One the greatest scientific achievements of
the 1990's was the development of a clear understanding of HIV
pathogenesis aided by simple mathematical models. These simple
formulations were enough to interpret clinical data, determining
quantitative characteristics of the interaction between the HIV
virus and the cells targeted by it. AIDS used to be considered a
slow disease, however, simple mathematics clarified that even when
AIDS takes place on a time scale of around 10 years, there are
underlying fast (on time scales of hours to days) and slower (on
time scales of weeks to months) dynamical processes. Understanding
of the actual time scales paved the way to design treatment with
powerful antiretroviral drugs.
The models used to describe virus dynamics inside a single host share similarities with models employed in theoretical ecology for interacting species, because virus particles interact with uninfected cells, infected cells, and other cells from the immune systems. The interaction between hosts (individuals in a human population) is the basis of studying epidemiological processes. A disease can be thought as a pathogen invading a population of hosts (separate patches): an epidemic is successful if the number of occupied patches (infected hosts) increases over time after the initial introduction of the pathogen.
Mathematical descriptions of epidemics constitute an active area of applied mathematics. Traditionally, epidemiological models concentrate on the dynamics of "traits" transmitted between individuals, communities or regions (within specific temporal or spatial scales). Mathematical epidemiology facilitates the understanding of underlying mechanisms influencing the spread of disease. One of the most fundamental results in mathematical epidemiology is that the majority of models support a threshold behavior: If the average number of secondary infections caused by a typical infective is less than one a disease will die out, while if it exceeds one there will be an epidemic. This theoretical principle has been applied to estimate effectiveness of vaccination policies and the likelihood that a disease may be eradicated.
The founding fathers of mathematical epidemiology were public health physicians. In 1760 Daniel Bernoulli, a trained physician and member of a well-known family of mathematicians, derived the first known mathematical result defending the practice of inoculation against smallpox. Between 1900 and 1935 a group of physicians, including R.A. Ross, A.G. McKendrick, and W.O. Kermack, proposed the compartmental model approach that became the backbone of deterministic and stochastic models of epidemics. Compartmental modeling considers a population divided into compartments, with assumptions about the nature and time rate of transfer from one compartment to another. The transfer rates between compartments are expressed as time derivatives of the sizes of compartments. There are also models with transfer rates depending on the size of compartments over the past and at the instant of transfer, leading to formulations that involve differential-difference equations and integral equations.
Sir R.A. Ross was awarded a Nobel Prize in Medicine for his demonstration of the transmission dynamics of malaria between humans and mosquitoes. Even though his work received immediate acceptance in the medical community, his conclusion that malaria could be controlled by controlling the mosquitoes was dismissed because it seemed impossible to rid a region of mosquitoes completely. After Ross formulated a mathematical model predicting that malaria outbreaks could be avoided if the mosquito population were reduced below a critical threshold level, field trials supported his conclusions leading to successful control strategies for malaria.
Today in a highly interconnected world, epidemic outbreaks become instant potential global health or economic threats with increasing segments of the population playing active roles in the transmission patterns of infectious diseases, like influenza. Individual and group actions, decisions, or activities can affect the effectiveness of intervention or control measures in the Information Era. Travel, social distancing, the availability of medical supplies (antiviral drugs and vaccine) and timely diagnostic tools, and the access to quality medical care are but some of the factors that have been identified as significant during the 2009 influenza pandemic. Varying levels of participation in the implementation of policies aimed at reducing a population's risk of infection are important components not only of disease dynamics but also of their evolutionary potential. Hence, we must study and assess the role of changes in the levels of antiviral drug resistance, the impact on transmission patterns, cross-immunity, and the intrinsic ability of diseases, like H1N1, that circle the globe, to evolve in response to a multitude of selective pressures.
Policy makers have been asked to assess the impact of disease dynamics, evolving in changing social landscapes, and driven by conflicting sets of individuals' decisions during a pandemic that has fortunately not live up to its expected potential. Demands of guarantees for timely access for every individual to the H1N1 vaccines and high levels of refusal to get vaccinated, particularly among emergency medical personnel, are but some of the challenges that have been faced. What can mathematics do to help understand disease dynamics and develop effective control measures? What can mathematics do to help develop effective approaches for the management of complex adaptive systems of this type? Despite the myriad of complexities associated with disease transmission dynamics, macroscopic epidemic patterns emerge and ways of making use of this knowledge in real time can be critically important.