Dynamical behaviour of SIR model with coinfection: The case of finite carrying capacity

Ghersheen, Samia; Kozlov, Vladimir; Tkachev, Vladimir G.; Wennergren, Uno

doi:10.1002/mma.5671

Cited by 8 publications

(36 citation statements)

References 35 publications

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“…Because the interaction between coinfected and susceptible classes results a single infection transmission or a simultaneous transmission of two infections,

\begin{matrix} alignleft \\ align-1 β_{1} S I_{12} & align-2 \to I_{1} \\ align-1 β_{2} S I_{12} & align-2 \to I_{2} \\ align-1 α_{3} S I_{12} & align-2 \to I_{12}, \end{matrix}

it follows from Figure that the corresponding reproduction/threshold number of the coinfected class is determined by

σ_{3} = \frac{μ_{3}}{\overset{α_{3}}{true}} = \frac{μ_{3}}{α_{3} + β_{1} + β_{2}},

where

\overset{α_{3}}{true} : = α_{3} + β_{1} + β_{2}

is the total transmission rate of infection from coinfected class to susceptible class. Note that in the Lotka‐Volterra case

σ_{3} = \frac{μ_{3}}{α_{3}}

, which is completely consistent with the notation of Ghersheen et al…”

Section: Formulation Of the Modelsupporting

confidence: 89%

“…Let us consider the determinantal condition in more detail. In the Lotka‐Volterra case treated in the work of Ghersheen et al, one has γ i = β j =0; hence, the corresponding determinant condition

\det B_{0} = \det [\begin{array}{cc} - α_{2} false(σ_{2} - σ_{1} false) & 0 \\ 0 & α_{3} σ_{1} + η_{1} I_{1}^{*} - μ_{3} \end{array}] > 0

becomes equivalent to a simpler inequality (cf ),

I_{1}^{*} = \frac{b}{K α_{1}} (S^{* *} - σ_{1}) < \frac{μ_{3} - α_{3} σ_{1}}{η_{1}} .

Coming back to the general case , the determinant

Δ (λ) : = \det [\begin{array}{cc} - α_{2} false(σ_{2} - σ_{1} false) - γ_{2} λ & β_{2} σ_{1} \\ false(γ_{1} + γ_{2} false) λ & α_{3} σ_{1} + η_{1} λ - μ_{3} \end{array}]

is a quadratic polynomial in λ with a negative leading coefficient. Further, by virtue of and , we have

...

…”

Section: Equilibrium Points: the Local Stability Analysismentioning

confidence: 99%

“…Note also that, because the variable R is not present in the first four equations, we may consider only the first four equations of system . In the work of Ghersheen et al, the particular case of was completely treated, when the parameters β i and γ j vanish, ie,

β_{1} = β_{2} = γ_{1} = γ_{2} = 0 .

The corresponding system has the Lotka‐Volterra type, and an approach based on the linear complimentary problem was suggested. The latter allows to obtain an effective description of the transition dynamics of for any admissible values of the fundamental parameters.…”

Section: Formulation Of the Modelmentioning

confidence: 99%

“…Finally, in the global stability analysis given below in Section 5, we shall need the following result established recently in the work of Ghersheen et al…”

Section: General Facts On the Sir Modelmentioning

confidence: 99%

“…In the work of Ghersheen et al, a very striking result has been established asserting that, for any K >0 and each admissible choice of the fundamental parameter,

\hat{p} = (b, μ_{0}, μ_{1}, μ_{2}, μ_{3}, α_{1}, α_{2}, α_{3}, η_{1}, η_{2}) \in R_{+}^{10},

there exists exactly one stable equilibrium point

E = E false(K, \overset{p}{true} false)

, which depends continuously on the data

false(K, \overset{p}{true} false)

. It is natural to consider the transition dynamics of the stable equilibrium point

E false(K, \overset{p}{true} false)

as a function of the carrying capacity K keeping other parameters in

\overset{p}{true}

fixed.…”

Section: Transition Dynamicsmentioning

confidence: 99%

See 4 more Smart Citations

Mathematical analysis of complex SIR model with coinfection and density dependence

Ghersheen

Kozlov

Tkachev

et al. 2019

Comp and Math Methods

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An SIR model with the coinfection of the two infectious agents in a single host population is considered. The model includes the environmental carry capacity in each class of population. A special case of this model is analyzed, and several threshold conditions are obtained, which describes the establishment of diseases in the population. We prove that, for small carrying capacity K, there exists a globally stable disease‐free equilibrium point. Furthermore, we establish the continuity of the transition dynamics of the stable equilibrium point, that is, we prove that, (1) for small values of K, there exists a unique globally stable equilibrium point, and (b) it moves continuously as K is growing (while its face type may change). This indicates that the carrying capacity is the crucial parameter and an increase in resources in terms of carrying capacity promotes the risk of infection.

