STAT601.29:BayesianInfectiousDiseaseModellingAssignment1-Winter2023

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1. SIRModel

The file 'Q1Cases.csv' below contains weekly case reports of an emerging disease in a city with a populationof 12,000 people. The disease is believed to have an infectious period of 1-3 weeks, and a very shortlatent period of a day or two, which for the purposes of modelling we shall ignore.We shall alsoignoreunderreporting,births,deathsandimmigration.

week	newcases
1	0
2	2
3	5
4	9
5	8
6	13
7	25
8	41
9	71
10	138
11	238
12	367

i. WithaviewtofittinganODE-based,population-levelSIRmodel,writeafunctionin R toconvert the new case data to prevalence data (i.e., a time series containing the number ofindividualsintheinfectious stateoneachday)undertheassumption thattheinfectiousperiodis two weeks long, and that new cases begin their infectious period the week they are recordedinthedata.

ii. Useleast-squaresestimationtofitthepopulationlevelSIRmodeltothisdata,undertheassumption that the infectious period is two weeks long. Assume that there is one unrecordedcasewhobecomes infectiousatthebeginningofweek0(weekbefore thefirstcasereportinthedata) and is the source of the epidemic. The remaining population is assumed to be susceptibleat this time.Report your estimate of the transmission rate and a plot showing how well themodelfitsthedata.

iii. Repeat part ii), first assuming the infectious period is one week long, and then again assumingtheinfectiousperiodisthreeweekslong.

iv. Fromtheabovewhatdoyouthinkismostlikely thecorrectinfectiousperiod(ofthethreevalues tried)?Do you see any potential problems using this process to identify the infectiousperiod?

2. SIRModelwithVaccination

Adapt your ODE-SIR function from Question 1 to allow for the effect of a introducing a vaccinationprogram during the epidemic by allowing for susceptibles to move into the removed category at a rate.Assume the vaccination program will begin at the beginning of week 13.Assuming a two weekinfectious period, use the transmission rate estimated in Question 1 part (ii), and simulate forwardfromthelastweekofdatacollection, usingthestatespredicted bythemodelatthatpoint.

Now determine to two significant figures of accuracy, the raterequired to bring the epidemicunder control within 10 weeks of the vaccination policy being introduced.For the purposes of thisexercise we will define "under control" as there being fewer than 20 infectious individuals at 20 weeksafterthepolicyisintroduced.Provide evidencethatyouransweriscorrect.

3. TwoPopulations(Parasite andHost)

We are now going to code up an ODE-based population-level model of two interacting species pop-ulations, in which one of the populations represents a host (let's call this HUMAN), and the otherpopulationrepresentsaparasite(let's callthisMOSQUITO).

WewillassumethatinthehumanpopulationthediseasefollowsanSIRcompartmentalmodel;andin the mosquito population the disease follows an SI compartmental model.We will also assumemosquitoes can infect humans, but not other mosquitoes;and that humans can infect mosquitoes, but not other humans.Further we will allow the infection rates between the two species to differ.Bothtypeofinfectioncanoccurwhenamosquito bitesahuman.

In addition, we will assume the human population has a constant birthrate, and a constant naturaldeath rate which affects susceptible, infectious and removed individuals at an equal rate. However,wewillallowforthebirth rate anddeathrateforhumanstodiffer.

For the mosquito population, we will also assume a constant birthrate, a constant death rate affectingsusceptiblesandinfectiousindividualsatanequalrate,andallowforthebirthrate anddeathrates todiffer.

(Note here, the disease itself is assumed not to kill either host, and the removed state in humans canbethoughtofasbeingrecoverywithacquiredimmunity).

We will assume that the infectious period for humans is 10 time units. Additionally, we will assumethatattimezero(t=0)wehaveapopulationof1000humans,999ofwhicharesusceptible,andone of which has just become infectious, as well as a population of 1000 mosquitoes all of which aresusceptible.

(Note that, human-mosquito analogy breaks down here since in reality the population sizes would notbeanywherenearequal,butitwillbeeasiertoplotthedynamicsonasinglegraph.Youcouldthinkof each mosquito in this system representing many thousands in reality, or imagine we are on an alienworld wheremosquito-likeparasites andhuman-likehostsdohavesimilarsizedpopulations!)

We will also assume a density-dependent model, in which both the number of susceptible, and thenumberofinfectious,individualshasaneffectontransmissionrates.

Findparametervalueswhichresults in:

i. an equilibrium state in which the prevalence (number of infectious individuals) of the diseaseremains between 5 and 995 individuals in both populations,but is higher in the mosquitopopulationthanthehumanpopulation;

ii. an equilibrium state in which the prevalence (number of infectious individuals) of the disease re-mains between 5 and 995 individuals in both populations, but is higher in the human populationthanthemosquito population.

Record down your model (ODEs), code, the parameter values you chose and show the dynamics ofeach system in a plot.Describe qualitatively why the systems you created produce the outcomesobserved.