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## Application of a fractional SIR model built with Mittag-Leffler distribution

### Sandro Rodrigues Mazorche

Fractional Calculus has proven to be an especially useful tool in capturing the dynamics of the physical process of several scientific objects, being in general related to the “memory effect”. Generally, it consists of making a classic model more flexible by replacing an entire order derivative with an arbitrary one. Notably, compartmental models have been widely studied with arbitrary orders (e.g., [2], [4]).

Throughout the first author's master's research, we sought to investigate the use of arbitrary derivatives in SIR-type models, theoretically, analytically also numerically. We are interested in the questions of persistence of characteristics, starting from the classical modeling to discuss the difficulties in the construction of a non-artificial arbitrary model. What characteristics are maintained when exchanging orders? Are consistent models automatically established, regarding the definition of parameters, physical significance, conservation, and units? What about non-negativity, monotonicity (if any), among other issues? The analytical and numerical techniques provide an interesting field for research. However, from the modeling point of view, it is important to try to verify how, where, and why Fractional Calculus interfere in the model.

We believe that derivatives of an arbitrary order may arise through power-laws time-since-infection dependence in the infectiousness and removal functions. Thus, we present in [3] a physical derivation following in the footsteps of Angstmann, Henry & McGann [1], where they use the probabilistic language of Continuous Time Random Walks (CTRW) and Mittag-Leffler functions. The Riemann-Liouville derivative appears throughout the construction and the SIR model, with $1 \geq \beta \geq \alpha> 0$, is given by $\dfrac{\omega(t)S(t)\theta(t,0)}{N\tau^\beta}D^{1-\beta}\bigg(\dfrac{I(t)}{\theta(t,0)}\bigg)-\gamma(t)S(t),$ $\dfrac{dI(t)}{dt}=\dfrac{\omega(t)S(t)\theta(t,0)}{N\tau^\beta}D^{1-\beta}\bigg(\dfrac{I(t)}{\theta(t,0)}\bigg)-\dfrac{\theta(t,0)}{\tau^\alpha}D^{1-\alpha}\bigg(\dfrac{I(t)}{\theta(t,0)}\bigg)-\gamma(t)I(t),$ $\dfrac{dR(t)}{dt}=\dfrac{\theta(t,0)}{\tau^\alpha}D^{1-\alpha}\bigg(\dfrac{I(t)}{\theta(t,0)}\bigg)-\gamma(t)R(t),$ where $\gamma (t)$ is the vital dynamic; $\omega (t)$, extrinsic infectivity; $N$, the total population; $\tau$, a scale parameter and, $\theta (t, t ')$, the probability that an infectious person since $t'$ would not die of natural death until $t$. If $\beta = \alpha = 1$, $\gamma (t) \equiv \gamma$ and $\omega (t) \equiv \omega$, we get the traditional SIR model. In [3], we revisited the authors' work, started the discussion of the reproduction number, and used optimization to apply the model to the data of the Brazilian and Italian pandemic of COVID-19. As explained in more detail in [3], the time of removal of the individual from the infectious compartment follows a Mittag-Leffler distribution related to $\alpha$, while the parameter $\beta$ relates to the law of the function of infectivity.

Here we pretend use this basis to explore current pandemic data of each state of Brazil, comparing similarities and differences about the laws of infectivity and recovery and equilibrium points. An interesting discussion would be a proposal to optimize the distribution of resources such as vaccines.

References

[1] Angstmann, C. N., Henry, B. I. and McGann, A. V. A fractional-order infectivity and recovery SIR model, Fractal and Fractional, 1(1), 11, 2017. DOI: 10.3390/fractalfract1010011.

[2] Cardoso, L. C., Santos, F. L. P. dos, and Camargo, R. F. Analysis of fractional-order models for hepatitis B, Computational and Applied Mathematics, 37(4), 4570-4586, 2018. DOI:10.1007/s40314-018-0588-4.

[3] Monteiro, N. Z. and Mazorche, S. R. Fractional Derivatives Applied to Epidemiology. TCAM,Trends in Computational and Applied Mathematics, 2021. (to appear)

[4] Santos, J. P. C. dos, Monteiro, E. and Vieira, G. B. Global stability of fractional SIR epidemic model, Proceeding Series of the Brazilian Society of Computational and Applied Mathematics,5(1), 2017. DOI: 10.5540/03.2017.005.01.0019.

Joint work with Noemi Zeraick Monteiro (Universidade Federal de Juiz de Fora, Brasil).