Expression as Differential Equations

Figure 10.1 defines some basic pathways between the compartments. Using these definitions, a set of differential equations can be written describing the rates of transition of the segments of the population:

(10.1)

Susceptible individuals (N) are converted to infecteds (I) at a force of infectivity β(t). This would be a function of the instantaneous exposure to pathogens.

The symptomatic and asymptomatic individuals convert to the postinfected state. Two rates R(t) and S(t) are indicated here: the number of symptomatic infected individuals per unit time and the number of asymptomatic individuals per unit time who enter the postinfected state, respectively. It is assumed that there is only a single postinfected state; particular illnesses may require multiple postinfected states.²

The function Q(t) describes the rate of conversion of infected people to either symptomatic or asymptomatic infectious states—this is the incubation time for the effect. Incubation time distributions describe the fraction of persons who first become ill at a given time after the initial exposure. However, we first need to define the number of individuals who are on a route to become infected. In a given population at risk, N will be the time-dependent number of persons who are potentially at risk. The parameter β(t) is defined as the instantaneous rate of infection at time t. In other words, N(t) β(t) is the instantaneous rate of persons entering the “pool” of persons who will ultimately become either symptomatically or asymptomatically infected.

The general incubation distribution, F(t), is defined as the cumulative fraction of persons who become ill on or before t days after sustaining an exposure, among those who will become ill. This can be differentiated to define the density function of incubation times, f(t). Then, the instantaneous rate of illnesses at time t can be written as the convolution:

(10.2)

The constant λ is the fraction of infected individuals who become ill, that is, a morbidity ratio (and note that 1 − λ is the fraction of persons who are infected but not becoming ill). This equation is derived by considering that the quantity β(τ)N(τ) represents the instantaneous rate (number/time) of newly infected persons who enter the infected state at time τ. Of these, a fraction λ will ultimately become ill. These individuals then will enter the diseased state with a frequency distribution f(t − τ), where t is the time of entry into the diseased state. Hence to determine the total number of individuals who enter at time t, it is necessary to integrate over those who entered the pool of infected individuals at all prior times. For the simple case of an instantaneous exposure at time 0, this integral recovers the result that the rate of new illnesses is equal to a constant multiplied by the incubation time itself.³ In Equation (10.2), N(t) is the number of persons at time t who are exposed to the risk of the infectious agent. This framework allows us to estimate the rate at which new cases appear (which is given by dY/dt) as a function of the instantaneous rate of infection, the proportion of ill persons, and the incubation time distribution. This does not allow us to predict the total number of ill or asymptomatically infected individuals in the population at any time, since we have not yet incorporated description of the duration of illness or residence in the asymptomatically infected state.

As a simple example, consider an incident in which exposure occurs over a 1-h period (e.g., a single meal). The instantaneous rate of infection can then be modeled as a square wave (assuming the exposed persons ate uniformly over the 1 h) as follows (assuming a particular microbial level and thus transformed into an instantaneous rate of infection):

(10.3)

For example, if the infectious organism had the characteristics of a Salmonella typhimurium, then the incubation curve might be lognormal with a median of 2.4 (57.6 h) with a 16th percentile of 57.6/1.49 = 39 h based on the analysis by Sartwell [2]). Using the nomenclature of Chapter 6, the parameters of the lognormal distribution would be (with time in hours) ζ = ln(57.6) = 4.05 (computed from the median) and δ = 0.388 (computed from the lower percentile).

Using these lognormal parameters, Equation (10.1) can be integrated numerically as a function of time.⁴ For simplicity, we will take N to be 1000 persons; in other words, 100 persons have consumed the microbially laden meal. Integration of these equations, which represent an integro-differential equation system, with initial values given (e.g., N = 1000 at t = 0, and I = X = Y = 0 at t = 0) must be performed numerically. For the examples of epidemic models in this chapter, we have used the differential equation solvers in MATLAB and modified the program to evaluate the integral in Equation (10.2) during each call of the integrator by the trapezoidal rule.

For the above assumptions, this gives the results in Figure 10.2 for the instantaneous rate of cases occurring, for the total number of infected persons at any one time, and for the cumulative number of cases. Note that the number of total cumulative cases is simply the integral of the product of β(t), λ, and N(t) with respect to time; and in the example, this is simply equal to 1000 × 0.02 × 1 × 0.5 = 10.

Clearly, complex shapes for the disease attack rate can be derived if, for example, the infection rate is complex. Consider the proceeding scenario, with a slight change: rather than a single 1-h exposure from 0 to 1 h, there are two 1-h exposures, at 0–1 and 24–25 h (e.g., a contaminated meal on each of two consecutive days). Figure 10.3 presents the results of this scenario. Comparison with Figure 10.2 shows that the curve of attack rate is displaced somewhat to the right and is also broadened. The latter finding reinforces the point made earlier namely that the breadth (or variance) of the incubation time distribution observed in an outbreak is influenced by both the intrinsic multiplication of pathogens in vivo, as well as the exposure of the susceptible population to the pathogens. Hence, to determine the intrinsic incubation time distribution from epidemiological data, it is necessary to remove the effect of the exposure distribution. This “inverse” problem is considerably more difficult; however, it is nonetheless solvable.

