptotically-linked traditional and delay models are given.

The Covid-19 epidemic has aroused great interest in pandemic modelling. This paper addresses two types of compartmental deterministic models. Such models compute the evolution of various populational compartments following initial infection Considered here are the classic SIR [7] and SEIR Models which address Susceptible, Exposed, Infectious, and Recovered (including deceased) populations. For a summary and details of these models and their many variations, see [8] and references within. Most [2, 6, 7, 8] of these models are expressed in terms of ordinary nonlinear differential equations local in the time.

Less common are models [1, 5, 10, 11, 12] which specify the rates of change in the populations at a given time in terms of populations at an earlier time. Specifically, this paper considers a pandemic model similar to the traditional SEIR Model (including SIR) constructed by specifying the incubation time ! (for example, 5 days) an individual spends prior to becoming infectious and the total time " (e.g., 20 days) to recovery. The traditional instantaneous differential equations of the SEIR Model are replaced by delay or functional nonlocal nonlinear differential equations [3] solvable numerically in terms of the prior history of the initially exposed or infectious. Since in reality, both ! and " may vary widely between individuals, this type of model, together with the instantaneous or local models, must be considered 0-th order approximations.

The plan for this paper is as follows. First, the delay differential equations are presented, together with various simple relationships among the populations. Asymptotic limits are given. Comparison of the instantaneous and delay models is possible by linking the parameters of the two types of models together requiring asymptotic coincidence. The models are also comparable through the finding of an accurate solution to the delay SIR model in analogy to solution of the usual instantaneous model with populations and time in parametric form from now linear equations, obviating addressing the differential equations numerically. Boundary conditions for the delay models are established that permit specifying the initial distribution of the pandemic instigators in terms of their initial disease progress, unlike the instantaneous models. Finally, several numerical comparisons are made.

As with the instantaneous SEIR Model, a function new[t] approximates the rate of new infections per unit time by:

where is a rate constant, i[t] is the number of infectious individuals at t, s[t] is the number of those susceptible to infection, and n is the total number of individuals. Following exposure, suppose that an incubation time ! is required until an individual becomes infectious and that a further time interval ( " -! ) remains until the individual either recovers or dies.

Suppose the pandemic begins at t = 0 with the introduction of a small number of exposed or infectious individuals. By the time t = " , all the instigators will have recovered (or died), after which the following conditions hold. The s[t] susceptible individuals at time t > " consist in all present ( − [0] − [0]) at t = 0+ minus those who have become infected after t = 0. Therefore,

At time t, the e[t] exposed individuals are all who suffered new exposures between t − ! and t. Therefore,

Similarly, the i[t] infectious individuals at t (> " ) consist exactly in those who suffered new exposures between t − " and t -! . Therefore,

At t > " , the r[t] recovered or deceased individuals at t are those who suffered new exposures before t − " plus the ( [0] + [0]) initially exposed or infectious. Therefore,

The above equations imply, of course, that:

Differentiating the above equations with respect to time t results in an equivalent set of non-local non-linear differential equations:

[⟹ 

The equations are valid for t > " and depend for solution on the functions i[t] and s [t] given over the interval, 0 < t < " as the initial values, unlike the boundary conditions at a single point with ordinary differential equations, hence the aptness of the term, functional differential equation. As detailed below, the boundary conditions, i[t] and s[t] over this early interval, are determined by the distribution in the exposure times of the initially exposed or infected who trigger the pandemic.

The above equations are analogous to the local SEIR Model in widespread use:

, [ ] = SEIR:

In the case of the instantaneous model, Equations (1 and 4') imply that -.

which satisfies s[0] ~ n at r = 0 (at t = 0).

With the less familiar delay model, Equation (7) holds only asymptotically (at t → ∞ or 0). In addition to

there are two independent nonlocal relations among s, e, i, and r. At t > " , new infections created at t result in new recoveries at t + " , so the recovery rate must lag behind the creation rate by " . Explicitly, Equations (3 and 6) imply that

where the integration constant n is determined by the → ∞ limit. Equations (4 and 5) then imply:

Similarly, infectiousness lags infection by ! , and Equations (3 and 4) give:

as the → ∞ limit ⟹ the integration constant = 0. As easily seen, the above relations hold for the simpler SIR Models as well (setting e[t] and ! equal to zero). Equations (2, 3, 8, and 9) together with boundary conditions are then sufficient to determine the evolution of the four populations.

Asymptotic limits SEIR:

Another similarity between the delay and instantaneous model is found in the asymptotic limits ( → ∞) of s[t] and r[t]. As → ∞ , i[t] and e[t] ⟶ 0, and so

In the familiar case of the local SEIR model [6, 9] , Equation (7) implies: Delay SEIR:

The Delay SEIR Model is somewhat more complicated, but results in an expression of the same form as Equation (10) . One approach is to expand the various functions in " and ! . Equation (8) implies:

Equations (9 and 6) imply:

Therefore, keeping only terms linear in ! or " ,

Equations (1 and 3) then give

which integrates to:

Finally, in the limit t → ∞, where i and e → 0,

Again, the asymptotic value 1 is given in terms of the Lambert W function.

In the sections below, the delay and instantaneous models are placed on an equal footing for comparison by identifying the infectious-time constant (7 with ( " -! ), the time an individual is infectious, and the exposed-time constant (7 with ! , i.e., with the means of the corresponding exponential distributions. Also, as indicated by Equations (10 and 10'), the former identification forces the asymptotic limits to be identical. 

