Modeling polio as a disease of development

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Abstract

Poliomyelitis is a disease which began to appear in epidemic proportions in the late 19th century, paradoxically, just at the time when living conditions and developments in health were transforming enormously for the better. We present a simple age-class model that explains this “disease of development” as a threshold phenomenon. Epidemics arise when improved conditions in hygiene are able to reduce disease transmission of polio amongst children below a critical threshold level. This generates a large susceptible adult population in which, under appropriate conditions, epidemics can propagate. The polio model is analysed in terms of its bifurcation properties and in terms of its non-equilibrium outbreak dynamics.

Introduction

“Poliomyelitis” is an epidemiological disease that has accompanied humankind throughout history. The earliest identifiable reference to paralytic poliomyelitis is an Egyptian stone engraving that dates back to more than 3500 years ago, and depicts a crippled young man, apparently a priest, with all the characteristic features of polio (Paul, 1971). The name itself is derived from the Greek words “polios”, or grey (referring to the grey matter of the nervous system) and “myelos” for marrow (referring to the myelin sheath membrane that surrounds the spinal cord) (Thomas and Robbins, 1997). Polio is also a prominent example of what is now referred to as a disease of development (Miller and Gay, 1997; Krause, 1998; Sutter et al., 1999). This is because in the late 19th and early 20th centuries, during a period of intense industrial development, social revolution and increased hygiene, there was a large increase of poliomyelitis with epidemics of a scale never seen previously (Fig. 1, Fig. 2).

Poliomyelitis is caused by poliovirus, which invades local lymphoid tissue and enters the blood stream. Poliovirus enters through the mouth, attaches to receptors on the epithelium of the throat and intestine, and replicates inside these cells. Polioviruses are spread directly or indirectly from person to person by droplets or aerosols and by fecal contamination of hands, eating utensils, milk, food and water (Dowdle and Birmingham, 1997). Exposure to poliomyelitis results in one of the following consequences: inapparent infection without symptoms (72% of people), minor illness (24%), non-paralytical poliomyelitis (4%) or paralytic poliomyelitis (<1%) (Sutter et al., 1999). Paralytic poliomyelitis is a severe form of the disease which occurs when a systemic infection moves to the central nervous system (CNS) and destroys neuronal cells. Although the paralytic form is an infrequent manifestation of polio, obviously a large-scale outbreak of the disease with tens of thousands of polio cases, can give rise to a large number of paralytic cases.

There is no simple all-encompassing theory that is capable of explaining the history and dynamics of polio epidemics. The best known theory is based on the somewhat controversial observation that the ratio of paralytic cases to the total number of infectives (case:infection ratio) increases with age (Nathanson and Martin, 1979; Miller and Gay, 1997). Thus age tends to increase the dangers of poliomyelitis, with adults more likely to be paralyzed or killed by the virus than children. Before the developments associated with the 20th century, almost all children were exposed to poliovirus during infancy, largely due to poor sanitation conditions. Sewage entered watersheds without treatment transporting the polio virus into rivers, lakes, streams and thus direct into the water supplies. Indirectly, polio virus passed through the food chain and could be traced even in milk supplies. Due to the low case:infection ratio of infants, and due to protection from transplacentally acquired maternal antibodies, paralysis was rare amongst young children, although the disease itself was endemic. Because of their exposure to polio at an early age, infected infants acquired immunity to the disease thereby protecting them in later life.

The transformation of poliomyelitis from endemic to epidemic occurred, paradoxically, just at the time of major hygienic improvements in the late 19th century. This period saw developments in technologies such as waste disposal, widespread use of indoor plumbing, and careful separation of sewage from drinking water. Improvements in sanitation reduced the transmissibility of the disease. As such, children were no longer exposed to the polio virus at an early age, and thus remained susceptible to the disease as older children or even adults. The average age of first-infection in the United States population was less than 5 years in mid 19th century to early 20th century (Sutter et al., 1999), and is likely to have been under 2 years of age similar to the situation in many developing countries (Sutter et al., 1999; Melnick, 1994). By the 1940s the average age increased to 9 years (Sutter et al., 1999). The increase in the average age of infection led to a large and growing pool of older unprotected susceptible individuals—the perfect setting for epidemics to ignite. As the case:infection ratio is larger in higher age brackets of the population (no longer protected by maternal antibodies), this further increased incidences of paralytic polio.