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“…Because the interaction between coinfected and susceptible classes results a single infection transmission or a simultaneous transmission of two infections,

\begin{matrix} alignleft \\ align-1 β_{1} S I_{12} & align-2 \to I_{1} \\ align-1 β_{2} S I_{12} & align-2 \to I_{2} \\ align-1 α_{3} S I_{12} & align-2 \to I_{12}, \end{matrix}

it follows from Figure that the corresponding reproduction/threshold number of the coinfected class is determined by

σ_{3} = \frac{μ_{3}}{\overset{α_{3}}{true}} = \frac{μ_{3}}{α_{3} + β_{1} + β_{2}},

where

\overset{α_{3}}{true} : = α_{3} + β_{1} + β_{2}

is the total transmission rate of infection from coinfected class to susceptible class. Note that in the Lotka‐Volterra case

σ_{3} = \frac{μ_{3}}{α_{3}}

, which is completely consistent with the notation of Ghersheen et al…”

Section: Formulation Of the Modelsupporting

confidence: 89%

\det B_{0} = \det [\begin{array}{cc} - α_{2} false(σ_{2} - σ_{1} false) & 0 \\ 0 & α_{3} σ_{1} + η_{1} I_{1}^{*} - μ_{3} \end{array}] > 0

becomes equivalent to a simpler inequality (cf ),

I_{1}^{*} = \frac{b}{K α_{1}} (S^{* *} - σ_{1}) < \frac{μ_{3} - α_{3} σ_{1}}{η_{1}} .

Coming back to the general case , the determinant

Δ (λ) : = \det [\begin{array}{cc} - α_{2} false(σ_{2} - σ_{1} false) - γ_{2} λ & β_{2} σ_{1} \\ false(γ_{1} + γ_{2} false) λ & α_{3} σ_{1} + η_{1} λ - μ_{3} \end{array}]

is a quadratic polynomial in λ with a negative leading coefficient. Further, by virtue of and , we have

...

…”

Section: Equilibrium Points: the Local Stability Analysismentioning

confidence: 99%

β_{1} = β_{2} = γ_{1} = γ_{2} = 0 .

Section: Formulation Of the Modelmentioning

confidence: 99%

“…Finally, in the global stability analysis given below in Section 5, we shall need the following result established recently in the work of Ghersheen et al…”

Section: General Facts On the Sir Modelmentioning

confidence: 99%

“…In the work of Ghersheen et al, a very striking result has been established asserting that, for any K >0 and each admissible choice of the fundamental parameter,

\hat{p} = (b, μ_{0}, μ_{1}, μ_{2}, μ_{3}, α_{1}, α_{2}, α_{3}, η_{1}, η_{2}) \in R_{+}^{10},

there exists exactly one stable equilibrium point

E = E false(K, \overset{p}{true} false)

, which depends continuously on the data

false(K, \overset{p}{true} false)

. It is natural to consider the transition dynamics of the stable equilibrium point

E false(K, \overset{p}{true} false)

as a function of the carrying capacity K keeping other parameters in

\overset{p}{true}

fixed.…”

Section: Transition Dynamicsmentioning

confidence: 99%

See 3 more Smart Citations

Mathematical analysis of complex SIR model with coinfection and density dependence

Ghersheen

Kozlov

Tkachev

et al. 2019

Comp and Math Methods

Self Cite

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show abstract

BPSL: a new rumor source location algorithm based on the time-stamp back propagation in social networks

2021

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Effect of density dependence on coinfection dynamics

et al. 2021

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In this paper we develop a compartmental model of SIR type (the abbreviation refers to the number of Susceptible, Infected and Recovered people) that models the population dynamics of two diseases that can coinfect. We discuss how the underlying dynamics depends on the carrying capacity K: from a simple dynamics to a more complex. This can also help in understanding the appearance of more complicated dynamics, for example, chaos and periodic oscillations, for large values of K. It is also presented that pathogens can invade in population and their invasion depends on the carrying capacity K which shows that the progression of disease in population depends on carrying capacity. More specifically, we establish all possible scenarios (the so-called transition diagrams) describing an evolution of an (always unique) locally stable equilibrium state (with only non-negative compartments) for fixed fundamental parameters (density independent transmission and vital rates) as a function of the carrying capacity K. An important implication of our results is the following important observation. Note that one can regard the value of K as the natural ‘size’ (the capacity) of a habitat. From this point of view, an isolation of individuals (the strategy which showed its efficiency for COVID-19 in various countries) into smaller resp. larger groups can be modelled by smaller resp. bigger values of K. Then we conclude that the infection dynamics becomes more complex for larger groups, as it fairly maybe expected for values of the reproduction number $$R_0\approx 1$$ R 0 ≈ 1 . We show even more, that for the values $$R_0>1$$ R 0 > 1 there are several (in fact four different) distinguished scenarios where the infection complexity (the number of nonzero infected classes) arises with growing K. Our approach is based on a bifurcation analysis which allows to generalize considerably the previous Lotka-Volterra model considered previously in Ghersheen et al. (Math Meth Appl Sci 42(8), 2019).

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Dynamical behaviour of SIR model with coinfection: The case of finite carrying capacity

Cited by 8 publications

References 35 publications

Mathematical analysis of complex SIR model with coinfection and density dependence

Mathematical analysis of complex SIR model with coinfection and density dependence

BPSL: a new rumor source location algorithm based on the time-stamp back propagation in social networks

Effect of density dependence on coinfection dynamics

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