**2. Reference telecommunication network definition and decomposition**

At any fixed time, the telecommunication network can be defined in terms of its topological description as well as in terms of its traffic pattern. As far as network topology is concerned, we consider the network nodes *n* ∈ Ν = {*n*1,*n*2,…,*nN*} and the network links defined as ordered pairs of nodes *l* ∈ Λ = {*l*1,*l*2,…,*lL*}. To describe the network traffic request we first define a path *v* ∈ Ω = {*v*1,*v*2,…,*vV*} as a collection of consecutive links, denoted by Λ*v*, from an ingoing node *i* to an outgoing node *j* (where *i*,*j* ∈ **N**). Moreover a certain set of different Service Classes *k* ∈ Κ = {*k*1,*k*2,…,*kK*}, is defined, each one characterized by a set of Quality of Service (QoS) parameters. According to the most recent trends, the QoS control is performed on a per flow basis, where a flow *f* ∈ Φ = {*f*1,*f*2,…,*fF*} is defined as the triple *f* = (*ni,nj,kp*), with *ni* denoting the ingoing node, *nj* denoting the outgoing node and *kp* denoting the service class. The traffic associated with a given flow *f* may possibly be routed on a set Ω*f* of one or more paths. We further introduce the set of indices {*a*(*l*,*v*), *l* ∈ Λ*, v* ∈ Ω}, defined as follows:

$$a(l,v) = \begin{cases} 1, \text{ if } l \in v \\ 0, \text{ otherwise} \end{cases} \tag{1}$$

For each link *l* ∈ Λ, at the given time, we may consider its occupancy level *c*(*l*) defined as the sum of all contributions to the occupancy due to the flows routed on the link itself. Each contribution of this type will be quantified by the bit rate *R*(*l*,*f*) which, in turn, is the sum of bit rates of all in-progress connections going through the link *l* and relevant to the flow *f*, possibly weighted by a coefficient α(*l*,*f*) which accounts for the specific need of the flow itself. Therefore we have:

$$\mathcal{L}(l) = \sum\_{f \in \Phi} \alpha(l, f) R(l, f) \tag{2}$$

where α(*l*,*f*) are positive known coefficients which take into account the fact that some technologies differentiate the classes of service by varying modulation, coding, and so on. For each link *l*, we consider the so-called nominal capacity *cNOM*(*l*), that is the value of the occupancy level suggested for a proper behaviour of the link (typically in terms of QoS)1.

1 *c*(*l*) and *cNOM*(*l*) can be interpreted as generalizations of "load factor" and "Noise Rise" in UMTS (see Holma and Toskala, 2002).

Optimal Control Strategies for Multipath Routing:

explicitly account for the network overloading.

several paths the function (9) can be rewritten as follows:

keeps increasing, then function (10) is approximated by:

arriving at the functions:

where:

over the whole sub-network:

From Load Balancing to Bottleneck Link Management 409

The optimal choice for *u* within its (convex) admissible set may be performed according to a cost function which assesses the network loading. In a previous work Bruni *et al.*, 2010, the control goal was the normalized load balancing in the sub-network, evaluated by the function:

> 1 () *NOM* ( ) *l*

with *L* denoting the cardinality of Λ. In Bruni *et al.*, 2010, and Bruni *et al.*, 2010 (to appear), a shortcoming of (7) was enlightened, which is due to the partial controllability property (therein defined) of some of the links. These links, in the following referred to as "ballast", are such that they are bound to accept traffic flows not controlled by the components of the control vector *u*. Thus other choices of the cost function might be considered which more

One first possibility is to assess the link overflow setting *k* = 0 in (7), thus more generally

*m*

*m*

( ) ( ) ( )

*l NOM c l J u c l* ∈Λ

for some integer *m* > 1. If the target is to give more importance to the links belonging to

( ) ( ) ( ) *v*

*v l NOM c l J u c l* ∈Ω ∈Λ

According to (9), (10) we try to distribute the total load in the network in such a way that the higher the normalized load for a link is, the stronger is the effort in reducing it. This selective attention to the most heavily loaded links progressively increases with *m*. As *m*

> ( ) ( )*m v*

( ) max*v* ( ) *v l NOM c l G*

Thus for each path *v* the optimization attention is just focused on the most heavily loaded link of the path itself (bottleneck). Eventually we can consider the worst bottleneck load

