Many models for the spread of infectious diseases in populations have been analyzed mathematically and applied to specific diseases. Threshold theorems involving the basic reproduction number $R_{0}$, the contact number $\sigma$, and the replacement number $R$ are reviewed for the classic SIR epidemic and endemic models. Similar results with new expressions for $R_{0}$ are obtained for MSEIR and SEIR endemic models with either continuous age or age groups. Values of $R_{0}$ and $\sigma$ are estimated for various diseases including measles in Niger and pertussis in the United States. Previous models with age structure, heterogeneity, and spatial structure are surveyed.

https://epubs.siam.org/doi/pdf/10.1137/S0036144500371907

The Mathematics of Infectious Diseases

If the host population is taken to be a dynamic variable (rather than constant, as conventionally assumed), a wider understanding of the population biology of infectious diseases emerges. In this first part of a two-part article, mathematical models are developed, shown to fit data from laboratory experiments, and used to explore the evolutionary relations among transmission parameters. In the second part of the article, to be published in next week's issue, the models are extended to include indirectly transmitted infections, and the general implications for infectious diseases are considered.

Population biology of infectious diseases: Part II

An approach is presented for making short-term predictions about the trajectories of chaotic dynamical systems. The method is applied to data on measles, chickenpox, and marine phytoplankton populations, to show how apparent noise associated with deterministic chaos can be distinguished from sampling error and other sources of externally induced environmental noise.

Nonlinear forecasting as a way of distinguishing chaos from measurement error in time series

Seasonal variations in temperature, rainfall and resource availability are ubiquitous and can exert strong pressures on population dynamics. Infectious diseases provide some of the best-studied examples of the role of seasonality in shaping population fluctuations. In this paper, we review examples from human and wildlife disease systems to illustrate the challenges inherent in understanding the mechanisms and impacts of seasonal environmental drivers. Empirical evidence points to several biologically distinct mechanisms by which seasonality can impact host-pathogen interactions, including seasonal changes in host social behaviour and contact rates, variation in encounters with infective stages in the environment, annual pulses of host births and deaths and changes in host immune defences. Mathematical models and field observations show that the strength and mechanisms of seasonality can alter the spread and persistence of infectious diseases, and that population-level responses can range from simple annual cycles to more complex multiyear fluctuations. From an applied perspective, understanding the timing and causes of seasonality offers important insights into how parasite-host systems operate, how and when parasite control measures should be applied, and how disease risks will respond to anthropogenic climate change and altered patterns of seasonality. Finally, by focusing on well-studied examples of infectious diseases, we hope to highlight general insights that are relevant to other ecological interactions.

/pdf/seasonality-and-the-dynamics-of-infectious-diseases-1nbx7q5bgi.pdf

Seasonality and the dynamics of infectious diseases.

When the traditional assumption that the incidence rate is proportional to the product of the numbers of infectives and susceptibles is dropped, the SIRS model can exhibit qualitatively different dynamical behaviors, including Hopf bifurcations, saddle-node bifurcations, and homoclinic loop bifurcations. These may be important epidemiologically in that they demonstrate the possibility of infection outbreak and collapse, or autonomous periodic coexistence of disease and host. The possible mechanisms leading to nonlinear incidence rates are discussed. Finally, a modified general criterion for supercritical or subcritical Hopf bifurcation of 2-dimensional systems is presented.

Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models

London, W. P. (Mathematical Research Branch, National Institute of Arthritis, Metabolism, and Digestive Diseases, Bethesda, Md. 20014) and J. A. Yorke. Recurrent outbreaks of measles, chickenpox and mumps. I. Seasonal variation in contact rates. Am J Epidemiol 98:453-468, 1973. —Recurrent outbreaks of measles, chickenpox and mumps in cities are studied with a mathematical model of ordinary differential delay equations. For each calendar month a mean contact rate (fraction of susceptibles contacted per day by an infective) is estimated from the monthly reported cases over a 30to 35-year period. For each disease the mean monthly contact rate is 1.7 to 2 times higher in the winter months than in the summer months; the seasonal variation is attributed primarily to the gathering of children in school. Computer simulations that use the seasonally varying contact rates reproduce the observed pattern of undamped recurrent outbreaks: annual outbreaks of chickenpox and mumps and biennial outbreaks of measles. The two-year period of measles outbreaks is the signature of an endemic infectious disease that would exhaust itself and become nonendemic if there were a minor increase in infectivity or a decrease in the length of the incubation period. For populations in which most members are vaccinated, simulations show that the persistence of the biennial pattern of measles outbreaks implies that the vaccine is not being used uniformly throughout the population.

http://yorke.umd.edu/Yorke_papers_most_cited_and_post2000/1973_03_London_I_AMJEpi.pdf

Recurrent outbreaks of measles, chickenpox and mumps i. seasonal variation in contact rates

Yorke, J. A. and W. P. London (Mathematical Research Branch, National Institute of Arthritis, Metabolism, and Digestive Diseases, Bethesda, Maryland 20014). Recurrent outbreaks of measles, chickenpox and mumps. I I . Systematic differences in contact rates and stochastic effects. Am J Epidemiol 98:469-482, 1973.—The mean monthly contact rates for measles, chickenpox and mumps estimated from the monthly reported cases show systematic differences between the years with many cases and the years with few cases. In New York City, the mean contact rates for chickenpox were different during the years 1931-1945 than during 1946-1960. The clustering of cases within social groups is proposed to account for these differences in the contact rates and for other empirical observations. The irregularity of outbreaks of measles in cities of fewer than two million people can be explained by stochastic effects. Outbreaks of measles in distant large metropolitan areas are highly correlated in time, but the reasons for the correlation are not clear.

Recurrent outbreaks of measles, chickenpox and mumps ii. systematic differences in contact rates and stochastic effects

One hundred thirty-six alcoholic men and 48 alcoholic women admitted consecutively to an adult alcohol and substance abuse unit were studied in an attempt to replicate previous reports of an association between alcoholism and handedness. Each individual received a structured admission interview, and handedness was determined by a modification of the Edinburgh Inventory. Left-handedness was more frequent in men with alcoholic fathers and in first-born men. These data are discussed in the context of a recent theory relating left-handedness, immune disease, developmental learning disorders, and fetal testosterone.

Handedness and alcoholism.

It is shown that if the matrix of an n -compartment system is upper Hessenberg (zero entries below the subdiagonal) and nonzero initial conditions exist in the first compartment only, then the solution for the n th compartment is β e 1 ∗ e 2 ∗…∗ e n where β is a constant, e i ( t )=exp(λ it ), λ i is the i th root of the matrix and ∗ denotes convolution. Conversely, the solution is a compartment can take this form only if the matrix of the system is Hessenberg (up to a permutational similarity). These results depend upon the relations which obtain among the distribution of zero entries in sequential powers of the matrix, the initial derivatives in the compartments, and the path length between compartments in the directed diagram of the system. These relations are developed and employed in an assessment of the product precursor criterion as proposed by Rescigno and Segre.

Wayne P. London

Papers

Recurrent outbreaks of measles, chickenpox and mumps i. seasonal variation in contact rates

Recurrent outbreaks of measles, chickenpox and mumps ii. systematic differences in contact rates and stochastic effects

Handedness and alcoholism.

Path lengths and initial derivatives in arbitrary and Hessenberg compartmental systems

Steady state kinetics of an enzyme reaction with one substrate and one modifier