Viruses (mathematics)
Viruses in mathematics refers to the study of viral structures and the modeling of virus spread within populations using mathematical frameworks. A virus is a parasitic entity that cannot reproduce independently; it must invade a host cell to replicate, often causing damage to the host in the process. The mathematical analysis of viruses includes examining their geometry, such as the structure of their protein coats, or capsids, which can be highly symmetric, like the helical forms of the Tobacco Mosaic Virus or the icosahedral shapes of the poliovirus.
Mathematical models, particularly the SIR model, are essential tools for analyzing the transmission dynamics of viral infections. This model categorizes individuals into three groups: susceptible, infected, and recovered, allowing researchers to simulate the spread of diseases like measles and influenza. Understanding the growth patterns of viral infections, especially their potential for exponential spread, is critical for public health planning and intervention strategies. By utilizing these mathematical insights, scientists and public health officials aim to devise effective responses to viral outbreaks, as evidenced during the COVID-19 pandemic, which highlighted the importance of timely and informed decision-making in managing infectious diseases.
Subject Terms
Viruses (mathematics)
Summary: The spread of viruses in a population—and the internal structure of viruses themselves—can be analyzed mathematically to help epidemiologists study viral infections.
A virus is a parasite. It cannot reproduce on its own. Instead, it must invade a cell of another organism and use the host cell’s machinery to make copies of itself. The newly replicated viruses then leave the host cell and infect other cells. In the process, the virus often damages the host. For example, different virus types cause measles, polio, and influenza in people; hoof-and-mouth disease in cattle; and leaf curl in many vegetables. Mathematics provides a language to describe viral structures. Furthermore, mathematical models of the spread of a virus in a population are powerful tools in public health policy.
Capsid Geometry
A virus consists of genetic material (either DNA or RNA) surrounded by a protein coat called a “capsid.” Viruses have much less genetic material and are much smaller than single-celled organisms like bacteria. With the limited genetics of a virus, it can encode only a few proteins of its own, and so must use them efficiently. Often, the entire capsid is assembled from many copies of a single protein, which means the capsid should be highly symmetric.
One of the first virus structures to be determined was that of the Tobacco Mosaic Virus (TMV). Copies of the TMV capsid protein are arranged in a helix around the viral RNA. Many other viruses have helical capsids as well. In contrast, poliovirus, the Hepatitis B virus, tomato bushy stunt virus, and other viruses have icosahedral capsids. Figure 1 shows a computer-generated image of the poliovirus capsid with protein subunits colored to highlight the icosahedral symmetry. Other, more complicated capsid shapes are possible.
While the capsids do not have flat triangular faces, they have axes of five-fold rotational symmetry, like those through the vertices of the icosahedron; axes of three-fold rotational symmetry, like those through the centers of the triangular faces of the icosahedron; and axes of two-fold rotational symmetry, like those through the centers of the edges of the icosahedron.
Much research has continued to focus on the geometry of viruses in the hope that better understanding of how viruses assemble themselves will lead to better methods of destroying them. In 2017, for example, multiple mathematical insights into viral structure were announced to considerable fanfare in the scientific community, including studies of hepatitis B virus (HBV) assembly and a study of RNA folding in human parechovirus associated with the common cold. In 2019, a publication by researchers at the University of York and San Diego State University presented a new mathematical framework for understanding the shaping of viral protein containers, and was hailed as an important step toward potential new antiviral treatments.
Modeling the Spread of Viruses
Mathematical models of infectious disease transmission in a population help researchers understand which interventions might slow the spread of a virus. The SIR model, first proposed by W. O. Kermack and A. G. McKendrick in 1927, is one of the simplest and is suitable for viruses such as measles and influenza. Each person in a population is in one of three categories: (1) susceptible to the virus, (2) infected and infectious, or (3) recovered and immune.
Let S, I, and R be the proportion of the population that is susceptible, infected, and recovered, respectively. The SIR model is given by the following system of differential equations:
where the constant β depends on the probability that an infected person transmits the virus to a susceptible person, and the constant γ depends on how long it takes an infected person to recover. This model does not lead to simple expressions for S, I, and R as functions of time but it can be explored computationally. One simple way to do so is to treat time discretely and approximate
by (St+1-St), where St is the value of S at time step t. This method yields the difference equations
The basic SIR model can be modified to fit other scenarios. For example, immunity might wear off over time, or some part of the population might be at higher risk of infection, or a vaccination campaign might begin.
The SIR model assumes that all possible contacts between infected people and susceptible people are equally likely (hence the factor of SI in dS/dt). Modifying the model to reflect the social structure of the population allows researchers to ask crucial questions. If the supply of influenza vaccine is limited, is it more effective to vaccinate school children, who spread the disease, or the elderly, who may suffer more complications from infection? Will closing airports slow an epidemic enough to justify the costs to travelers? In such situations, mathematical models allow public health officials to test the effects of different interventions before choosing a course of action.
One of the major risks with viral and other infectious diseases is that they can spread exponentially rather than linearly. This means that infection rates may seem low (and manageable) initially, before suddenly spiking to epidemic or even pandemic proportions. The outbreak of coronavirus 2019 (COVID-19) in 2019 and 2020 is a strong example of this trend, as the number of cases grew rapidly, overwhelming some early quarantine efforts and causing major social and economic disruption around the world. Efforts to "flatten the curve" of infections referred to the importance of stopping the exponential increase in cases, as even a small drop in the infection rate has a significant impact from a statistical perspective.
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