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Modelos con Vectores
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ID:(350, 0)


Malaria case
Description

In the case of malaria, it is necessary to model not only the infection but also the evolution of the carrier.In the case of malaria it is a parasite that is transmitted by mosquitoes. In the process the mosquito females transmit the parasite to the human being and vice versa the infected human being can infect the mosquito.2.7 million people die annually from this disease.

ID:(877, 0)


Vectors Models
Description

ID:(874, 0)


Mosquito
Description


ID:(3023, 0)


Vector model simulation
Description

With the equation for the fraction of infected humans

$\displaystyle\frac{di}{dt}=p_bp_I\Lambda v(1-i)-\gamma i$



and the fraction of infected mosquitoes is

$\displaystyle\frac{dv}{dt}=p_bp_Vi(1-v)-\mu v$

You can run a simulation that shows the dynamics of both populations.

ID:(8208, 0)


Modelos con Vectores
Description

Variables
Symbol
Text
Variable
Value
Units
Calculate
MKS Value
MKS Units
$I_{crit}$
I_crit
Asymptotic Infected
-
$V_{\infty}$
V_t
Asymptotic Vectors
-
$\mu$
mu
Death of Mosquito Factor
$\Lambda$
L
Fraction Females Mosquitos
-
$N_I$
N_I
Human Population
-
$I_t$
I_t
Infected
-
$dI$
dI
Infected Variation
-
$dt$
dt
Infinitesimal Variation of Time
s
$N_V$
N_V
Mosquito Population
-
$p_b$
p_b
Probability of being Bite by Time
1/s
$p_I$
p_I
Probability of Infecting bite Human Being
-
$p_V$
p_V
Probability of Infecting the Mosquito Bite
-
$R_0$
R_0
Propagation Factor
$\gamma$
gamma
Recovery Time Factor
1/s
$V$
V
Vector
-
$dV$
dV
Vector Variation
-

Calculations

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Symbol
Equation
Solved
Translated

Calculations

Symbol
Equation
Solved
Translated

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Equations

Examples

In the case of malaria, it is necessary to model not only the infection but also the evolution of the carrier.In the case of malaria it is a parasite that is transmitted by mosquitoes. In the process the mosquito females transmit the parasite to the human being and vice versa the infected human being can infect the mosquito.2.7 million people die annually from this disease.

(ID 877)


(ID 3023)

The equation that describes the evolution of infected people I must include a factor that describes the infection and another that considers the recovery or death of those infected.\\n\\nIn the first case you should consider the total number of population N_I those who are not yet infected, that is N_I-I. Then we must consider the probability that p_b is stung and that it really leads to the disease p_I. We must also consider the fraction of the mosquitoes that infected V on which there are N_V and that the mosquito is female for which we have a factor \Lambda. With them the increase factor of infected people will be\\n\\n

$\left(\displaystyle\frac{dI}{dt}\right)_{infectar}=p_bp_I\Lambda\displaystyle\frac{V}{N_V}(N_I-I)$

\\n\\nIn the second case it corresponds to the fraction of those who recover that behaves the same as in the 'SIR' and 'SEIR' models:\\n\\n

$\left(\displaystyle\frac{dI}{dt}\right)_{muere}=-\gamma I$



with what the equation for the infected is

$\displaystyle\frac{dI}{dt}=p_bp_I\Lambda\displaystyle\frac{V}{N_V}(N_I-I)-\gamma I$

(ID 4094)

The equation that describes the evolution of infected mosquitoes V must include a factor that describes the infection and another that considers the death of those infected.\\n\\nIn the first case you should consider the total number of insects N_V those that are not yet infected, that is N_V-V. Then we must consider the probability that p_b is stung and that it really leads to the disease p_V. We must also consider the fraction of mosquitoes that infected I on which there are N_I. With them the increase factor of infected people will be\\n\\n

$\left(\displaystyle\frac{dV}{dt}\right)_{infectar}=p_bp_V\displaystyle\frac{I}{N_I}(N_V-V)$

\\n\\nIn the second case it corresponds to the fraction of those who die that behaves the same as in the SIR and SEIR models:\\n\\n

$\left(\displaystyle\frac{dV}{dt}\right)_{muere}=-\mu V$



with what the equation for the infected is

$\displaystyle\frac{dV}{dt}=p_bp_V\displaystyle\frac{I}{N_I}(N_V-V)-\mu V$

(ID 4095)

With p_b the probability of being bitten, p_I the probability is that the bite generates a disease in humans, p_V the probability that the mosquito is infected, \Lambda is the proportion of mosquitoes be female, \gamma the reproduction factor and \mu the death of the mosquito reproduction factor will be

$R_0=\displaystyle\frac{p_b^2p_Vp_I\Lambda}{\mu\gamma}$

(ID 4098)

In the event that the system enters a stationary phase, the temporal derivative will be in both equations equal to zero. This gives us the equations

$\displaystyle\frac{dI}{dt}=p_bp_I\Lambda\displaystyle\frac{V}{N_V}(N_I-I)-\gamma I$



