The sick population fraction (y) is considerably smaller than either the naive (x) or immunized (z) populations (o(10−5)). Thus in the equation for z, the term ψy has a negligible effect. Furthermore, the illness probability in the immune state (z) is much smaller than in the naive state (x). We thus also neglect λ′σ(T)z in the equations for y and z. If indeed λ′ << λ (by at least a 2-order factor) then the removal of this term has no effect. If, on the other hand, λ′ ≈ λ, then the only way to reduce listeriosis is by reducing the exposure to L. monocytogenes; in that case, the results presented in the current manuscript would not be valid. Finally one can ignore the −λσ(T)x in the equation for x (but not in the equation for y), since λ << λv, again since the listeriosis rate is of the order of o(10−5). One is thus left with a simplified model (all the terms with no brackets in Fig. 1):
Given the above-mentioned assumptions, the solution of the simplified and original model are similar, as is shown in the results section.
The equations are solved for the lifetime of a patient from birth to death (0 → Tend). Typical listeriosis frequencies are of o (10−5 to 10−6). We thus set λ << 1. The total probability of a person to be sick during his lifetime, independent of the outcome of the disease, isL(λ) = ʃTλσ(T)x(λT)dT. Since x and z are not a function of y, one can solve their dynamics without taking y into account and obtain:
For this simple case, we can compute L(λ) analytically and numerically.
Given the lifetime axis of an individual T:0 → Tdeath, and the year of birth tbirth, the life of an individual can be adapted to an absolute time—t through T = t − tbirth.
A model more realistic than the simple model presented in the previous section (and therefore more detailed) can be computed using the following functions: λ(t), σ(T), N(T, t). λ(t) represents the exposure probability to L. monocytogenes in a given year t. σ(T) is the probability of being sick following an exposure at a given age T, and N(T, t) is the population at age T in the year t. Using these functions, one can compute the listeriosis frequency in a given year L(t) (which is different from the total listeriosis probability of an individual L(λ)) as:
We here assume that N(T, t) = N(T). In other words we assume that the age distribution of the population has not changed drastically in the last few years. We then normalize N(T) → N(T)/NTotal;ʃTN(T)dt = 1 to obtain: L(t = ʃTN(T)σ(T)x(T, t)λ(t)dT. This value is compared with the listeriosis frequency in the population in a year t.
To build the argument, we start the analysis from very simple functions and then, step by step, elaborate these functions.
At the first stage we assume the simplest model where σ(T) is a step function after 60 years,
, N(T, t)=const, λ(t)=const. This model can be fully solved analytically.
At the second stage, we maintain the assumptions on σ(T) and N(T, t), but vary λ(t) either as a step function or as a linear decrease, ie,
The possibility of a sharp and significant reduction in λ is realistic. There was a substantial shift in hygienic and storage conditions during the 1950s and the 1960s with the purchase of refrigerators in England that resulted in a significant change in parameter space.19 Hence, λ(t) depends on the absolute chronological time (t).
At the third stage, we include a complex population based on recent population estimates for England and Wales.20 We determine N(T) as a decay linear function: N(T) = 400 − 20(T − 60)(thousands) fitted to the observed data, and the observed values of σ(T).11 We ignore at the current stage the population over age 80 since it contributes a negligible number of listeriosis cases.
Finally, a realistic model is designed that enlarges the generic features observed in the simplified model. It considers a detailed adaptation of the simplified model to the observed data.
If one assumes that boosting may be required to maintain the immune state, a slightly more complex model can be built in which another state (w) is introduced. This model can be written as:
This additional state represents people originally immunized (by exposure to L. monocytogenes), who will become immune again following a boosting of the immune system, with a rate of λB. The probability of losing immunity is again β but, in this model, the transition from the immune population (z) is to the postimmune population (w) and not to the naive population (x). This model can again be solved using the simplifying assumptions mentioned above:
We show below that, as long as λ′′ ≤ λ, this model reproduces the maximum listeriosis frequency for intermediate values of λ, and it can actually produce the same results even for values of β higher than in the 3-compartments model.
