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Modelos con Vectores

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Malaria case

Description

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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

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ID:(874, 0)

Mosquito

Image

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ID:(3023, 0)

Equation of infected humans

Equation

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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, 0)

Equation of infected mosquitoes

Equation

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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, 0)

Reproduction factor

Equation

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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, 0)

Number of stationary infected humans

Equation

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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

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

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

\\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, 0)

Number of stationary infected mosquitoes

Equation

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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

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

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

\\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, 0)

Fraction of infected humans

Equation

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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, 0)

Infection of infected mosquitoes

Equation

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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, 0)

Equation of the fraction of infected humans

Equation

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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, 0)

Equation of the fraction of infected mosquitoes

Equation

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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, 0)

Fraction of stationary infected humans

Equation

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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, 0)

Fraction of stationary infected mosquitoes

Equation

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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, 0)

Vector model simulation

Html

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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)