Keynesian economics cover theories of the recession. So, to avoid duplication, only the theoretical framework of the Lockdown policy are stated in this section.
Each model of an epidemic is usually represented by letters that indicate the type of model. The "SI model" (sometimes called the simple epidemic model), For example, is a model in which any susceptible person (S) will never recover (R) if they are Infected (I). However, in the "SIS model", infected people will be recovered and are still susceptive to infection (e.g. gonorrhea). In the "SIR model", infected people will be recovered and become safe; That is, they do not become Infected again (for example, measles or the flu). The "SEIR model" is the same as the previous model, except that the disease has a Latent (or Exposed) period (E). Information about Covid-19 is not yet conclusive. For example, we do not know whether people who have recovered will become infected again or not. So we have to assume that the recovered are immune (or at least immune for a while, or maybe infected with other mutations in the virus). For this reason, we consider the Covid-19 epidemic as a SIR model.
Suppose that, St People are susceptible to Covid, at time t; It people affected, Rt is the number of people improved, and N is the size of the population6.so:
$${N}_{t}={S}_{t}+{I}_{t}+{R}_{t}$$
If we divide the number of these three groups by N:
$$1={s}_{t}+{i}_{t}+{r}_{t}$$
st is the percentage of susceptible individuals of the total population, it is the percentage of infected individuals, and rt is the percentage of improved individuals.
When a susceptible person is in contact with an infected person, everyone in the susceptible group can become infected and go into the infected group. The contact variable varies in different diseases. In the case of Covid-19, contact means bringing people closer together so that the disease is transmitted through breathing. In addition, being in contact with a sick person does not necessarily guarantee to get sick. Let α show the probability of infection to Covid-19 due to contact with an infected person.
In period 0, we have I0. This is called the initial state of the system. Assume that each infected person is in contact with γ healthy person at any given time. Therefore, the number of potential new patients is γI0. However, as mentioned, not all contacts lead to disease transmission. Therefore, each infected person can develop αγ new infection at any time. We show this value as β. Β is the average number of possible transmissions from an infected person in each period. So in any period, an infected person can develop βst new case. We also assume that k percent of patients will be recovered.
Now we need a set of differential equations to show the dynamics of the system. Due to the transmission of the disease, βstIt person decreased from group "s", in each period; for this reason, we have in period t + 1:
$${S}_{t+1}={S}_{t}-{\beta S}_{t}{I}_{t}$$
Also, the equation for the recovered is:
$${R}_{t+1}={R}_{t}+k{I}_{t}$$
In each period, k percent of an infected person recovered and fell into the "R" group.
Finally, the changes in the population of the affected group in each period increase as new cases rise and decrease as the number of recovered in each period:
$${I}_{t+1}={I}_{t}+{\beta S}_{t}{I}_{t}-k{I}_{t}={I}_{t}(1+{\beta S}_{t}-k)$$
Of course, each of these three equations can be written in terms of proportional variables:
$${s}_{t+1}={s}_{t}-{\beta s}_{t}{i}_{t}$$
$${r}_{t+1}={r}_{t}+k{i}_{t}$$
$${i}_{t+1}={i}_{t}(1+{\beta s}_{t}-k)$$
With the constrain that
$${s}_{t+1}+{i}_{t+1}+{r}_{t+1}={s}_{t}+{i}_{t}+{r}_{t}=1$$
As long as \({i}_{t+1}>{i}_{t}\), we are above the epidemic threshold, and the number of infections will increase. This means\({\beta S}_{t}>k\). The parameter k is a function of biological and physiological characteristics of individuals and is in the field of specialization of physicians and medical researchers. However, β is the function of the social behaviour of the population and can be controlled. It may be recommended, for example, that sick children and workers stay at home so that others stay less infected. Therefore, people can be quarantined as long as there is a Coved-19 pandemic.
6Here we assume that both the population and its turnover are constant.