Delay-Differential Equations. Tools for Epidemics Modelling

Delay-Differential Equations. Tools for

Epidemics Modelling

𝑏𝑦 𝐼𝑔𝑛𝑎𝑠𝑖𝐺𝑟𝑜𝑠

Contents

Delay-Differential Equations

• Differential Equations• Example• Stability• Initial Conditions

• Delay-Differential Equations• Solution• Behaviour of Solutions• Stability

Model Types with Example

• SIR Model• Incubation Period Considered

• With Delay-Differential Equations• By Compartments (SEIR Model)

• Symptoms Considered (SEAIR Model)• Space Spread

Conclusions

Delay-Differential Equations

• Differential Equations• Example• Stability• Initial Conditions

• Delay-Differential Equations• Solution• Behaviour of Solutions• Stability

Differential-Delay Equations

The PendulumDifferential Equations

https://en.wikipedia.org/wiki/Rheonomous


The PendulumDifferential Equations (Stability)


Equilibrium Points:• º• º


The PendulumDifferential Equations (Stability)


Equilibrium Points:• º Stable• º Unstable


A Differential Equation:Differential-Delay Equations (Initial Conditions)


A Differential Equation:With one initial value, it’s enough!

Differential-Delay Equations (Initial Conditions)


A Differential Equation:With one initial value, it’s enough!At time ,



A Differential Equation:With one initial value, it’s enough!At time , So And the function can be “constructed”



A Differential-Delay Equation:Differential-Delay Equations


A Differential-Delay Equation:If at , ,



A Differential-Delay Equation:If at , , then



A Differential-Delay Equation:If at , , thenSo we cannot “construct” our function. And if we specify



A Differential-Delay Equation:If at , , thenSo we cannot “construct” our function. And if we specify Still not enough, because we can “construct” just a little and then we don’t know .



A Differential-Delay Equation:If at , , thenSo we cannot “construct” our function. And if we specify Still not enough, because we can “construct” just a little and then we don’t know .We need all values of for between -1 and 0. (Infinite Values)



A Differential-Delay Equation:To solve it, we try ,

Differential-Delay Equations (Solution)


A Differential-Delay Equation:To solve it, we try , and we find



A Differential-Delay Equation:To solve it, we try , and we findHence



A Differential-Delay Equation:To solve it, we try , and we findHenceWith infinite complex solutions.



A Differential-Delay Equation:To solve it, we try , and we findHenceWith infinite complex solutions. A solution will be a linear combination, for example:



A Differential-Delay Equation:To solve it, we try , and we findHenceWith infinite complex solutions. A solution will be a linear combination, for example:Or



A Differential-Delay Equation and its Characteristic Equation:Let’s consider one posible solution:

Differential-Delay Equations (Behaviour of Solutions)


A Differential-Delay Equation and its Characteristic Equation:Let’s consider an example:And let

.




.If for example and ,




.If for example and ,Whereas if and ,



A Differential-Delay Equation:Let’s consider an example:And let

.If for example and ,Whereas if and ,

Differential-Delay Equations (Stability)

Model Types with Example

• SIR Model• Incubation Period Considered

• With Delay-Differential Equations• By Compartments (SEIR Model)

• Symptoms Considered (SEAIR Model)• Space Spread

Model Types

In this model population is divided in 3 classes:SIR Model

Model Types

In this model population is divided in 3 classes: Susceptibles: People who are not infected but susceptible of being infected. Infecteds: People infected by the disease who can transmit it. Recovereds: People once infected, but now recovered and not susceptible of becoming infected again.

SIR Model

Model Types

In this model population is divided in 3 classes: Susceptibles: People who are not infected but susceptible of being infected. Infecteds: People infected by the disease who can transmit it. Recovereds: People once infected, but now recovered and not susceptible of becoming infected again.The Model is described by the following system:

SIR Model

Model Types

In this model population is divided in 3 classes: Susceptibles: People who are not infected but susceptible of being infected. Infecteds: People infected by the disease who can transmit it. Recovereds: People once infected, but now recovered and not susceptible of becoming infected again.The Model is described by the following system:for some , which are called the contagion and recovery parameters, respectively.

SIR Model

Susceptibles Infecteds

Recovereds

Model Types

This time we consider that the infected individuals don’t become infectious until a time of seconds passes. SIR Model with Incubation Period

Model Types

This time we consider that the infected individuals don’t become infectious until a time of seconds passes. Therefore the infectious individuals are those infecteds seconds before, minus the ones that have recovered in this period. To substract these, we add a factor of .

SIR Model with Incubation Period

Model Types

This time we consider that the infected individuals don’t become infectious until a time of seconds passes. Therefore the infectious individuals are those infecteds seconds before, minus the ones that have recovered in this period. To substract these, we add a factor of .At the same time, we consider that in the incubation period they cannot recover.


Model Types

This time we consider that the infected individuals don’t become infectious until a time of seconds passes. Therefore the infectious individuals are those infecteds seconds before, minus the ones that have recovered in this period. To substract these, we add a factor of .At the same time, we consider that in the incubation period they cannot recover.The SIR Model was:for some .