*v Ju G* ∈Ω

*c l k Lc l* ∈Λ

( ) ( ) *NOM* ( ) *l c l J u k c l* ∈Λ

with *k* a given constant. If, for any given *u*, we optimize (7) with respect to *k*, we get:

2

= − (7)

= (8)

= (9)

= (10)

= (11)

∈Λ *c l* = (12)

that indicates the fraction of *R*(*f*) to be routed on path *v* ∈ Ω*f*. Then, due to the bit conservation law, we have:

$$R(l,f) = \sum\_{v \in \Omega\_f} \alpha(l,f) R(f) \mu(f,v) \tag{3}$$

where obviously:

$$u(f, v) \in [0, 1], \,\forall f \in \Phi, \,\forall v \in \Omega\_f$$

$$\sum\_{v \in \Omega\_f} u(f, v) = 1, \,\forall f \in \Phi \tag{4}$$

As shown in Bruni *et al.*, 2010, with reference to the routing control problem, the link set Λ might be decomposed into separated subclasses Λ(*j*), *j* = 0,1,2,…,*P*, each of them involving separate subsets of control variables, where Λ(0) is the set, possibly empty, of links that cannot be controlled by any control variable and which therefore they are not involved in any routing control problem.

For every communicating class of links Λ(*j*) ⊂ Λ, there exists the (uniquely) corresponding communicating class of flows Φ(*j*) ⊂ Φ defined as the set of flows such that, for each *f* ∈ Φ(*j*), there exists (at least) a link *l* ∈ Λ(*j*), and therefore a pair of links (generally depending on *f* itself), which are controllable with respect to *f*. Clearly, the set Φ(0) coincides with the empty set. We now observe that the set {Φ(*j*), *j >* 0} of flow communicating classes forms a partition of Φ, corresponding to the fact that the set {Λ(*j*), *j >* 0} of link communicating classes forms a partition of Λ. This partition for Λ and Φ immediately induces a partition of the network. Note that each *j*-th part of the network is controlled by a corresponding subvector of control variables, later defined as *u*(*j*) independently of the other parts; the components of the vector *u*(*j*) are the variables *u*(*f*,*v*), *f* ∈ Φ(*j*), *v* ∈ Ω*f*. In the following {Λ(*j*), Φ(*j*)} will denote a sub-network. We will use the detailed network decomposition procedure described in Bruni *et al.*, 2010, facing the routing control problem in each sub-network (but in Λ(0)).

#### **3. A rationale for the network loading**

In the following, we will focus attention on the routing problem for any given sub-network {Λ(*j*), Φ(*j*)}. As mentioned above, any such problem is characterized by a set *u*(*j*) of control variables, which may be (optimally) selected independently of the other ones. As stated in Bruni *et al.*, 2010, the admissible set for *u*(*j*) is defined by the constraints:

$$
\mu(f, v) \in [0, 1], \,\forall f \in \Phi^{(j)}, \,\forall v \in \Omega\_f \tag{5}
$$

$$\sum\_{v \in \mathfrak{Q}\_f} u(f, v) = 1, \; \forall f \in \Phi^{(j)} \tag{6}$$

so that the set itself turns out to be convex. From here on, for sake of simplicity the apices *j* will be dropped.

that indicates the fraction of *R*(*f*) to be routed on path *v* ∈ Ω*f*. Then, due to the bit

(, ) (, ) ( ) ( , )

∈ ∀ ∈Φ ∀ ∈Ω

= (3)

*f*

= ∀ ∈Φ (4)

*Rl f l f R f u f v* α

*f v*

∈Ω

( , ) [0,1], , ( , ) 1,

*ufv f v ufv f*

As shown in Bruni *et al.*, 2010, with reference to the routing control problem, the link set Λ might be decomposed into separated subclasses Λ(*j*), *j* = 0,1,2,…,*P*, each of them involving separate subsets of control variables, where Λ(0) is the set, possibly empty, of links that cannot be controlled by any control variable and which therefore they are not involved in

