How Diseases Spread, SIR Models

Describing changes in populations with equations

A note about notation: N(t) or N(t-1) is read as "N at t" or "N at t minus 1", and should not be understood as " N multiplied by t" or "N multiplied by t minus 1". -N(t) is value at this time point -First N(t-1) is the population size at the previous time point -k is rate/fraction affected -± based on whether the effect of k is to increase or decrease the population --Arrow towards population in S&F: increase, + --Arrow out of a population in S&F: decrease, - --Tip: Draw a circle around your population to focus only on arrows that are coming or going -Second N(t-1) is the population that the rate applies to --k can depend on multiple populations, e.g. k·N(t-1)·M(t-1) --k can be independent of population, e.g. k

A Stock and Flow Diagram: SIR (epidemiology)

A stock is a function that outputs the size of a population at a specific time: stock(t). A flow measures the change of a stock over a period of time: flow(t).

Vaccines

All vaccines work by training the acquired immune system to recognize and attack pathogens, enabling the host to mount a successful, targeted defense to the disease if it is encountered in the future. Critically, vaccines accomplish this without infecting the recipients with the disease without them becoming ill! Vaccines present the human immune system with small bits of a pathogen, known as antigens. These antigens are typically specialized proteins from the surface of the pathogen (typically a virus or bacterium). Most historical vaccines were made from dead or weakened bacteria or viruses, or from pieces that were removed from pathogen particles. D

Bacteria

Bacteria are single-celled prokaryotic organisms, which lack membrane-bound nuclei. Bacteria are small, abundant, and diverse. Humans host between 500 and 1,000 species of bacteria. 1 There are nearly as many bacterial cells living on or in a person as there are human cells in the body. 2 • Most bacteria hosted by humans do not cause disease, and some even aid human health. Binary fission Bacteria reproduce through simple cellular division in a process known as binary fission, as shown on the right. Pathogenic bacteria fuel their growth and binary fission by consuming the host's resources. Binary fission allows bacteria to reproduce very quickly.

SIR model for a vector born disease (math)

Blue arrows indicate the movement of people between compartments in the model, which occurs as their health status changes from susceptible to infected to recovered. The yellow arrow represents the transfer of Plasmodium pathogens from infected mosquitoes to susceptible humans. It is this interaction that causes the disease to spread, rather than contact between susceptible and infected humans. As indicated by the yellow arrow, the pathogen (Plasmodium) is transferred to susceptible mosquitoes when they bite infected humans. The resulting change in health status of these mosquitoes is indicated by the blue arrow connecting susceptible and infected vector compartments.

Closer look at SIR model

Compartment models use rules to determine how many individuals should move between compartments during each time step. For instance, the number of infected individuals (1) could change because susceptible individuals become infected (increasing 1), and because infected individuals recover (decreasing I). The corresponding rule is: change in I = new infections - new recoveries Rules are then formalized as mathematical equations. To see how, think about the first term in the rule above. The number of new infections per day increases when: - There are more sick people around to infect others. - There are more susceptible people to get sick. - The disease is easily transmitted. new infections = BIS where B is the transmission rate you have been playing with-the average rate at which infected-susceptible encounters result in new infections; and I and S are the number of infected and susceptible individuals. New infections affect the size of both S and I compartments. A similar equation describes new recoveries, which affects both I and R compartments.

ocean acidification

Direct correlation between levels of atmospheric CO, and seawater CO, Inverse correlation with pH (i.e. direct correlation with increased dissolved H+)

Pathogen type matters

Fighting infection is a key to an organism's survival. The effectiveness of the defensive strategy taken by an organism-or your doctor- depends on the type of pathogen. Below are some of the more common strategies used to combat infections. Antibiotics are chemicals that target processes unique to bacteria (see diagram at right), which, as prokaryotes, differ dramatically from the eukaryotes they infect. Many eukaryotic pathogens can be combatted using poisons (chemicals) that are toxic to a given pathogen but not its host. Poisons work best if the pathogen's cells are distinct from the host cells in some crucial way. Antifungals, antiprotozoals, and anthelmintics all attempt to exploit such differences. Antivirals attempt to disrupt the function of viruses, for example by blocking receptors needed to enter a host cell, or by disrupting viral replication. The cellular function of the host is not affected.