If the RHS is approximated as 0 th order & [ ] as,

Equation (3') is a neutral delay differential equation (i.e., with delay in the derivative [3] ) which integrates to

where the constant is determined by initial conditions, avoiding discontinuities [3] in Although the above delay differential equations are non-linear, near the start of a pandemic with a small number of infected individuals, a linearized version of the equations is extremely accurate. The corresponding solutions for t < " are simple and transparent. Furthermore, superposition makes possible the combination of the initially infected as distributed over a range of infection stages. Therefore, the boundary conditions (for time t < " ) that reflect the history of the initially infected are easily determined and permit solution of the nonlinear delay equations at t > " using the same numerical methods as with instantaneous nonlinear equations.

The earliest times in a pandemic are special in that only the initially infected can recover at t < " , within which explicit solution of s[t] is possible. For example, with the Delayed SIR Model, as long as no recovery occurs by time t,

These equations are easily solved in terms of the logistic function, since i + s + r = n (and r = 0 and n are constants). However, for the reasons outlined above, at t < " the linearized equation is adopted: Then following recovery of the initial infection at t = tr,

In other words, Incidentally, the density [ > , ] at later times (t > 2 " ) of the infectious vs the length of time > infectious (i.e, " − > = tr is the time to recovery) is given by:

which is justified by:

Any "old" case prior to − " has "recovered" by the time t; Equation (15) expresses the fact that only the new infections need to be considered for the density at time t. The boundary conditions for the delayed SEIR Model can be expressed similarly to Equations (13), but are somewhat more complicated and are often more simply determined by numerical solution.

The differences between the delay and local models are best illustrated by means of numerical examples. Parameters for the above models were selected as follows:

= 0.20 Note the identical asymptotic behavior of the populations in accordance with Equations (10 and 10'). Also, the local curves lag behind those of the delay model, by 1-2 months. The curves for the new cases/day attain maxima at 87 days and 118 days for the delay model and local model, respectively. This lag is not surprising as the probability of continued infectiousness of an individual falls off as ('/) " for the local model, whereas the probability remains equal to 1 from t = 0 until recovery at t = " . Also, note the nearsymmetry of the curves from the delay model and close relation to the logistic function.

At the opposite extreme from an even density, calculations were also done for sharply peaked densities at a variety of initial recovery times of the instigating individuals. The results for the SIR models are shown in Fig. 4 . The time required for attaining a maximum (as in Fig. 3 ) in the new cases/day was determined for each initial recovery time. The curve for the delay model is remarkably flat, despite upward turning for the nearly recovered instigator, and, of course, the approach to infinity in the limit of an initially recovered individual. The difference between local and delay models is close to that of the even distribution over the initial recovery times, despite the nonlinearity of the equations. In this case, the linearized boundary conditions were determined numerically. This required attention to the three separate situations: tr < ! , ! < tr < " − ! , and " − ! < tr < " where the initial instigators were only exposed prior to becoming infectious. Again, the increase in the time to maximum in the new cases/day on approaching nearly recovered instigators is apparent, yet is limited to the lowest few days prior to recovery. The difference between local and delay models is similar to that of Fig. 4 for the SIR model, although time from pandemic start to maximum in the new cases/day is naturally longer for the SEIR model with time required for infectiousness to begin.

Similarities and differences are found between basic simple deterministic pandemic models-instantaneous vs delay. Both types of models expressed in terms of differential equations are readily addressed using established techniques of numerical analysis. Interestingly, though nonlinear, both can be expressed in terms of accurate closed-form solutions. Just as the instantaneous equations are very simply linearized in terms of population compartments and time in parametric form, the delay equations admit a simply-evaluated recursive solution. Further research into this solution is merited, for example, explaining the accuracy attained over a wide range of model parameter values.

Both model types share the functional form of asymptotic values relevant to the pandemic winding down. This allows linkage between model types for comparison. Equivalently, parameters can be chosen to equate growth at the pandemic start.

A difference between the models exists in the form of the initial boundary conditions. The delay equations depend on an initial function. This function can be expressed in terms of the distribution of pandemic instigators as to initial disease progress. This expression is facilitated by adopting accurate linear boundary conditions permitting superposition even though the equations valid during the progress of the pandemic are nonlinear. The results of this work show significant difference between the usual local and the delay models between the start and death of the pandemic. The difference no doubt relates to the individual's probability of remaining infectious. With the usual local model, this probability falls rapidly as ('/) " or ('/() " () ! ) where t is the time from the start of infectiousness. In contrast, with the delay model considered here, the probability of infectiousness lasting until time t remains high at 1.00 until the time " or " − ! is reached.

This difference may be significant to the progress of a pandemic. The ultimate result as the pandemic winds down is identical for the two types of models considered here. Of course, it is the progress up to the time of the maximum in the new cases/day that is necessary to understand in order to adopt appropriate mitigating measures.

These results may provide help in choosing between the models. From the point of view of calculation, the delay model is not significantly more difficult to analyze. Which model is appropriate depends on the details of recovery or death of the individual following infection.

A Delay Differential Model for Pandemic Influenza with Antiviral Treatment

Mathematical epidemiology: Past, present, and future

Functional Differential Equations

Exact analytical solutions of the Susceptible-Infected-Recovered (SIR) epidemic model and of the SIR model with equal death and birth rates

An epidemiological model with a delay and a nonlinear incidence rate

The Mathematics of Infectious Diseases

A Contribution to the Mathematical Theory of Epidemics

Modelling and simulation of COVID-19 propagation in a large population with specific reference to India

A note on the derivation of epidemic final sizes

Delay equation formulation for an epidemic model with waning immunity: an application to mycoplasma pneumoniae

A modified discrete SIR model

Stability of a delayed sirs epidemic model with a nonlinear incidence rate