# The contacts multiplied by the proportion of susceptibles

The second one will be the proportion of people in the different groups. Then N will be the total population. s(t)=S(t)N which is those who are susceptiblei(t)=I(t)N those who are infectedAnd r(t)=R(t)N those who have recoveredUsually in a perfect situation, it would be expected that those who are susceptible decrease overtime, those have recovered increase overtime, and those who are infected increase and then decrease overtime. And no matter at what point in time,  s(t) + i(t) + r(t)= N.  Keep in mind that this investigation is still choosing to ignore births and migration.  So the only manner in which someone can leave one group is by becoming infected, dying, and or recovering. Something else that should considered is that both the proportion of susceptibles and infected depends on the amount of contact between each of them.  For the investigation one should suppose the infected people have a set number of contacts per day, b. And if the population is continuously interacting each infected person should create a set number of new infected individuals per day, which will be shown as  bs(t). For terms of simplicity the investigation will also assume the number of infected who recover on any day is a fixed proportion, k. Now one can observe the derivatives of the dependent variables. The first equation will be the change in the amount of susceptibles equals the negative number of contacts multiplied by the proportion of susceptibles out of the total population multiplied by the amount of infected people. dSdt = -bs(t)I(t)The amount of infected was included in this equation because they directly affect those who can still get infected. The negative sign was included because there shouldn’t be any contact. Then we can obtain the differential equation for the change in the suscpetible proprtion of people from teh total population which should equal the negative number of contacts multiplied by the product of the susceptible and infected proportions. dsdt=-bs(t)i(t)The second equation is the change of the amount recovered should equal the fixed number of people infected that will recover multiplied by the proportion of infected from the total population. drdt=ki(t)This follows what the investigation has assumed as it includes the the fixed fraction of those that will recover and the proportion of infected from the population. The third equation is the change of the amount infected should equal the fixed fraction of contacts multiplied by the product of the proportion of susceptibles and infected subtracted by the set amount of people that will recover and then finally multiplied by the proportion infected. didt=bs(t)i(t)-ki(t)