Fungi

Fungi are eukaryotic organisms that have two growth forms. Some grow as single-celled yeasts, while others are multicellular and grow as branched, weblike structures known as mycelia. Fungi are important sources of food and medicine for humans. People eat mushrooms and rely on yeasts to make fermented foods like bread and beer. Fungi are also used to make antibiotic medicines like penicilin. Hundreds of fungal species live on or in the human body, and most of these cause no damage most of the time. 1

Helminths

Helminths are a diverse group of parasitic worms that include flukes, tapeworms, and roundworms. Helminths are internal parasites and exclude parasitic worms that attach to a host's surface. Helminths are animals and, as such, are multicellular and eukaryotic. While all helminths are parasitic, not all cause disease.In fact, some are hypothesized to help boost the immune system.

Birth and death

How did mask wearing affect the three values measured? Mask wearing decreased all of the values measured, having 50% mask wearing decreased all of the values a bit, and then having 100% mask wearing decreased all of the values by a lot, as seen in the large decrease in the number of people infected, number of deaths, and peak number of infections. What are the benefits of "flattening the curve?" You aren't stopping the amount of ppl who get sick overall, instead you are slowing down the spread of the disease and slowing down how many people get it over time, so that not everyone gets the disease all at the same time (sharp curve), rather we want the curve flatter and longer so that the people getting sick is spread out.B

Plasmodium life cycle

Plasmodium that infects both the liver and red blood cells of human hosts. The infection's blood stage brings the worst of malaria's symptoms. Like many vector-borne pathogens, Plasmodium has a complex life cycle (below) involving multiple hosts. One strategy for treating malaria is antimalarial drugs. For example, artemisinin-based combination therapies (ACTS). target the life-stage of the Plasmodium parasite that infects the blood and produces the disease's symptoms. Their primary benefit is that they help infected people recover from the disease. Might these drugs reduce the overall prevalence of malaria, too? B ACTs have been widely adopted because they are a safe and effective means of combatting the blood stages of the Plasmodium parasite. However, as your data suggest, ACTs can also slow the spread of the disease. They do this by reducing the abundance of the parasite's germ cells, which are the type of cells that infect mosquitoes when they take a blood meal. In other words, ACTs reduce the likelihood that mosquitoes become infected when they bite an infected human. Mosquito nets are very effective for slowing the spread of malaria. Besides reducing transmission, insecticide-treated nets can help fight malaria in another way: When widely used throughout a community these nets can reduce the size of the mosquito population, which helps protect everyone in the community.

Three determinants of disease spread

Population density_ (N): the density of individuals in the population, which determines how often individuals contact each other. Because your population occupies a fixed area, the population density is directly related to population size, and so here both are referred to as N. Transmission rate (B): the rate (per unit of time) of disease transmission between infected and susceptible individuals in the population. This rate depends both on how frequently susceptible and infected people make contact in a way that can transmit disease, and on the probability that such a contact actually produces a new infection. • Infectious period (L): the average period of time that an infected person can transmit the disease to a susceptible person.

Protozoans

Protozoans are single-celled eukaryotes that, unlike prokaryotic bacteria, have a well-defined, membrane-bound nucleus. Protozoans are incredibly diverse. Many are free-living while many others are parasitic, infecting hosts across the animal and plant kingdoms. Some, but not all, parasitic protozoans cause disease.

Above is a graph that shows progression of COVID-19 among the four subpopulations described above (see legend), but the model does not include waning immunity or use of the drug. Predict how the "S" and "A" lines would change once both interventions are added into the model, and justify your predictions.