Model Types

This time we consider that the infected individuals don’t become infectious until a time of seconds passes. Therefore the infectious individuals are those infecteds seconds before, minus the ones that have recovered in this period. To substract these, we add a factor of .At the same time, we consider that in the incubation period they cannot recover.SIR when adding infectiousness delayed:for some .


Model Types

This time we consider that the infected individuals don’t become infectious until a time of seconds passes. Therefore the infectious individuals are those infecteds seconds before, minus the ones that have recovered in this period. To substract these, we add a factor of .At the same time, we consider that in the incubation period they cannot recover.SIR when adding recovery delayed:for some .


Model Types

This time we consider that the infected individuals don’t become infectious until a time of seconds passes. Therefore the infectious individuals are those infecteds seconds before, minus the ones that have recovered in this period. To substract these, we add a factor of .At the same time, we consider that in the incubation period they cannot recover.SIR Model with Incubation Period Considered:for some .


Model Types

We consider again incubation period but by just introducing a new class: The Exposed. SEIR Model

Model Types

We consider again incubation period but by just introducing a new class: The Exposed. They are infected but not infectious, and they canot recover.

SEIR Model

Model Types

We consider again incubation period but by just introducing a new class: The Exposed. They are infected but not infectious, and they canot recover.SIR Model was:for some

SEIR Model

Susceptibles Infecteds

Recovereds

Model Types

We consider again incubation period but by just introducing a new class: The Exposed. They are infected but not infectious, and they canot recover.The Model consists on the following system:for some

SEIR Model

Susceptibles

Exposed

Infected

Recovered

Model Types

Now, following our last model, for some

SEAIR Model

Susceptibles

Exposed

Infected

Recovered

Susceptibles

Exposed

Asymptomatic

Infected

Recovered

Model Types

Now, following our last model, we introduce the class of the Asymptomatics.for some

SEAIR Model

Susceptibles

Exposed

Asymptomatic

Infected

Recovered

Model Types

Following our last model, we introduce the class of the Asymptomatics. They are infectious but with a different rate of contagion because they don’t have symptoms.The Model consists on the following system:for some

SEAIR Model

Susceptibles

Exposed

Asymptomatic

Infected

Recovered

Model Types

Following our last model, we introduce the class of the Asymptomatics. They are infectious but with a different rate of contagion because they don’t have symptoms. And they recover at a different rate, as their infection is unknown.The Model consists on the following system:for some

SEAIR Model

Susceptibles

Exposed

Asymptomatic

Infected

Recovered

Model Types

Following our last model, we introduce the class of the Asymptomatics. They are infectious but with a different rate of contagion because they don’t have symptoms. And they recover at a different rate, as their infection is unknown. There’s probability of getting symptoms.The Model consists on the following system:for some

SEAIR Model

Susceptibles

Exposed

Asymptomatic

Infected

Recovered

Model Types

Following our last model, we introduce the class of the Asymptomatics. They are infectious but with a different rate of contagion because they don’t have symptoms. And they recover at a different rate, as their infection is unknown. There’s probability of getting symptoms. And we define as the rate at which asymptomatic develop symptoms.The Model consists on the following system:for some

SEAIR Model

Model Types

Our last model is the simplest model that considers space as a factor.Space Spread

Model Types

Our last model is the simplest model that considers space as a factor. We have 2 places.Space Spread

Model Types

Our last model is the simplest model that considers space as a factor. We have 2 places. And a disease (Let’s say Ebola).Space Spread

Model Types

Our last model is the simplest model that considers space as a factor. We have 2 places. And a disease (Let’s say Ebola).Space Spread

Barcelona Sierra Leone

Model Types

Our last model is the simplest model that considers space as a factor. We have 2 places. And a disease (Let’s say Ebola).We have travel between both places and disease is spread from place 1 to place 2, and viceversa.

Space Spread


Model Types

Our last model is the simplest model that considers space as a factor. We have 2 places. And a disease (Let’s say Ebola).We have travel between both places and disease is spread from place 1 to place 2, and viceversa. Disease is spread inside the planes (or the cars, or the trains, etc.) as well.

Space Spread


Model Types

Our last model is the simplest model that considers space as a factor. We have 2 places. And a disease (Let’s say Ebola).We have travel between both places and disease is spread from place 1 to place 2, and vice-versa. Disease is spread inside the planes (or the cars, or the trains, etc.) as well. We have an SEAIR system in every region and in every plane, with interaction between them.

Space Spread


(𝑆𝐸𝐴𝐼𝑅)𝐵 (𝑆𝐸𝐴𝐼𝑅)𝑆𝐿

(𝑠𝑒𝑎𝑖𝑟 )𝑆𝐿→𝐵

(𝑠𝑒𝑎𝑖𝑟 )𝐵→𝑆𝐿

Delay-Differential Equations. Tools for Epidemics Modelling

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