The integral of Equation (10.2) defines the instantaneous rate of new cases. We may be interested in, or may have data on, the total number of cases at any one time. Hence, we need to extend the model to consider the length of time individuals spend in a disease state.

If functions for R(t) and S(t) are available, representing the exit rate of individuals in symptomatic and asymptomatic states, it then becomes possible to solve the full model, including X and Y, at any one time. The function R(t) is dependent upon the duration of illness.

If we define the cumulative probability distribution function, G(t) as the fraction of persons who exit the disease state on or before time t, and if the corresponding density function is g(t), then by analogy with Equation (10.2), the rate of exit from the disease state can be defined by:

(10.4)

Information on g(t), at least in a crude sense, is available in standard references [3], typically given as a modal value and a range. In the absence of other information, therefore, a triangular distribution might be used to model this distribution.

Information available to support a determination of S(t) is less available. The loss of asymptomatic infections has rarely been measured. One such measurement, in the case of rotavirus infection from children exposed via the day-care environment, has been reported by Pickering et al. [4]. This study measured the excretion of rotavirus by children subsequent to the acquisition of disease. This is not directly a measurement in support of S(t), since it pertains to the loss of infection from initially symptomatic individuals. However, it might be reasonable to presuppose that the loss of infectivity in symptomatic individuals is similar to that in asymptomatic individuals. Figure 10.4 summarizes measurements versus time postillness, compared to a simple exponential fit. For this data set, an exponential loss of infectivity provides good agreement with the data. Hence, if we define H(t) as the cumulative fraction of asymptomatic individuals who convert to a postinfected state within t days of becoming infected, and h(t) is its derivative, then we would for this particular case take h(t) to be an exponential function, that is, h(t) = [exp(−0.24t)/0.24].

Then we can compute S(t) by the following integral:

(10.5)

Note that this formulation might be modified if we assume a different incubation time distribution for the asymptomatic state than for the symptomatic state.

With the above definitions of Q, R, and S, we can define formal mass balances corresponding to the model in Figure 10.1. By considering the rates of exit and entry into each of the boxes, and equating time derivatives to the net rates of exit and entry, the following five differential equations describe the system dynamics:

(10.6)

This system can be treated as an initial value problem, since in the usual case, the values for population size in each of the compartments will be known or assumed at time 0 (i.e., N = N₀, X = Y = Z = 0). However, given the fact that Q, R, and S are integrals (or multiple integrals—that is, R and S involve integrals of Q, which in turn is another integral), the problem is not a simple differential equation system, but rather a system of integro-differential equations, which must be solved numerically.

Figure 10.5 is the results of computing the model given by Equation (10.6) using the incubation time distributions assumed in the earlier examples. A time-variable attack rate (upper panel of Figure 10.5) over the first 20 h is assumed. The distributions for duration in the symptomatic state (density function g) and asymptomatic state (density function h) are both assumed to be lognormal with respective means of 48 and 36 h, and respective standard deviations of 10 and 72 h. The value for λ is assumed to be 0.2. Note that the maximum number of symptomatic and asymptomatic individuals is described (about 100 and 20, respectively).

There are a number of advantages to the use of these types of models, although there is a substantial data gap with respect to the parameters (and distributional forms) that are included. First, by depicting the time profile of cases (and asymptomatic individuals), it may allow a more precise estimation of impacts (economic and otherwise) from community level spread of infectious disease—particularly if the impact is related both to the number of cases and to their duration. Second, as will be illustrated in the following text, a particularly unique aspect of microbial infectious agents (as compared to chemical agents) is their ability to induce secondary cases—additional infections and illnesses in persons who were not exposed to the contaminated water, food, etc. A modest extension of the framework of Equation (10.6) and Figure 10.5 allows formal quantitative consideration of this impact.

Stochastic Approach

In our modeling approach, we have deliberately chosen a deterministic formulation to investigating dynamics. In other words, the number of cases and their changes are described by solutions to coupled (integro-differential) equations. This implicitly treats number of cases as a continuous variable. This is a reasonable assumption if the total population, as well as the number of cases, is reasonably large. However, for small numbers of cases, it is necessary to reformulate the problem in stochastic terms.

A full stochastic model treats the time evolution of probabilities that the “compartments” (boxes in Figure 10.1) will assume a set of particular numerical values. For example, p_i,j,k,l,m would be the joint probability that the following conditions would all simultaneously be true:

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