The local stability of a modified multi-strain SIR model for emerging viral strains
We study a novel multi-strain SIR epidemic model with selective immunity by vaccination. A newer strain is made to emerge in the population when a preexisting strain has reached equilbrium. We assume that this newer strain does not exhibit cross-immunity with the original strain, hence those who are vaccinated and recovered from the original strain become susceptible to the newer strain. Recent events involving the COVID-19 virus shows that it is possible for a viral strain to emerge from a population at a time when the influenza virus, a well-known virus with a vaccine readily available, is active in a population. We solved for four different equilibrium points and investigated the conditions for existence and local stability. The reproduction number was also determined for the epidemiological model and found to be consistent with the local stability condition for the disease-free equilibrium.
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[00:00:01.860]Good day everyone, I am Miguel Fudolig
[00:00:04.010]and I am here to present my study titled the local stability
[00:00:07.520]of a modified multi-strain SIR model
[00:00:10.270]for emerging viral strains.
[00:00:12.110]Which I worked on under Dr. Howard.
[00:00:15.270]This research was published in 2020 in PLOS One.
[00:00:20.640]When we first conceptualize this project
[00:00:22.990]we were thinking about studying the dynamics
[00:00:25.510]of selective immunity through immunization
[00:00:28.050]for different flu vaccines.
[00:00:30.230]However, when we were in the middle of writing the paper,
[00:00:33.640]COVID-19 spread everywhere around the world.
[00:00:36.720]One of the things that are commonly overlooked
[00:00:39.200]in studying COVID-19 is that the Corona virus emerged
[00:00:43.200]during the flu season.
[00:00:45.380]In the 2019-2020 season, there were a lot of cases
[00:00:50.570]of influenza-like illnesses that were also inflated
[00:00:53.240]with COVID-19 cases.
[00:00:55.480]If you take these two viruses
[00:00:58.230]that shared similar symptoms with one of those strains
[00:01:02.310]having an existing vaccine,
[00:01:03.920]that is the flu virus.
[00:01:05.520]Or the emergent strain,
[00:01:06.417]the COVID-19 virus is unaffected by the vaccine.
[00:01:11.350]This begs the question of whether
[00:01:13.120]these two viruses could co-exist in a population
[00:01:16.780]and what the conditions are for this to happen.
[00:01:19.920]This can also be relevant
[00:01:21.300]to the recent emergence of vaccine related resistant strains
[00:01:25.220]of COVID-19 in some of the countries around the world.
[00:01:31.000]And so there are a lot of ways
[00:01:33.020]to model the spread of a virus in a population.
[00:01:36.410]One of the most prominent approaches is
[00:01:38.340]the compartmental model approach,
[00:01:40.890]where the population is separated
[00:01:42.780]into exclusive compartments.
[00:01:45.190]And individuals moved from one compartment
[00:01:47.573]to another based on specified transition rates.
[00:01:51.060]The most common compartmental model used
[00:01:53.490]for viral epidemics is the susceptible infected
[00:01:57.400]and recovered model, commonly known as the SIR model.
[00:02:01.440]Where there are three compartments corresponding
[00:02:03.580]to individuals susceptible in contacting the virus: S.
[00:02:08.130]Infected individuals: I.
[00:02:11.340]And recovered individuals: R.
[00:02:13.290]So that's where the SIR compartments come in.
[00:02:18.500]This model has been modified by adding
[00:02:20.660]or removing compartments,
[00:02:21.770]so it counts for different scenarios
[00:02:23.510]such as vaccination and the existence of multiple strains.
[00:02:28.920]In this paper we aim to modify
[00:02:31.110]the existing SIR compartmental model.
[00:02:35.130]that will describe the spread
[00:02:36.874]of an emergence training the population
[00:02:40.460]that has been affected by an existing virus
[00:02:43.470]with a vaccine available.
[00:02:45.300]We also solve for the basic reproduction number
[00:02:47.900]of the model, which quantifies how contagious a virus is
[00:02:51.560]in a population.
[00:02:53.440]This is an important parameter
[00:02:55.160]in creating compartmental models for epidemics.
[00:02:58.640]We'll investigate the different cases
[00:03:00.320]for the long-term behavior of the system
[00:03:02.320]as well as the conditions for their existence and stability.