S (Predicted change due to waning immunity): Presently the slope of the line of S is negative, and by adding in w, the slope would in turn become more positive (see example drawing below). This is because the variable w increases the amount of susceptible individuals since it sends a proportion of the recovered subpopulation back to S. S( Predicted change due to addition of drug) :The addition of the drug alone has no direct effect on the S sub-population since the drug halves the recovery time which does not impact susceptible individuals. A (Predicted change due to waning immunity): no direct effect A( Predicted change due to addition of drug): flatten slope

Describing differences in S,I, and R with math Write an equation to describe the change in the susceptible population at time t relative to the previous time point Write an equation to describe the change in the infected population at time t relative to the previous time point Write an equation to describe the change in the recovered population at time t relative to the previous time point

S--> I depends on an infected individual and finding a susceptible individual and spreading disease Susceptible people can only become infected, infected people can only become recovered S(t)=S(t-1) - BS(t-1)xI(t-1) I(t)=I(t-1)+BS(t-1)xI(t-1)-vI(t-1) R(t)=R(t-1)-vI(t-1) (this one is correct)

Create an SIR model that describes the progression of COVID-19 in a population. In addition to the basic SIR model, it should include: An asymptomatic population "A." All individuals who eventually become infected first enter an asymptomatic state. Some asymptomatic individuals progress to an infectious state; others progress straight to the recovered population without ever experiencing symptoms. Both infected and asymptomatic individuals can infect susceptible individuals. A small proportion of the recovered population "w" that has their immunity wane each day, and they become susceptible to infection again. Write an equation to describe the change in the asymptomatic subpopulation over time. You may use Python syntax or regular mathematical syntax, whichever you prefer.

S= Susceptible subpopulation A=Asymptomatic, infected subpopulation I=Infected, symptomatic subpopulation R=Recovered subpopulation β=Likelihood of an infected or asymptomatic individual transmitting to a susceptible individual ν=Frequency of an infected or asymptomatic human recovering (1/L) w=Proportion of the recovered population whose immunity wanes each day p=Frequency of an asymptomatic human progressing to an infectious state A[t]= A*[t-1] + β*S[t-1]*A[t-1]+β*S[t-1]*I[t-1] - p*A[t-1] - v*A[t-1}

Update the basic SIR equations below to describe how each subpopulation in your model changes with time accounting for births (birth rate b) and deaths (death rate d).

S[t] = S[t-1] - beta*S[t-1]*I[t-1] + b*(S[t-1]+I[t-1]+R[t+1]) - d*S[t-1] I[t] = I[t-1] + beta*S[t-1]*I[t-1] - v*I[t-1] - d*I[t-1] R[t] = R[t-1] + v*I[t-1] - d*R[t-1]

mosquitos and humans (FULL SIR equation)

Sh[t] = Sh[t-1] - beta*Iv[t-1]*Sh[t-1] Ih[t] = Ih[t-1] + beta*Iv[t-1]*Sh[t-1] - v*Ih[t-1] Rh[t] = Rh[t-1] + v*Ih[t-1] Sv[t] = Sv[t-1] - alpha*Ih[t-1]*Sv[t-1] Iv[t] = Iv[t-1] + alpha*Ih[t-1]*Sv[t-1]

What do S, I, and R stand for? Describe the characteristics of each population.

Susceptible (A susceptible individual is capable of being infected by a pathogen, if the pathogen is transmitted by an infected individual), infected (A susceptible individual is capable of being infected by a pathogen, if the pathogen is transmitted by an infected individual), recovered (A recovered individual is a formerly infected individual that is still alive but no longer capable of infecting another. Often a recovered individual has immunity against repeated infection for at least a short time) Key SIR model assumptions: -Once infected and recovered, an individual enjoys permanent immunity. -Everyone has an equal chance of contacting everyone else in the population. -As soon as individuals become infected, they are instantly infectious to others. The basic SIR model does not account for evolution. Neither does it account for diseases that are indirectly transmitted or infect more thar one species. Also, sometimes recovered people lose their immunity to a disease.

