Question

Consider a colony in which an infectious disease (such as the common cold) is present. The population consists of three "species" of individuals. Let \(s\) represent the susceptibles-healthy individuals capable of contracting the illness. Let \(i\) denote the infected individuals, and let \(r\) represent those who have recovered from the illness. Assume that those who have recovered from the illness are not permanently immunized but can become susceptible again. Also assume that the rate of infection is proportional to \(s i\), the product of the susceptible and infected populations. We obtain the model $$ \begin{aligned} &s^{\prime}=-\alpha s i+\gamma r \\ &i^{\prime}=\alpha s i-\beta i \\ &r^{\prime}=\beta i-\gamma r \end{aligned} $$ where \(\alpha, \beta\), and \(\gamma\) are positive constants. (a) Show that the system of equations (11) describes a population whose size remains constant in time. In particular, show that \(s(t)+i(t)+r(t)=N\), a constant. (b) Modify (11) to model a situation where those who recover from the disease are permanently immunized. Is \(s(t)+i(t)+r(t)\) constant in this case? (c) Suppose that those who recover from the disease are permanently immunized but that the disease is a serious one and some of the infected individuals perish. How does the system of equations you formulated in part (b) have to be further modified? Is \(s(t)+i(t)+r(t)\) constant in this case?

Short answer

Explain your answer. Answer: In the first modified model (with permanent immunity), the population size remains constant because the sum of the time derivatives of susceptibles (s), infected (i), and recovered (r) is equal to zero, which indicates that the sum of s(t), i(t), and r(t) is a constant N. In the second modified model (with permanent immunity and deaths), the population size does not remain constant, as the sum of the time derivatives of s, i, and r is not equal to zero, indicating that the sum of s(t), i(t), and r(t) is not constant and decreases over time due to deaths among the infected individuals.

Step-by-step solution

_a) Show that population size remains constant in time with given model.#

First, we sum up the given equations in the model to see if \(s(t)+i(t)+r(t)\) remains constant: $$ \begin{aligned} &(s' + i' + r') = -\alpha si + \gamma r + \alpha si - \beta i + \beta i - \gamma r \end{aligned} $$ Upon simplification, we get: $$ \begin{aligned} & (s' + i' + r') = 0 \end{aligned} $$ Since the time derivatives of s, i, and r sum up to zero, their sum must be a constant \(N\). This shows that the population size remains constant in time. $$ s(t) + i(t) + r(t) = N $$

_b) Modify the model for permanent immunity and check if population size remains constant.#

Since those who recover are permanently immunized, they cannot become susceptible again. The modified system of equations becomes: $$ \begin{aligned} &s' = -\alpha si \\ &i' = \alpha si - \beta i \\ &r' = \beta i \end{aligned} $$ Now let's sum up the modified equations to see if the population size remains constant over time: $$ \begin{aligned} &(s' + i' + r') = -\alpha si + \alpha si - \beta i + \beta i \\ \end{aligned} $$ Upon simplification, we get: $$ \begin{aligned} & (s' + i' + r') = 0 \end{aligned} $$ Again, since the time derivatives of s, i, and r sum up to zero, their sum must be a constant. In this modified model, the population size still remains constant in time. $$ s(t) + i(t) + r(t) = N $$ So, \(s(t)+i(t)+r(t)\) is constant in this case.

_c) Modify the model with permanent immunity and deaths, and check if population size remains constant.#

Let's suppose that a proportion \(d\) of infected individuals perish. Then the modified system of equations becomes: $$ \begin{aligned} &s' = -\alpha si \\ &i' = \alpha si - \beta i - di \\ &r' = \beta i \end{aligned} $$ Now let's sum up these modified equations and see if the population size remains constant over time: $$ \begin{aligned} &(s' + i' + r') = -\alpha si + \alpha si - \beta i - di+ \beta i \\ \end{aligned} $$ Upon simplification, we get: $$ \begin{aligned} & (s' + i' + r') = - di \end{aligned} $$ Since the sum of the time derivatives of s, i, and r is not zero in this case, the population size is not constant over time. The population decreases due to the deaths of some infected individuals. $$ s(t) + i(t) + r(t) \neq N $$ So, \(s(t)+i(t)+r(t)\) is not constant in this case.

Key concepts

SIR Model

The SIR model is a classic framework used in the field of epidemiology to understand the spread of infectious diseases within a population. Its name derives from its categorization of individuals into three compartments, Susceptible (S), Infected (I), and Recovered (R).

The model is based on a set of ordinary differential equations that track the number of individuals in each compartment over time. The susceptibles (S) are those not yet infected but are vulnerable to the disease, the infected (I) are currently experiencing the infection and can spread it, and the recovered (R) are those who have overcome the disease and are no longer infectious. It is assumed that the transition from susceptible to infected is proportional to the product of the susceptible and infected individuals, represented by the rate of infection parameter.

The crucial aspect of the SIR model is its dynamism—individuals can move from one compartment to another based on disease dynamics and biological factors, such as immunity, recovery rate, and the rate at which people lose immunity, if at all. This flow is governed by parameters \(\alpha\), \(\beta\), and \(\gamma\), which represent the rate of infection, the rate of recovery, and the rate of losing immunity, respectively.

Rate of Infection

In the context of the SIR model, the 'rate of infection' is a critical parameter that influences the speed and scope of the disease's spread. It is denoted by the symbol \(\alpha\) and refers to the frequency at which susceptible individuals become infected upon contact with an infected person.

The rate of infection often depends on factors such as the pathogen's infectiousness, social behavior, population density, and public health interventions like vaccination or quarantine. For simplicity, the model assumes a proportional relationship, implying that the rate at which susceptibles become infected is directly proportional to the number of contacts between susceptible and infected individuals, given by the term \(\alpha si\).

Understanding and controlling the rate of infection is vital for managing an outbreak. High rates indicate rapid disease spread, leading to more aggressive interventions needed to curb the epidemic. Conversely, lower rates suggest a slower spread, allowing for different public health strategies.

Population Dynamics

Population dynamics in epidemiology are complex and involve understanding how a disease impacts the number of susceptible, infected, and recovered individuals over time. The dynamics are shaped by biological processes such as birth, death, infection, recovery, and immunity changes.

In the SIR model, these dynamics are outlined by equations representing changes in the compartments over time. The model assumes that individuals can transition from being susceptible to infected, and from infected to recovered. If varying levels of immunity are considered, recovered individuals may become susceptible again.

Disease Impact on Population Size

The total population size is often assumed to remain constant when the disease doesn't cause death. However, when modifying SIR equations to include death or permanent immunity, the assumption may vary. For instance, if infected individuals can die from the disease, this results in a decreasing population over time. On the other hand, if individuals gain permanent immunity after recovery, this does not necessarily affect population size but alters the distribution among the S, I, and R compartments.

The step-by-step solution exemplifies these dynamics by demonstrating various scenarios in which the SIR model's assumptions are modified, such as incorporating permanent immunity or accounting for the death of infected individuals. Each modification directly affects the equations' terms and outcomes for the population size and composition over time.