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When a disease spreads in a population

When a disease spreads in a population, it in simplest scenario splits the population into three classes (Figure 1); susceptible individuals(S), infected individuals (I) and recovered individuals (R). Susceptible individuals are people, who are healthy. Infected individuals are people, who have been infected by the disease and are now sick. Recovered individuals are people, who have recovered and cannot be infected by the disease again (Keeling, 2001).

Figure 1: Dynamic of disease.
The people in all of those three classes represents the whole society (Equation 1). According to Hethcote (1994) the number of people in each with time (t), which is measured in days, changes. So, sustainable (St) individuals become infected (It) and infected individuals recover (Rt). In other words, the number of people is time dependent (Hethcote H. W., 2000).
Equation 1: Population size.
N=S+I+R
Equation 2: Number of susceptible, infected and recovered individuals are time dependent.
S=S_t I=I_t R=R_t
So, to prove this time-dependent variables are inserted in the equation of the population size and then divided with N (presented below). So, we get the same equation, which is expressed in the form of the derivates. They are fractions that represent the number of individuals at the specific time. The sum of these derivatives from Equation 3 equals to 1, which means that their sum equals to the size of the population, which is constant (Hethcote H. W., 1994).
Equation 3: Equation of the population size in the form of the derivatives.
N =S_t+I_t+R_t /:N
1=S_t/N+I_t/N+R_t/N
1=dS/dt+dI/dt+dR/dt

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Figure 1: Diagram of movement of an individual from sustainable group to infected group of individuals by disease transmission rate and to recovered group of individuals by disease recovery rate.
The movement of individual from one group to another group is like in the diagram (Figure 1) dependent on the disease transmission rate (?) and recovery rate (?). For individual from sustainable group to come into infected group needs to get infected. This movement depends on the disease transmission rate (?), which is rate at which sustainable individuals become infected by infected individuals (Feng, Towers, & Yang, 2011).
The disease transmission rate can in some diseases – like for example seasonal influenza -depend on seasonal variation. Feng et. al. (2011) say that seasonal transition rate is expressed as sum of background transmission rate (?_0) and transmission rate (?) that is dependent on the season, which is a product of cosine periodic function with a 1-year period (cos?((2?×t)/365)), oscillation amplitude (?) and background transmission rate (Feng, Towers, & Yang, 2011).
Equation 4: Seasonal transmission rate (Feng, Towers, & Yang, 2011).
?(t)=?_0+ ?_0 × ?×cos??((2?×t)/365)= ?_0×(1+ ?×cos?((2?×t)/365) ?
Infected individuals then can come into recovered group of individuals by recovery rate. Recovery rate (?) is rate at which individual is recovered from the diseases and according to the assumption gets permanent immunity (Feng, Towers, & Yang, 2011).
CHANGES IN THE NUMBER OF GROUPS
The number of individuals of sustainable group ((dS(t))/dt) is calculated by the negative total infection rate. That is product of I_t×S_t×?(t), which is divided by population size(N) (Equation 5) (Huppert & Katriel, 2013).
The new number of sustainable individuals (S_(t+1)) is calculated by sum of the number of sustainable individuals (S_t) and the difference in the number of sustainable people ((dS(t))/dt), like it is presented in Equation 7 (Hethcote H. W., 1994).
Equation 5: The change in the number of sustainable individuals (?S_t).
(dS(t))/dt=-?(t)×(I_t×S_t)/N
Equation 6: The new number of sustainable individuals (S_(t+1))..
S_(t+1)= S_t+ (dS(t))/dt=S_t- ?(t)×(I_t×S_t)/N
The difference in the number of individuals of recovered group ((dR(t))/dt) is calculated by multiplying the recovery rate (?) and number of infected individuals (I_t), like in the Equation 8. The new number of recovered individuals is calculated by sum of recovered individuals (R_t) and the difference in the number of recovered individuals ((dR(t))/dt) (Hethcote H. W., 1994).
Equation 7: The difference in the number of recovered individuals.
(dR(t))/dt= ?×I_t
Equation 8: The number of recovered individuals.
R_(t+1)= (dR(t))/dt+ R_t= R_t+ ?×I_t
The difference in the number of individuals of infected group ((dI(T))/dt) is calculated by the difference between the sustainable individuals (S_t) and the recovered individuals (R_t), like in Equation 10. The new number of infected individuals is calculated by sum of infected individuals (I_t) and the difference in the number of infected individuals ((dI(t))/dt) (Hethcote H. W., 1994).
Equation 9: The difference in the number of infected individuals.
dI(t)/dt=dS(t)/dt- dR(t)/dt= -?(t)×(I_t×S_t)/N- ?×I_t
Equation 10: The new number of infected individuals.
I_(t+1)= I_t+ (dI(t))/dt= I_t+-?(t)×(I_t×S_t)/N- ?×I_t

