# Objective:

• To study the interaction between host and pathogen.

• To predict the spread of infectious disease in a population.

• Understanding the dynamics of disease transmission.

## Introduction:

Throughout history, devastating epidemics of infectious disease have wiped out large percentages of the human population. For example, the epidemics such as plague, flu and AIDS had killed millions of peoples worldwide. Millions of people all over the world are currently infected with human immunodeficiency syndrome (HIV) virus, about 95% of them in developing counties. The modeling of infectious diseases is a tool which has been used to study the mechanisms by which diseases spread, to predict the future course of an outbreak and to evaluate strategies to control an epidemic and has some enormous potential to help improve human lives. The ability to make predictions about diseases could enable scientists to evaluate inoculation or isolation plans and may have a significant effect on the mortality rate of a particular epidemic. This field of study called mathematical epidemiology is a subject of growing importance in medicine.

The first scientist who systematically tried to quantify causes of death was John Graunt in his book *Natural and Political Observations made upon the Bills of Mortality*, in 1662. The earliest account of mathematical modeling of spread of disease was carried out in 1766 by Daniel Bernoulli. He created a mathematical model to defend the practice of inoculating against smallpox. More systematic work on modeling disease was done in early 20th century by Hamer, who was interested in the regular recurrence of measles epidemics, and Ross. They put forward hypotheses about transmission of infectious disease and investigated their consequences through mathematical modeling. Based on their work, Kermack and McKendrick published a classic paper in 1927 that discovered a threshold condition for the spread of a disease and gave a means of predicting the ultimate size of an epidemic. Epidemic diseases are prevalent in a population only at particular times or under particular circumstances, while endemic diseases are habitually prevalent. The prevalence of a disease in a population is the fraction infected. The incidence is the rate at which infections occur.

While modeling an epidemic process we need to make some assumptions about,

- the population affected,

- the way the disease is spread, and

- the mechanism of recovery from the disease.

Based on these parameters we can model the population dynamics and the disease status structure of the population. In modeling the dynamics of an infectious disease, we focus on the population in which it occurs.

To begin formulating our model, at each time t, we divide the population N into three categories; the *susceptible* class, those who may catch the disease but currently are not infected; the* infective* class, those who are infected with the disease and are currently contagious; and the *removed* or *resistant *class, those who cannot get the disease, because they either have recovered permanently, are naturally immune, or have died.

The infective class may be split up further depending on whether the disease is *microparasitic *or *macroparasitic *and epidemic or endemic diseases.

Microparasitic diseases are caused by a virus (e.g. measles), a bacterium (e.g. TB), or a protozoan (e.g. malaria). And macroparasitic disease are caused by a helminth (e.g. a tapeworm), or an arthropod.

## Microparasite – Host Dynamics

A microparasite is an organism that can complete its life cycle within a single host. Most of the microparasites are viruses, bacteria, or fungi; a few are protists also.

### • The Kermack- McKendrick Model

###

The classic model for microparasites was developed in 1927, by the British epidemiologists W. O. Kermack and A. G. McKendrick. So this model is known as the Kermack- McKendrick model or SIR model (since it uses three classes). This model divides the host population into three compartments: susceptible, infected and resistant (immune).We call these three subgroups as *x1*, **x2** and **x3** in the equations. Individuals move from one group to another as the disease progresses. The rate at which individuals move from the susceptible to the infected compartment is described by *bx1x2*, where **b** is the probability that contact between a susceptible and an infected individual produces another infected individual and **x1x2** reflects the rate at which the two types of people come into contact. The rate at which individuals move from infected to resistant (**u**) is inversely related to the length of the infectious period. For some diseases, resistant individuals can become susceptible again (e.g. the flu and the common cold), and so there is a parameter, called *p*, that describes this rate of flow. The diagrammatic representation for the Kermack- McKendrick model of microparasite – host dynamics is illustrated in Figure.1.

Figure 1. Flow diagram for the Kermack- McKendrick model of microparasite – host dynamics.

Where, d is the Birth / Death rate,

'b' is the Probability of disease transmission,

'u' is the Rate at which infected individuals become resistant,

'p' is the Rate at which resistant individuals become susceptible.

Here, the Kermack- McKendrick model assumes the birth rate is equal to the death rate, i.e; zero population growth of the host population. Both the birth and death rates can be denoted as *d*. The total population size is N and is the sum of three *x *values.

The population dynamics of each compartment (**x1**, **x2** and **x3**) are described by differential equations. From the diagram, we can see that the change in number of individuals in each compartment and that for the susceptible class is described mathematically as,

The change in number of individuals in the infected compartment is described by

Since the total population size is constant, which is the sum of the three x values, we can find the number of individual in the resistant class (**x3**).

So the change in number of individuals in the resistant compartment can also describe mathematically as,

The parameters of the model can be varying from one disease to another.

## Macroparasite – Host Dynamics

A macroparasite spends only a part of its life cycle within one host. The rest of their life cycle is spends either as a free-living individual or within an individual of another host species. Most of the macroparasites are arthropods (fleas, mites, lice, etc.) or worms. Here we are considering the infections by worms. A macroparasite particularly a worm can infect any part of the body, most commonly parts such as the small intestine, where they feed on digested materials. They weaken the host by reducing its reproductive output and increasing the host’s vulnerability to other sources of mortality. The effect of infections increases with the increase in number of worms within the host.

### • The Anderson – May Model

The classic model of worm – host dynamics was developed in 1978 by Roy Anderson (a British epidemiologist) and Robert May (an Australian theoretical ecologist). The Anderson – May model has three populations. They are, the host population, the worm population within the hosts, and the worm population outside the hosts. There are different parameters which control the population dynamics within the three population groups. The movement of worms from one population to another is shown in Figure 2.

Figure 2. Flow diagram for the Anderson – May model of macroparasite – host dynamics.

Where, b is the Birth rate of host in absence of parasite, d is the Death rate of host in absence of parasite, a is the Effect of parasite on birth and death rate of host (birth rate of host decreases with larger worm burden and death rate of host increases with larger worm burden), c is the Rate at which parasite enter a host from the environment, u is Natural mortality rate of adult parasites in the hosts, p is the Rate at which immature parasites released into the environment and m is the Death rate of immature parasite in the environment outside the host.

From the diagram, we can figure out the change in number of individuals in each compartment and can be written as three differential equations:

The term, -a[(k+1)/k] (x2)^(2/x1) in the equation describing the population group 2 (x2) controls the spatial distribution of worms within the host population. Where k = Spatial distribution / clumping factor.

## Note:

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