Basic reproductive number

The basic reproductive number for an infectious disease in a population, symbolized R, is the average number of individuals infected by an infected individual, assuming a population composed only of susceptible individuals. Ro depends on the initial density of the susceptible population, S, as well as on the transmission rate, B, and infectious period, L, of the disease: R0 = SBL If R. > 1, the disease will spread through the population. which can predict whether and how quickly a disease will spread in a population. Ro is the average number of individuals that a sick person infects in a population of susceptible individuals. Importantly, the value of Ro is specific to a given pathogen, population, and environment. -R0 describes the average number of people any one infected person will infect -R0 = SβL -If R0 < 1 Each infected person will infect 0 or 1 other The disease will eventually stop spreading through the population If the entire population is susceptible initially, the disease will spread if the basic reproductive number of the disease, Ro, is greater than 1. When people are vaccinated, the proportion of the population that is initially susceptible drops, and the number of contacts each infected person has with susceptible individuals declines. If enough people are vaccinated, each sick person will infect less than one person on average before recovering, and the disease will not be able to spread. The more contagious (high β, high L) a disease is, the higher R0 is The higher R0 is, the higher pc is — you need to vaccinate a larger proportion of people to achieve herd immunity

A pharmaceutical company has secured FDA approval for a COVID-19 drug, which when given to an infected or asymptomatic individual, halves the amount of time they take to recover. Which variable(s) in your SIR model above would you modify in order to incorporate the use of this drug into your model? How would it/they change?

The drug when administered affects the variable "v". This is because v represents the recovery rate and the drug increases (doubles) the rate by which individuals recover from either the asymptomatic or infected populations.

Human immune response

The innate immune response is rapid and generic, responding to different classes of pathogens in the same way. It includes barriers that help prevent pathogens from entering the body as well as internal defenses that respond when an invader gets inside. The acquired immune response is slower and more specific than the innate response. Using this system, the host "learns" to identify a pathogen when first exposed to it. Afterward the host rapidly responds to subsequent exposures, preventing new infections. For example, after you recover from being sick with chickenpox you are unlikely to get chickenpox again your immune system can effectively attack the type of invaders that made you sick before. In other words, you developed immunity. to that pathogen.

Herd Immunity

The level of vaccination that prevents disease spread indicates herd immunity. A whole "herd" is immune when enough individuals are resistant to the disease that the epidemic threshold is not reached. Even if some susceptible individuals remain in the herd, they are unlikely to get sick. All widespread immunization programs are built on this fundamental idea. Herd immunity. is established when the proportion of people who have been vaccinated is sufficiently high to ensure that each infected person infects less than one susceptible person, on average, before they recover (or die). Once herd immunity is established, the disease will no longer be able to spread.

Critical immunization threshold

The minimum proportion of the population that must be vaccinated to establish herd immunity is called the critical immunization threshold, Pc• This threshold varies from disease to disease. For example, when you simulated smallpox and measles outbreaks and varied the proportion vaccinated, you discovered that pc is higher for measles than for smallpox. This is because measles spreads so much more readily, having a much greater Ro. The critical immunization threshold depends on Ro in a precise way: pc = (1- (1/R0)) Importantly, herd immunity is achieved whenever the proportion vaccinated is greater than Pc• This relationship is graphed on the right. It shows that herd immunity is established if the critical immunization threshold (yellow line) has been exceeded, and that the disease will continue to spread if it has not. p is the proportion of the population (a fraction between 0 and 1) that is immune to a disease pc is the critical immunization threshold, the proportion of the population that needs to be immune to achieve herd immunity (R0 < 1) Also known as Herd Immunity Threshold (HIT) Note: pc is lower than the number of people who need to be vaccinated -- vaccines are not 100% effective, so therefore: Vaccination effectiveness (E or Ve) must be considered.

What subpopulation(s) in the SIR model do(es) p affect? Why?