To test the possibility that a reduction in the exposure rate to L. monocytogenes can indeed lead to an increase in the rate of listeriosis, we first solved a simple model [Eq. (2)]. Here we assumed that both the probability of immunization (λv) and the probability of being sick (λ) are constant and are actually a function of the exposure rate to L. monocytogenes. One can thus write λv = ελ and compute the listeriosis rate (L(λ)) as a function of λ. If one assumes that immunization is lifelong (β = 0), the model shows a maximal incidence of listeriosis for intermediate values of λ (Fig. 2). Increasing or decreasing λ toward this maximal value increases listeriosis incidence. For very low values of λ, the population is not immunized, but the L. monocytogenes prevalence is low enough to induce low disease levels. If, on the other hand, the L. monocytogenes prevalence is very high, by the age of 60 years, practically all the population is immunized and old-age listeriosis rarely occurs despite high exposure to L. monocytogenes. There is an in-between range where not all the population is immunized by the age of 60 years and exposure to L. monocytogenes is high enough to induce disease in parts of the population. The reduction of L. monocytogenes prevalence in many European countries may have led to this situation. Note that one can relax the assumption that immunity is life-long. However, the maximal listeriosis rate at intermediate exposure to L. monocytogenes is valid if immunity is long-lived (at least 20 years; β = 0.05 [Fig. 2 inset]). The ratio ε = λv/λ determines the worst value of λ (Fig. 2). This ratio is proportional to the ratio between the L. monocytogenes dose required for immunization and the dose eliciting disease in the susceptible population. The model at this stage contains only 2 parameters: β, ε; all other parameters have no effect on the model results. A maximum incidence is obtained for all values of ε.
Chronological Increase in Listeriosis
The generic mathematical model shows that, assuming a constant prevalence of L. monocytogenes, there is a value maximizing listeriosis. Raising or lowering the prevalence toward this value merely leads to a higher incidence. To further examine our specific hypothesis, we then considered the evolution from a high L. monocytogenes prevalence to a lower value as a result of a sharp drop in exposure to the pathogen following the utilization of new refrigeration technologies in the 1960s.17 Immediately following the decrease in exposure, the susceptible population is still immunized but is substantially less exposed to L. monocytogenes. The incidence of listeriosis thus drops sharply (thick solid line and triangles in Fig. 3). Then, for a long period, a part of the population slowly loses its immunity, and eventually this part of the population gets old (30–40 years later, ie, now) and becomes susceptible to clinical listeriosis. Following the argument presented in Figure 2, listeriosis frequency in the population can then increase even beyond its prior level, until stabilized at a new high value (thick solid line and triangles in Fig. 3).
The shape of the curve depicted in Figure 3 is generic and is not sensitive to the specific parameters used. The observed behavior is the result of 2 contrasting mechanisms; the immediate sharp drop in listeriosis following the decrease in L. monocytogenes, and the increase in the steady state of listeriosis incidence following the same drop in the exposure rate, and thus in the overall level of population immunity. As long as these 2 mechanisms occur, the same behavior will be observed. It can be further noted that if one starts to measure listeriosis after the initial drop, one would observe only the rapid increase as is actually the situation now. Similar results are obtained if one uses a gradual decrease over forty years in the exposure to L. monocytogenes [Eq. (5), dashed lines and circles in Fig. 3], and if one uses a realistic age distribution (dotted-dashed line and squares in Fig. 3) and a realistic function of the illness probability—σ(T)11 (thin dotted line and diamonds in Fig. 3).
We have simplified the model and neglected many elements. To validate that the simplifying assumptions have no important effect on the results, we have reproduced the curves in Figure 3 using the full model. As can be seen, there is practically no difference between the end result of the simplified model (lines) and the full model (symbols).
A more realistic model reproduces the features observed in the simplified model. Our realistic model contains a linear decrease in the L. monocytogenes exposure rate—λ(t), a linear decrease in the population after the age of 60 as observed in the population20—N(T, t), and the experimentally measured values of listeriosis probability—σ(T).11 Using realistic values for all of these functions reproduces the observed sharp rise in listeriosis cases (Fig. 4) for many arbitrary parameter sets. This model predicts, that even if no measures are taken, listeriosis incidence will reach a new plateau within a few years, with values higher than the ones obtained before the reduction in the exposure rates to L. monocytogenes. The long time lag between the decrease in L. monocytogenes results from the late onset of listeriosis in the elderly.
A similar model can be developed for pregnant women. The only difference between the model for pregnant women and for the elderly is the age distribution of disease susceptibility σ(T). While old persons start being sensitive around age 60 and keep increasing their sensitivity, pregnant women are sensitive between the ages of 20 and 40. We have therefore developed a model in which the disease susceptibility σ(T) is constant between the ages of 20 and 40, and zero otherwise. This model, as in the case for the elderly, has a peak listeriosis frequency in the middle range of exposure rates (data not shown). The main difference between the elderly and pregnant-women populations is that changes in the exposure rate to L. monocytogenes affect pregnant women faster than elderly. Therefore, pregnant women reach a steady state much faster, and are expected to show an increase in listeriosis long before the elderly. We may now be in a period after a new steady state has been achieved, and may not be able to see a reduction in the listeriosis rate among pregnant women (Fig. 4, dashed line).
