Science has a deep relation with mathematics. Today even life sciences depend on quantitative measurements and correlations. Charles Darwin, the father of evolution is said to have expressed his regrets when he said, ‘ I have not proceeded far enough at least to understand something of the great leading principles of mathematics; for men thus endowed seem to have an extra sense’. While mathematics has played pivotal role in physics, its role in biology is steadily becoming clear. But in this context, one has to bear in mind a small over reach of the scientific community. Unfortunately, mathematical models are thought to provide objectivity and are used even in human sciences. These positivist tendencies do have their short comings and have to be accepted as we apply mathematics to new domains like life sciences.
Joel E. Kohen says, ‘mathematics became to biology what microscope represented a few centuries ago: a tool of approach and that enabled us to interpret a new and fascinating world’. Among several mathematical applications, we can trace the study of the dynamics and evolution of infectious diseases like malaria, HIV, Ebora, and SARS. With the increase of our knowledge of non-linear dynamical systems (chaos), mathematics has steadily assisted us to understand the spread of these diseases and design regulatory measures. Mathematical modelling of infectious disease is highly advanced in the understanding, regulation of the transmission infectious disease and implementation of measures for prevention and control.
Infectious disease like Covid-19 is caused by virus. Besides viral infections, deadly infections are caused by bacteria or parasites (such as protozoans) and can endanger human life. This is why humanity is deeply interested in understanding the dynamics of its infection and transmission. Mathematical models that study this dynamics are found to be useful to understand and regulate the spread of infections. It is said that Benouilli studied the transmission of smallpox using mathematical model in 1760. But real work in epidemiological studies began in early twentieth century. We can find mathematical models being used to understand the spread of measles and malaria in the early part of the twentieth century.
There is a compartmental model that is often used to study the spread of a pandemic. This model divides human population into compartments. The individuals that are susceptible to the disease are put into the compartment S. Those that are infected are set into the compartment I while individual who are removed from the infected compartment because of being healed are put in a compartment called R. Let’s assume that at a time t we have S (t) in the compartment S, I (t) in the compartment I and R(t) in the compartment R. The total population (assuming that there is no emigration) will be K= S(t)+ I(t)+R(t). Usually infected persons are those that meet other infected persons and develop the chance of being infected (like the case of covid-19. There are also other vector borne infections). This chance of such persons being infected for a unit time (t) is given by what is called transmission coefficient which is Bs(t), where b is the transmission coefficient. The number of the persons who are removed from the infected compartment due to their recovery is rI(t) at a time (t) where r is rate constant for recovery. This model is modified to address other dynamic conditions depending on the rate at which the infection develops and spreads. Often graphs and charts become effective means to visualize the rate of transmission as well as healing.
Besides time, epidemic models are also associated with place/ space. Geographical considerations are also important to understand and respond to spreading infection. Covid-19 has familiarized us with lockdowns. Mathematical models assist us to understand the effect of population dispersal on the spread of an infectious disease. Basically, there are two common ways to factor in place and movement of people in the mathematical model of understanding the transmission of a infectious disease. The first one uses space variables and use diffusion equations that take into consideration of movement among individuals in a direct neighbourhood. Further, there are other factors like animal vectors like birds that move to different places and carry the infection (Bird flu). Scientists also have considered mathematical models to embrace vectors other than humans as agents of transmission of infections in their models of transmission of infection. Besides, to factor in mutating virus we require non-linear mathematical models to understand the dynamics of these virulent infections. We also use mathematical models to understand the cytotoxic immune responses of an infected cell.
Mathematics being a science of patterns assist us to trace patterns in the infection, transmission and healing of infectious disease and is indeed valuable to us humans. Besides, quantitative, statistical and computational models are growing in importance in several areas life sciences by the day. They help us to analyze several biological problems and trace or predict outcomes. Sciences like genomics, proteomics, pharmacogenomics etc are both biological as well as mathematical. This inter-disciplinary approach is fast exhibiting a natural link between mathematics and biology. The internal life of the organism as well as its interaction with the stimuli from the environment can be analyzed with the help of quantitative models from mathematics. This analysis can manifest the growth and decline of organisms in an ecosystem. The rate of growth or extinction can be mathematically calculated and can assist us to predict and retrodict the conditions of the ecosystem assisting us to understand and respond to several conditions of the ecosystem. We have almost a Newtonian first law in population genetics which says, ‘if no force is applied to a population (natural selection/ migration) the population will remain in a genetic equilibrium. .
Mathematics does provide us an approximation of our real world. Galileo who said that God created the world in the alphabets of mathematics is almost proved right. In fact mathematics is a powerful language of the cosmos. It has now begun speaking the language of life. There is another advantage. We can mathematically simulate complex variables of life and thus shed light on complex biological systems. Mathematics indeed is the next microscope of Biology and has opened new ways of understanding life.