[00:03:08.550]Here's the model that we came up with
[00:03:11.170]for the emergence of the new strain.
[00:03:14.200]Before the emergence only the susceptible,
[00:03:18.154]S, I1, R are the susceptible infected
[00:03:22.380]and recovered compartments.
[00:03:24.810]And then V is the vaccinated compartment,
[00:03:27.360]which is isolated.
[00:03:29.070]Upon emergence, we introduce a second compartment:
[00:03:33.220]I2 for the infected.
[00:03:35.140]Those infected by the emergent strain and R2
[00:03:38.480]which are the ones that recovered from the emergent strain.
[00:03:42.170]Note that the vaccinated compartment
[00:03:44.930]and the R1, the initial recovered
[00:03:48.800]compartment for the initial or existing strain
[00:03:52.470]can still be susceptible to contacting the emergent virus.
[00:03:57.720]But the other way around is not the case.
[00:04:00.360]One of the assumptions of this model
[00:04:02.180]is that once infected by the emergent strain
[00:04:04.990]individuals are unlikely to be affected
[00:04:07.120]by an existing strain.
[00:04:10.620]To determine how infectious a disease is within a model
[00:04:14.360]or how infectious the system is,
[00:04:17.430]we solve for the basic reproduction number.
[00:04:19.920]This is defined as how many individuals an infectious agent
[00:04:24.160]can infect as long as the individual is infectious.
[00:04:28.210]For this specific model we determined
[00:04:30.220]that the reproduction number
[00:04:31.420]is just the higher reproduction number
[00:04:35.880]of the existing strain which is this.
[00:04:39.050]And then this one is the reproduction number
[00:04:41.590]of the emergent strain, if they existed individually
[00:04:46.100]in a certain population.
[00:04:50.080]This means that whichever is more contagious on its own
[00:04:52.850]would dominate the infection of the population.
[00:04:58.530]Now we investigate the long-term behavior of the population.
[00:05:02.800]There are four cases that we should consider.
[00:05:04.900]The Disease Free Equilibrium known as DFE.
[00:05:08.090]The Original Strain Equilibrium
[00:05:09.610]where only the original strain survives.
[00:05:12.360]The New Strain Equilibrium,
[00:05:13.610]where only the emergent strain survives.
[00:05:16.560]And the Endemic Equilibrium,
[00:05:17.960]where the two strains coexist.
[00:05:21.550]Noteworthy instances are
[00:05:25.010]that the Original Strain Equilibrium
[00:05:27.530]and in Endemic Equilibrium depends on the removal rate
[00:05:31.060]of the original strain.
[00:05:33.880]And the Emergent Strain Equilibrium can exist
[00:05:36.000]even if the existing strain already exists on its own.
[00:05:41.090]This is also shown in this plot that we have
[00:05:47.150]where the four colors designate
[00:05:48.870]the four possible equilibrium states of the population.
[00:05:52.760]The large area for the Emergent Strain Equilibrium
[00:05:56.220]is expected because of its inherently bigger,
[00:05:58.890]susceptible pool of S, V and the initial recovered R1.
[00:06:05.554]The existence of the Endemic Equilibrium and the original strain,
[00:06:09.860]which I said earlier it depends on the recovery rate
[00:06:12.050]of the individuals infected by the original strain.
[00:06:15.920]Which is denoted by this interface.
[00:06:23.220]In summary, we modified the SIR model to describe the spread
[00:06:27.890]of an emergent strain in the population
[00:06:30.280]where an older strain already exists,
[00:06:32.730]with a vaccine that is ineffective against
[00:06:36.250]the emergent strain.
[00:06:37.800]The basic reproduction number was then determined
[00:06:40.630]to be the higher basic reproduction number
[00:06:43.370]of the two strains.
[00:06:45.460]And we discovered four different cases
[00:06:48.100]of long-term behavior: the Deceased Free Equilibrium,
[00:06:51.630]the Original Strain Equilibrium,
[00:06:53.530]the Emergent Strain Equilibrium and the Endemic Equilibrium,
[00:06:57.320]which we determined
[00:07:00.030]the existence and stability conditions
[00:07:02.620]for each of these cases.
[00:07:07.200]And that is the end of my presentation.
[00:07:11.370]And I hope you have a great day.
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