Y

$\displaystyle\frac{dV}{dt}=p_bp_V\displaystyle\frac{I}{N_I}(N_V-V)-\mu V$

\\n\\nwith what you have\\n\\n

$\displaystyle\frac{dI}{dt}\displaystyle\bigg|_{I=I_{\infty}}=p_bp_I\Lambda V_{\infty}(N_I-I_{\infty})-\gamma I_{\infty}N_V=0$

\\n\\nY\\n\\n

$\displaystyle\frac{dV}{dt}\displaystyle\bigg|_{V=V_{\infty}}=p_bp_VI_{\infty}(N_V-V_{\infty})-\mu V_{\infty}N_I=0$



so the solution for the human being will be

$I_{\infty}=N_I\displaystyle\frac{\Lambda p_b^2p_Ip_V-\mu\gamma}{p_bp_V(\gamma+\Lambda p_bp_I)}$

(ID 4096)

In the event that the system enters a stationary phase, the temporal derivative will be in both equations equal to zero. This gives us the equations

$\displaystyle\frac{dI}{dt}=p_bp_I\Lambda\displaystyle\frac{V}{N_V}(N_I-I)-\gamma I$



Y

$\displaystyle\frac{dV}{dt}=p_bp_V\displaystyle\frac{I}{N_I}(N_V-V)-\mu V$

\\n\\nwith what you have\\n\\n

$\displaystyle\frac{dI}{dt}\displaystyle\bigg|_{I=I_{\infty}}=p_bp_I\Lambda V_{\infty}(N_I-I_{\infty})-\gamma I_{\infty}N_V=0$

\\n\\nY\\n\\n

$\displaystyle\frac{dV}{dt}\displaystyle\bigg|_{V=V_{\infty}}=p_bp_VI_{\infty}(N_V-V_{\infty})-\mu V_{\infty}N_I=0$



so the solution for the mosquito will be

$V_{\infty}=N_V\displaystyle\frac{\Lambda p_b^2p_Ip_V-\mu\gamma}{\Lambda p_bp_I(\mu+p_bp_V)}$

(ID 4097)

To avoid working with very large numbers, it is convenient to transform the equations based on the fraction of human infected rather than the total number. This is why it is introduced

$ i =\displaystyle\frac{ I }{ N_I }$

(ID 8204)

To avoid working with very large numbers it is convenient to transform the equations based on the fraction of infected mosquitoes instead of the total number. This is why it is introduced

$ v =\displaystyle\frac{ V }{ N_V }$

(ID 8205)

With the equation the number of infected humans is

$\displaystyle\frac{dI}{dt}=p_bp_I\Lambda\displaystyle\frac{V}{N_V}(N_I-I)-\gamma I$



and the fraction of infected humans

$ i =\displaystyle\frac{ I }{ N_I }$



and infected mosquitoes

$ v =\displaystyle\frac{ V }{ N_V }$



is obtained

$\displaystyle\frac{di}{dt}=p_bp_I\Lambda v(1-i)-\gamma i$

(ID 8207)

With the equation for the evolution of the number of infected mosquitoes it is

$\displaystyle\frac{dV}{dt}=p_bp_V\displaystyle\frac{I}{N_I}(N_V-V)-\mu V$



and the fraction of infected humans

$ i =\displaystyle\frac{ I }{ N_I }$



and infected mosquitoes

$ v =\displaystyle\frac{ V }{ N_V }$



it has to

$\displaystyle\frac{dv}{dt}=p_bp_Vi(1-v)-\mu v$

(ID 8206)

Since the number of infected humans in the stationary limit is

$I_{\infty}=N_I\displaystyle\frac{\Lambda p_b^2p_Ip_V-\mu\gamma}{p_bp_V(\gamma+\Lambda p_bp_I)}$



you can with

$ i =\displaystyle\frac{ I }{ N_I }$



rewrite the limit for the infected fraction

$i_{\infty}=\displaystyle\frac{\Lambda p_b^2p_Ip_V-\mu\gamma}{p_bp_V(\gamma+\Lambda p_bp_I)}$

(ID 8209)

As the number of infected mosquitoes in the stationary limit is

$V_{\infty}=N_V\displaystyle\frac{\Lambda p_b^2p_Ip_V-\mu\gamma}{\Lambda p_bp_I(\mu+p_bp_V)}$



and you have that the fraction is

$ v =\displaystyle\frac{ V }{ N_V }$



a limit fraction can be estimated by

$v_{\infty}=\displaystyle\frac{\Lambda p_b^2p_Ip_V-\mu\gamma}{\Lambda p_bp_I(\mu+p_bp_V)}$

(ID 8210)

With the equation for the fraction of infected humans

$\displaystyle\frac{di}{dt}=p_bp_I\Lambda v(1-i)-\gamma i$



and the fraction of infected mosquitoes is

$\displaystyle\frac{dv}{dt}=p_bp_Vi(1-v)-\mu v$

You can run a simulation that shows the dynamics of both populations.

(ID 8208)

ID:(350, 0)