Mathematical Models in Epidemiology

Recorded history continuously documents the invasion of populations by infectious agents, some causing many deaths before disappearing, others reappearing in invasions some years later in populations that have acquired some degree of immunity, due to prior exposure to related infectious pathogens. The “Spanish” flu epidemic of 1918–1919 exemplifies the devastating impact of relatively rare pandemics; this one was responsible for about 50,000,000 deaths worldwide, while on the mild side of the spectrum we experience annual influenza seasonal epidemics that cause roughly 35,000 deaths in the USA each year.

Communicable diseases that are endemic (always present in a population ) cause many deaths). For example, in 2011 tuberculosis caused an estimated 1,400,000 deaths and HIV/AIDS caused an estimated 1,200,000 deaths worldwide. According to the World Health Organization there were 627,000 deaths caused by malaria, but other estimates put the number of malaria deaths at 1,2000,000. Measles, which is easily treated in the developed world, caused 160,000 deaths in 2011, but in 1980 there were 2,600,000 measles deaths. The striking reduction in measles deaths is due to the availability of a measles vaccine. Other diseases such as typhus, cholera, schistosomiasis, and sleeping sickness are endemic in many parts of the world. The effects of high disease mortality on mean life span and of disease debilitation and mortality on the economy in afflicted countries are considerable. Most of these disease deaths are in less developed countries, especially in Africa, where endemic diseases are a huge barrier to development.

In this chapter, we consider models for disease that may be endemic. In the preceding chapter we studied SIS models with and without demographics and SIR models with demographics. In each model, the basic reproduction number \(\mathcal {R}_0\) determined a threshold. If \(\mathcal {R}_0 < 1\) the disease dies out, while if \(\mathcal {R}_0 > 1\) the disease becomes endemic. The analysis in each case involves determination of equilibria and determining the asymptotic stability of each equilibrium by linearization about the equilibrium. In each of the cases studied in the preceding chapter the disease-free equilibrium was asymptotically stable if and only if \(\mathcal {R}_0 < 1\) and if \(\mathcal {R}_0 > 1\) there was a unique endemic equilibrium that was asymptotically stable. In this chapter, we will see that these properties continue to hold for many more general models, but there are situations in which there may be an asymptotically stable endemic equilibrium when \(\mathcal {R}_0 < 1\), and other situations in which there is an endemic equilibrium that is unstable for some values of \(\mathcal {R}_0 > 1\).

In this chapter we describe models for epidemics, acting on a sufficiently rapid time scale that demographic effects, such as births, natural deaths, immigration into and emigration out of a population may be ignored. The prototype epidemic model is the simple Kermack–McKendrick model studied in Sect. 2.​4.

To cope with annual seasonal influenza epidemics there is a program of vaccination before the “flu” season begins. Each year a vaccine is produced aimed at protecting against the three influenza strains considered most dangerous for the coming season. We formulate a model to add vaccination to the simple SIR model under the assumption that vaccination reduces susceptibility (the probability of infection if a contact with an infected member of the population is made).

Many diseases are transmitted from human to human indirectly, through a vector. Vectors are living organisms that can transmit infectious diseases between humans. Many vectors are bloodsucking insects that ingest disease-producing microorganisms during blood meals from an infected (human) host, and then inject it into a new host during a subsequent blood meal. The best known vectors are mosquitoes for diseases including malaria, dengue fever, chikungunya, Zika virus, Rift Valley fever, yellow fever, Japanese encephalitis, lymphatic filariasis, and West Nile fever, but ticks (for Lyme disease and tularemia), bugs (for Chagas’ disease), flies (for onchocerciasis), sandflies (for leishmaniasis), fleas (for plague, transmitted by fleas from rats to humans), and some freshwater snails (for schistosomiasis) are vectors for some diseases.

In this chapter we describe several models for tuberculosis (TB). The disease is endemic in many areas of the world. The models in this chapter will be extensions of the standard SIR or SEIR type of endemic models presented in Chap. 3. Depending on the typical characteristics of a specific disease, various modifications of the standard models will be considered.

Acquired immunodeficiency syndrome (AIDS) was first identified as a new disease in the homosexual community in San Francisco in 1981. The human immunodeficiency virus (HIV) was identified as the causative agent for AIDS in 1983. The disease has several very unusual aspects. After the initial infection, there are symptoms, including headaches and fever for 2 or 3 weeks. Transmissibility is high for about 2 months, and then there is a very long latent period during which transmissibility is low. At the end of this latent period, which may last 10 years, transmissibility rises, signaling the development of full-blown AIDS. In the absence of treatment, AIDS is invariably fatal. Now, HIV can be treated with a combination of highly active antiretroviral therapy (HAART) drugs, which both reduce the symptoms and prolong the period of low infectivity. While there is still no cure for AIDS, treatment has made it no longer a necessarily fatal disease. To describe the variation of infectivity for HIV, one possibility would be to use a staged progression model, with multiple infective stages having different infectivity. Another possibility would be to use an age of infection model.