MODELING THE SIR MODEL WITH VACCINATION
We also know vaccine for some diseases. When vaccine is used for curing or to stop spreading the disease, we also use the term herd immunity. Herd immunity (q_c), also known as population immunity or the critical immunization threshold, is a measure of protection of individuals who are not immune. For this to happen a large percentage of a population need to be immune to the infectious disease in order to form an indirect protection for unvaccinated individuals (Wikipedia, the free encyclopedia, 2018). When the individuals in the population get vaccinated, they get permeant immunity from disease. When enough people get vaccinated they create herd immunity. The herd immunity predicts if there is chance for disease to spread. Herd immunity depend on the reproduction rate (R_0) and the contact number (Nicho, 2010).
The reproduction rate (R_0) is defined as the expected number of secondary cases produced by a single (typical) infection in a completely susceptible population (Jones, 2007). According to the Jones basic reproduction number is product of probability of infection given contact between a susceptible and infected individual (?), the average rate of contact between susceptible and infected individuals (c) and is the duration of infectiousness (d). In this model probability of infection given contact between a susceptible and infected individual (?) and the average rate of contact between susceptible and infected individuals (c) equal to the disease transmission rate ?(t). Duration of infectiousness (d) is the inverse of the recovery rate (?) (Jones, 2007). Basic reproduction rate depends on the division of the seasonal disease transmission rate (?(t)) and recovery rate (?). If this product is less than one (R_0 0 then disease will spread and the higher the number the more infectious disease is (Nicho, 2010).
Equation 11: Basic reproduction rate.
R_0=(infection/contact)×(contact/time)×(time/infection)= ? × c ×d= ?(t)/?

From the basic reproduction number, we can calculate the critical immunization threshold (qc), which is the minimum proportion of the population that must be immunized in order for the infection to die out in the population (Feng, Towers, & Yang, 2011).
Equation 12: The critical immunization threshold (q_c) (Feng, Towers, & Yang, 2011).
q_c=1- 1/R_0
The critical immunization threshold than helps us to calculate vaccine coverage, V_c. Vaccine coverage is calculated by dividing critical immunization threshold, q_c, with vaccine efficacy, E. Vaccine efficacy is the percentage of people who receive vaccine and become immune. Vaccination is not always 100% effective and its effectiveness vary. In some infectious disease’s vaccine has high efficacy and in some little (Feng, Towers, ; Yang, 2011).
Equation 13: Vaccine coverage (V_c) (Feng, Towers, ; Yang, 2011).
V_c= q_c/E= (1- 1/R_0 )/E= (R_0-1)/(R_0 ×E)

ASSUMPTIONS
In these model there also are some assumptions. Those are:
Population size is constant. The birth rate is equal to the rate of death. It is also assumed that there is no disease that cause death. Also, immigration is ignored (Hethcote H. W., 1994).
Age, sex, status and race do not affect the probability of being infected (Hethcote H. W., 1994).
The only way an individual leaves the susceptible group is by becoming infected and infectious group is left when individual recovers (Hethcote H. W., 1994).
Recovered individuals are assumed to have immunity for the disease (Hethcote H. W., 1994).
Those individuals who get vaccinated have 100% immunity against the flu even thou that is not true.
At the begging of each year seasonal disease start again with high number of sustainable individuals, group of infected individuals and zero recovered individuals.
For infected individuals is also assumed that they are also infectious (Hethcote H. W., 1994).
All rates are constant (Hethcote H. W., 1994) with the exception of the disease transmission rate, which is periodic function.

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