The subpopulation in the SIR model that p affects is the susceptible, infected, and recovered populations because the susceptible people become recovered through vaccination. This directly decreases the susceptible population and increases the recovered population. Since there is a decrease in the susceptible population this decreases the number of individuals that can become infected.

p impact on SIR What subpopulation(s) in the SIR model do(es) p affect? Why? Does p affect these populations on a one-time or ongoing basis? If one-time, at which time? Add a dashed arrow representing a fraction of the population, p, being immune (due to vaccination) at time=0 to the stock and flow diagram below.

The subpopulation in the SIR model that p affects is the susceptible, infected, and recovered populations because the susceptible people become recovered through vaccination. This directly decreases the susceptible population and increases the recovered population. Since there is a decrease in the susceptible population this decreases the number of individuals that can become infected. 1-p because p is the amount of people vacccinated (not susceptible) so you must subtract these people from the total amount (p is a proportion so subtract from 1)

SIR model for a vector born disease

There are many susceptible humans (high Sh) There are many infected vectors (high Iv). The disease transmits easily from vector to human when bites occur (high Bh). Thus: new human infections = Bh Sh Iv In this model, B describes how frequently, on average, an infected vector (mosquito) transmits the pathogen to a human. An analogous equation describes new mosquito infections as the product of Bh, Sh, and Iv As with other SIR models, the transmission rates (Bh and Bv) depend on both the average frequency at which infected individuals contact susceptible individuals (i.e., frequency of mosquito bites), as well as the probability that a contact (i.e., mosquito bite event) leads to transmission of the parasite. This model assumes: The infectious period, L, is the average length of time infected people remain sick. Human and vector populations are maintained by constant birth and death rates (not shown)..

Viruses

Viruses are not really "alive". They are microscopic, ubiquitous parasites made up of tiny packets of genetic information--either RNA or DNA--wrapped in a protein coat. All viruses reproduce by hijacking the cellular machinery of their hosts. Viruses are not cellular-based life forms. They do not feed, grow, store energy, or metabolize on their own, but instead rely entirely on host cells. Like pathogenic bacteria, viruses also infect human hosts, but they use strikingly different methods. A virus is about 1/100th the size of a bacterial cell. Because a virus is not itself a cell, it cannot replicate on its own. To reproduce, it must penetrate a host cell, hijack the host cell's reproduction machinery, and force the host cell to make new virus particles. Then the new virus particles burst out of the host cell and move on to infect other cells. Most viruses cannot survive for very long outside a host's system.

Disease transmission modes

With direct transmission the pathogen moves directly from an infected individual to an uninfected one. For people, this might happen with a handshake, sexual contact, or getting sneezed or coughed on. Many common human diseases can be directly transmitted, such as influenza, colds, COVID-19, and sexually transmitted diseases like chlamydia and gonorrhea. By contrast, indirect transmission occurs when a pathogen travels from one host to another via some intermediate agent- a vehicle. In inanimate transmission, the agent might be a doorknob, an improperly sterilized medical device, food, or water. Most hospital-acquired infections are transmitted this way, as are food-borne pathogens like Salmonella, and diseases transmitted by the fecal-oral route like polio and cholera. In animate transmission, living organisms serve as the agents of indirect disease spread. The organism responsible for transmission is called a vector. Some vectors become infected themselves by a pathogen whose life cycle includes more than one host species. Malaria (vector: mosquito) and Lyme disease (vector: tick) are both spread this way.

Based on the data in Table 2, which specific conditions are required for gametocytes to differentiate into mature gametes? (4 pts)

XA of 0.1mM in temp of 32°C in a pH of 7.4

Links between increasing atmospheric CO levels and ocean acidification

reduced pH

SIR for birth and death rates -- math

s--> I -People move from S to I when they get infected. A transmission happens when an infected person (I) bumps into a susceptible person (S) with some probability (β). I-->R -People move from I to R when they recover. An infected person (I) has some probability (𝜈 = 1/L) of recovering.