Influenza causes more morbidity and more mortality than all other respiratory diseases together. There are annual seasonal epidemics that cause about 500,000 deaths worldwide each year. During the twentieth century there were three influenza pandemics. The World Health Organization estimates that there were 40,000,000–50,000,000 deaths worldwide in the 1918 pandemic, 2,000,000 deaths worldwide in the 1957 pandemic, and 1,000,000 deaths worldwide in the 1968 pandemic. There has been concern since 2005 that the H5N1 strain of avian influenza could develop into a strain that can be transmitted readily from human to human and develop into another pandemic, together with a widely held belief that even if this does not occur there is likely to be an influenza pandemic in the near future. More recently, the H1N1 strain of influenza did develop into a pandemic in 2009, but fortunately its case mortality rate was low and this pandemic turned out to be much less serious than had been feared. There were 18,500 confirmed deaths, but the actual number of deaths caused by the H1N1 influenza may have been as many as 200,000. This history has aroused considerable interest in modeling both the spread of influenza and comparison of the results of possible management strategies.

Another important infectious disease is Ebola virus disease (EVD). Ebola hemorrhagic fever is a very infectious disease with a case fatality rate of more than 70%. It was first identified in 1976 in the Democratic Republic of Congo and there have been more than a dozen serious outbreaks since then. The most serious outbreak to date occurred in 2014 in Guinea, Liberia, and Sierra Leone and caused more than 10,000 deaths. Response to this epidemic included the development of vaccines to combat the disease, currently being used to combat the most recent outbreak in the Democratic Republic of Congo.

Malaria is one of the most important diseases transmitted by vectors. The vectors for many vector-transmitted diseases are mosquitoes or other insects which tend to be more common in warmer climates. One influence of climate change in coming years may be to extend the regions where mosquitoes can thrive and thus to cause the spread of vector-transmitted diseases geographically.

While there have been cases of probable dengue fever more than 1000 years ago, the first recognized dengue epidemics occurred in Asia, Africa, and North America in the 1780s. There have been frequent outbreaks since then, and the number of reported cases has been increasing rapidly recently. According to the World Health Organization, approximately 50,000,000 people worldwide are infected with dengue. Symptoms may include fever, headaches, joint and muscle pain, and nausea, but many cases are very mild. There is no cure for dengue fever, but most patients recover with rest and fluids. There are at least four different strains of dengue fever, and there is some cross-immunity between strains. Dengue fever is transmitted by the mosquito aedes aegypti, and most control strategies are aimed at mosquito control.

Age is one of the most important characteristics in the modeling of populations and infectious diseases. Because age groups frequently mix heterogeneously it may be appropriate to include age structure in epidemiological models. While there are other aspects of heterogeneity in disease transmission models, such as behavioral and spatial heterogeneity, age structure is one of the most important aspects of heterogeneity in disease modeling.

With the advent of substantial intercontinental air travel, it is possible for diseases to move from one location to a completely separate location very rapidly. This was an essential aspect of modeling SARS during the epidemic of 2002–2003, and has become a very important part of the study of the spread of epidemics. Mathematically, it has led to the study of metapopulation models or models with patchy environments and movement between patches.

The work of Eubank et al., Sara del Valle et al., Chowell et al., and Castillo-Chavez and Song have highlighted the impact of modified modeling approaches that incorporate heterogeneous modes of mobility within variable environments in order to study their impact on the dynamics of infectious diseases. Castillo-Chavez and Song, for example, proceeded to highlight a Lagrangian perspective, that is, the use of models that keep track at all times of the identity of each individual. This approach was used to study the consequences of deliberate efforts to transmit smallpox in a highly populated city, involving transient sub-populations and the availability of massive modes of public transportation.

Lessons learned from the HIV pandemic, SARS in 2003, the 2009 H1N1 influenza pandemic, the 2014 Ebola outbreak in West Africa, and the ongoing Zika outbreaks in the Americas can be framed under a public health policy model that responds after the fact. Responses often come through reallocation of resources from one disease control effort to a new pressing need. The operating models of preparedness and response are ill-equipped to prevent or ameliorate disease emergence or reemergence at global scales. Epidemiological challenges that are a threat to the economic stability of many regions of the world, particularly those depending on travel and trade, remain at the forefront of the Global Commons. Consequently, efforts to quantify the impact of mobility and trade on disease dynamics have dominated the interests of theoreticians for some time. Our experience includes an H1N1 influenza pandemic crisscrossing the world during 2009 and 2010, the 2014 Ebola outbreaks, limited to regions of West Africa lacking appropriate medical facilities, health infrastructure, and sufficient levels of preparedness and education, and the expanding Zika outbreaks, moving expeditiously across habitats suitable for Aedes aegypti. These provide opportunities to quantify the impact of disease emergence or reemergence on the decisions that individuals take in response to real or perceived disease risks. The case of SARS 2003 in 2003, the efforts to reduce the burden of H1N1 influenza cases in 2009, and the challenges faced in reducing the number of Ebola cases in 2014 are the three recent scenarios that required a timely global response. Studies addressing the impact of centralized sources of information, the impact of information along social connections, or the role of past disease outbreak experiences on the risk-aversion decisions that individuals undertake may help identify and quantify the role of human responses to disease dynamics while recognizing the importance of assessing the timing of disease emergence and reemergence. The co-evolving human responses to disease dynamics are prototypical of the feedbacks that define complex adaptive systems. In short, we live in a socioepisphere being reshaped by ecoepidemiology in the “Era of Information.”