In the late 19th century and early 20th century, epidemics occurred in industrial countries, such as Sweden, Norway, and the United States. The three largest poliomyelitis outbreaks of that period occurred in Vermont in 1894 (132 cases), Sweden in 1905 (1031 cases) and New York in 1916 (9000 cases). Fig. 1 illustrates the sudden increased number of paralytic poliomyelitis cases at the beginning of 20th century in the United States. The incidence of polio rose steadily in the 1940s in developed nations peaking in the 1950s (Fig. 2). The worst recorded polio epidemic in US occurred in 1952 with 57 628 cases reported (Sutter et al., 1999). Two factors are considered responsible for this large-scale epidemic. Firstly, during Word War II, 50 million soldiers worldwide left their homes to be sent overseas. Many of these soldiers, nearly all susceptible to polio, traveled to developing countries where polio was endemic (Paul, 1971; Krause, 1998). Western soldiers became victims of domestic sanitary conditions and subsequently developed paralytic poliomyelitis. Secondly, at the same time, the western population increased drastically due to a post-war baby boom, thus creating an increased pool of susceptibles. Finally, that paralytic poliomyelitis became more prominent in developed countries is made clear in Fig. 3. Keeping in mind that developing countries are associated with higher infant mortality rates, the graph shows that countries with lower mortality rates generally have higher numbers of polio cases (per million population).

Although there have been numerous attempts at modeling the population dynamics of poliomyelitis (Hillis, 1979; Cvjetanovic et al., 1982; Anderson and May, 1991; Ranta et al., 2001), to our knowledge no model has successfully explained the effect seen in Fig. 1, Fig. 2 as a threshold phenomenon, as is attempted here. An interesting model of Coleman et al. (2001) shows that endemic infection rates can paradoxically increase with increasing disease control measures (analogous to sanitation here) as a result of the population's age structure. However, Coleman et al. (2001) do not attempt to explain the non-equilibrium outbreak dynamics. Our goal is to build a model taking into consideration previous theories, to explain these sudden and dramatic peaks in paralytic cases.

Section snippets

Two class age-structured epidemic model

We use the classical age-structured mathematical model of Schenzle (1984) as a framework for studying “diseases of development”, with poliomyelitis taken as a particular case-study. The model considers a constant population having a fixed number of host individuals N, each belonging to only one of four different possible groups: susceptible (S), exposed (E), infected (I) and recovered (R). When exposed to the infection, susceptible individuals are transferred to the exposed group and remain for

Equilibrium and stability analysis

The above model has an “infection-free” equilibrium in which Ic*=Ia*=0. Substituting these values into Eq. (3) one finds that the population consists entirely of susceptible individuals: Sc*=Nc*=μα+μ,Sa*=Na*=αα+μ.As newborns are not exposed to infections there are no recovered individuals and Rc*=Ra*=0. Inserting these values into the Jacobian of Appendix A, yields the matrix J having five eigenvalues, three of which are negative: λ1,2=-α-μ, λ3=-μ. The remaining two eigenvalues are solutions of

A simplified model

It is difficult to gain any further analytical insights into the dynamics of Eq. (3) directly. However, it is possible to examine the case βca=βac=0 where there is no significant contact between child and adult age classes. This is a reasonable first approximation that allows us to make important analytical insights regarding the mechanism underlying the model's dynamics. Eq. (3) becomes:dScdt=μ-(α+μ+βccNcIc)Sc,dSadt=αSc-(μ+βaaNaIa)Sa,dIcdt=βccNcIcSc-(γc+μ)Ic,dIadt=βaaNaIaSa-(γA+μ)Ia,dRcdt=γcIc-

Threshold effects in the simplified model

The above model offers an explanation for the unusual transition seen in Fig. 1 with regard to paralytic cases of polio, and more generally with regard to “diseases of development”. Fig. 1 shows that before the 20th century relatively few cases of polio were observed, while after 1906 there is a sustained epidemic of the disease. This may be understood in light of the model's different equilibria. It is believed that the pre-epidemic period was characterized by a relatively high contact rate

Indirect poliovirus transmission via environmental factors

The SIR model is based on the conventional direct transmission route of diseases in well mixed populations where infections are passed on through random contact between susceptible and infected individuals. This is controlled by the contact rates (βij) in Eq. (6), which in the main, covers direct disease transmission through the oral–oral and fecal–oral routes. However, poliomyelitis is also transmitted indirectly through a common vehicle or via the surrounding environment without direct

Discussion

The above model offers an explanation for the extraordinary jump in the number of paralytic polio cases that emerged at the beginning of the 20th century and the epidemics that followed with development. As Fig. 1 shows, the increase in cases appears to be similar to a threshold effect, a feature which is intrinsic to the epidemic model outlined here. The threshold is shown to be a specific outcome of an interplay between the dynamics of the two age classes within the population and governed by

Acknowledgements

We thank Professors Manfred Green, Danny Cohen and Tiberio Swartz for helpful comments and suggestions. We are grateful to Ronen Olinky, Eliezer Shochat, David Bunimovich and Shimon Zeldner for fruitful and stimulating discussions. In addition, we gratefully acknowledge and helpful suggestions of Vincent Jansen and three anonymous reviewers, one of whom suggested the “next generation matrix” approach. The work was supported by the James S McDonnell Foundation.

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