The model presented up to now has only 3 possible states. One could in principle assume that, to maintain immunity, the immune system has to be boosted by a de novo exposure to L. monocytogenes. We have developed a model to include the possibility of boosting. A new compartment of postimmunized population was introduced with a lower probability of being sick in the postimmunized state. This extended model has 4 compartments: naive, immunized, postimmunized and sick. We compute again the total number of listeriosis cases as a function of the exposure rate to L. monocytogenes—L(λ) or the total number of listeriosis cases in a given year—L(t). The introduction of a fourth state allows us to relax the requirement that the immune memory is eternal by adding a requirement that the illness probability following exposure is substantially higher in the first exposure than in the exposure following an immune period. A maximal listeriosis frequency at middle values of the exposure rate λ can be obtained even if immunity lasts for only 10 years (Fig. 5). The inclusion of this additional compartment replaces the requirements that immunity lasts for many years by the possibility of a graded-illness probability.
The interplay between pathogen exposure and the resultant development of protective immunity plays a strong role in determining the incidence of infectious diseases. In diseases associated with old ages there is a balance between the immunization obtained at younger ages and the pathogen prevalence. In the current study we discussed the influence of this balance on the old-age morbidity of listeriosis which is a noncontagious infection. We considered here the effect of hygienic improvement on the incidence of listeriosis, a food-borne infection affecting pregnant women, immune-compromised patients, and the elderly. However, the model we have constructed may be implemented for noncontagious infectious diseases in general (eg, food-borne diseases or vector-borne diseases for which human are considered dead-end hosts).
We have provided support for the hypothesis that the increase in listeriosis incidence observed in many countries in Europe in the last decade is explained by a reduction of exposure to L. monocytogenes a few decades ago due to changes in hygiene and food storage conditions.
Understanding the nature of immune response to L. monocytogenes is of high importance for assessing the validity of our model. Two important aspects are the existence of long-term immunity after infection by L. monocytogenes, and cross-protection by various L. monocytogenes strains. Studies in mice have shown that a long-term protective immunity was elicited after infection by L. monocytogenes,4,7 and have also shown cross-protection among the various strains.21 However, data are available only from animal studies. Furthermore, in most of the existing studies the infection route is parenteral, while natural infection occurs through the oral route. The nature of immunity acquired in humans by oral exposure to L. monocytogenes should be further studied. We have showed that our results are less sensitive to the length of the immune period if a boosting effect is assumed. However, there are no available data to show that such an effect exists.
So far as we are aware, none of the risk assessments of listerosis consider the possibility of acquired immunity elicited by exposure to L. monocytogenes. About 1 in 15,000 servings of ready-to-eat food contains very high doses of L. monocytogenes (approximately 105 colony-forming units of L. monocytogenes in 1 g of food).22 These levels have been associated with several outbreaks of listeriosis. Taking into consideration that in the United States the number of ready-to-eat food servings consumed by high-risk populations is estimated to be around 1010 per annum,23 the incidence of listeriosis should have been orders of magnitude higher than the current incidence of about 2500 cases a year.24 A possible explanation for this relatively low incidence of listeriosis may be in the immunity acquired through previous exposure to the organism. Thus, taking into account this fact and the data on immunity to L. monocytogenes, we believe that our explanation for the current upsurge of listeriosis observed among the elderly is plausible and should be considered when analyzing the current trends in incidence of this highly important disease.
Nevertheless, there are no longitudinal data on exposure to L. monocytogenes, and obviously no direct measures have been performed to reveal what is the exact level of exposure that leads to the development of immunity and what level leads to listeriosis. While our results show the plausibility of the suggested scenario in a reasonable parameter range, they cannot prove this assumption. Studies to examine the existence of cellular immunity to L. monocytogenes in various age groups, as well as prospective studies to correlate surrogates of immunity with actual protection from disease, will contribute significantly to assessments of exposure and immunity to L. monocytogenes, and to a better understanding of the dynamics of listeriosis in the population.
We thank Ran Nir-Paz (Department of Clinical Microbiology and Infectious Diseases, Hadassah Hebrew University Medical Center, Jerusalem, Israel) for taking part in the brain-storming during model construction.
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