SIR ModelsCoalter Palmer and Kailee SilverApril 20181 Differential EquationsA differential equation is one which involves derivatives of a function or functions. It represents therelationship between a continuously varying quantity and its rate of change. They usually represent aphysical quantity of a derivative. Differential equations, for this reason, are often used in biology, physics,and economics.2 Introduction to SIR ModelsSIR models, created by Kermack and McKendrick in 1927, provide a simple way to monitor and predictthe dynamics of an infectious disease in a large population. We assume the population exists in threetypes of individuals, of whose numbers are denoted by S, I, and R. These are all functions of time, t,and change according to a system of differential equations.S: stands for the number of susceptible individuals- those who are not infected but can be infected.I: is the number of infected individuals. These people have the disease and are able to transmit it toother susceptibles.R: represents the Recovered fraction of the community, assuming that as they are recovered, theywill not get the disease again (for some especially bad diseases this number will be extremely low or even0, showing that few people actually recover). These people cannot transmit the disease to others.R0 is another important variable, and is defined as ”the average number of secondary cases caused byan infectious individual in a totally susceptible population” So, if the R0 value of a disease is 2, then thatmeans that on average, if a single infected individual is added to a population, thats how many peoplewill be infected However, this R0 number can be lowered by the development of antibiotics or medicine,and/or by improving the conditions of the environment in which the disease is spreading. However, thisnumber can also be raised by unfavorable conditions. For example, lack of access to clean water canincrease the spread of waterborne illnesses and of diseases that spread through contact (washing hands,etc.), and thus increases the number of secondary cases typically caused by a single individual.SIR Models are considered under a short enough time scale that births and deaths (besides thosefrom the disease) can be neglected.3 Equations for SIR ModelsNew infections occur when there is an interaction between infected and susceptible individuals. The rateat which these new infections occur is SI, for some positive constant . When a new infection occurs,the infected individual is now removed from the susceptible population and into the infected population.There is no way other way other than getting infected by which individuals can enter or leave or enterthe susceptible class. This brings the first differential equation:ddtS=−βSIIn addition to the S and the I, you may notice theβin this equation. Thisβis a value for thetransmissivity of the particular disease.The other process that can occur is when the infected individuals enter the removed class. We assumethat this happens at the rateγI for some positive constantγ. That brings the other two differentialequations:1
ddtS=βIS−γIddtR=γINota Bene:The total populationS+I+Ris constant becausedSdt+dIdt+dRdt=−βIS+ (βIS−γI) +γI= 0For convenience purposes, the unit of population is chosen so that the total population is 1. IfI= 0then there are no infectives so the right sides of all three of the equations are 0 so nothing changes. Onemust start with a certain number of infectives.In the medical world, an epidemic is classified as a scenario whereddtS >0, so the second equationcan also be used to determine if something is an epidemic or not.Some questions to be considered when using SRI models:How long will the epidemic last?How will the epidemic end? Will there be any susceptibles left when it’s over?If a small portion of the population is infected and the majority is susceptible, will the number ofinfectives increase substantially and create an epidemic? Or will the disease deteriorate?4 Historical Examples: The Ebola Crisis of 2014/2015In 2014, the Ebola outbreak in West Africa made international headlines. Ebola is an ultra-deadly virusthat spreads very easily, so efforts immediately went towards containing the disease.Ultimately, the SIR model could only help predict the number of individuals who would be infectedin the future. While this one number may not seem very helpful, it turns out that predicting the numberof individuals who will have a virus is vital because it helps countries and organizations know whatresources they need to allocate (”doctors, medication, mobile clinics, money, or international aid”) andhow many. In brief, SIR models are helpful in determining just how bad a disease is, and how badit could potentially be if left unattended to; by simply having a somewhat accurate estimate for howmany individuals have been infected, mathematicians can use this model to create a tangible, real-worldimpact, making sure that organizations provide enough resources to quell the spread of the disease.Unlike many diseases, having Ebola once does not make you immune from potentially contracting itin the future. This must be factored into the SIR model.5 The Zombie ApocalypseTheoretically, the zombie apocalypse can also be used with the SRI model using biological assumptionsfrom popular zombie movies. The three basic classes of this model are:Susceptible (S)Zombie (Z)Removed (R)Note that susceptibles can die a non-zombie related death, denoted by the symbolδ. The removedclass in this case are the individuals who have died, either naturally or from a zombie attack. However,people in this class can resurrect and become ”undead”, as a zombie, which is represented byζ. Theonly way for susceptibles to become a zombie is to have an encounter with a zombie,β. So zombies canonly come from being resurrected from the newly deceased (removed group) or from susceptibles whobecome infected from a zombie. Birth rate is also held constant atπ.Additionally, zombies can be defeated and thus moved to the removed group, denoted byα.The basic model is given by the below equations:S′=π−βSZ−δSZ′=βSZ+ζR−αSZR′=δS+αSZ−ζR2
This model proves to be slightly more complicated than a basic SIR model given the two-masstransmissions leading to having more than one nonlinear term. A mass-action incidence is one wherean average member of the population makes a substantial amount of contact to transmit infection withβNothers per unit time, where N is the total population without infection. (Infection in this case iszombification.)The figure above shows the configuration and relationship between susceptibles, zombies, and theremoved populations.The probability that a contact is randomly made by a zombie with a susceptible is S / N. The numberof new zombies through this transmission in unit time per zombie is:(βN)(S/N)Z=βSZIt must also be assumed that a susceptible can avoid zombification through defeating the zombie.Each susceptible is capable of resisting infection at a rateα. Using the same idea as above, the probablyZ / N , the number of zombies destroyed by a susceptible is:(αN)(Z/N)S=αSZNota Bene: t−→∞and π6= 0The figure below shoes a basic model of a zombie outbreak scenario, with an R0 value>1 :In the figure above,α= 0.005,β= 0.0095,ζ= 0.001,andδ= 0.001.Susceptibles are quickly eradi-cated and zombies quickly infect everyone. Thus, coexistence between human and zombies is impossible.