For every communicating class of links Λ(*j*) ⊂ Λ, there exists the (uniquely) corresponding communicating class of flows Φ(*j*) ⊂ Φ defined as the set of flows such that, for each *f* ∈ Φ(*j*), there exists (at least) a link *l* ∈ Λ(*j*), and therefore a pair of links (generally depending on *f* itself), which are controllable with respect to *f*. Clearly, the set Φ(0) coincides with the empty set. We now observe that the set {Φ(*j*), *j >* 0} of flow communicating classes forms a partition of Φ, corresponding to the fact that the set {Λ(*j*), *j >* 0} of link communicating classes forms a partition of Λ. This partition for Λ and Φ immediately induces a partition of the network. Note that each *j*-th part of the network is controlled by a corresponding subvector of control variables, later defined as *u*(*j*) independently of the other parts; the components of the vector *u*(*j*) are the variables *u*(*f*,*v*), *f* ∈ Φ(*j*), *v* ∈ Ω*f*. In the following {Λ(*j*), Φ(*j*)} will denote a sub-network. We will use the detailed network decomposition procedure described in Bruni *et al.*, 2010, facing the

In the following, we will focus attention on the routing problem for any given sub-network {Λ(*j*), Φ(*j*)}. As mentioned above, any such problem is characterized by a set *u*(*j*) of control variables, which may be (optimally) selected independently of the other ones. As stated in

( ) ( , ) 1,

*ufv f*

so that the set itself turns out to be convex. From here on, for sake of simplicity the apices *j*

( ) ( , ) [0,1], , *j ufv f v* ∈ ∀ ∈Φ ∀ ∈Ω *f* (5)

= ∀ ∈Φ (6)

*j*

*f*

*v*

routing control problem in each sub-network (but in Λ(0)).

Bruni *et al.*, 2010, the admissible set for *u*(*j*) is defined by the constraints:

*f*

*v*

∈Ω

**3. A rationale for the network loading**

will be dropped.

∈Ω

conservation law, we have:

any routing control problem.

where obviously:

The optimal choice for *u* within its (convex) admissible set may be performed according to a cost function which assesses the network loading. In a previous work Bruni *et al.*, 2010, the control goal was the normalized load balancing in the sub-network, evaluated by the function:

$$f(\mu) = \sum\_{l \in \Lambda} \left( \frac{c(l)}{c\_{NOM}(l)} - k \right)^2 \tag{7}$$

with *k* a given constant. If, for any given *u*, we optimize (7) with respect to *k*, we get:

$$k = \frac{1}{L} \sum\_{l \in \Lambda} \frac{c(l)}{c\_{NOM}(l)} \tag{8}$$

with *L* denoting the cardinality of Λ. In Bruni *et al.*, 2010, and Bruni *et al.*, 2010 (to appear), a shortcoming of (7) was enlightened, which is due to the partial controllability property (therein defined) of some of the links. These links, in the following referred to as "ballast", are such that they are bound to accept traffic flows not controlled by the components of the control vector *u*. Thus other choices of the cost function might be considered which more explicitly account for the network overloading.

One first possibility is to assess the link overflow setting *k* = 0 in (7), thus more generally arriving at the functions:

$$f(\mu) = \sum\_{l \in \Lambda} \left( \frac{c(l)}{c\_{NOM}(l)} \right)^m \tag{9}$$

for some integer *m* > 1. If the target is to give more importance to the links belonging to several paths the function (9) can be rewritten as follows:

$$f(\mu) = \sum\_{v \in \Omega} \sum\_{l \in \Lambda\_v} \left( \frac{c(l)}{c\_{NOM}(l)} \right)^m \tag{10}$$

According to (9), (10) we try to distribute the total load in the network in such a way that the higher the normalized load for a link is, the stronger is the effort in reducing it. This selective attention to the most heavily loaded links progressively increases with *m*. As *m* keeps increasing, then function (10) is approximated by:

$$J(u) = \sum\_{v \in \Omega} \left( G\_v \right)^m \tag{11}$$

where:

$$G\_v = \max\_{l \in \Lambda\_v} \frac{c(l)}{c\_{NOM}(l)} \tag{12}$$

Thus for each path *v* the optimization attention is just focused on the most heavily loaded link of the path itself (bottleneck). Eventually we can consider the worst bottleneck load over the whole sub-network:

$$J(u) = \max\_{v \in \Omega} G\_v \tag{13}$$

Optimal Control Strategies for Multipath Routing:

*cNOM*

Λ(0) x x

denoted by 'x' if *fi* ∈ Φ(*j*).

Management.

From Load Balancing to Bottleneck Link Management 411

Λ(1) x x x x ◊ x x x x x x x Λ(2) x x x x x x x x x

*v*; the generic entry

) is

) is

Table 1. Network Topology and Decomposition; the first row shows the nominal link

denoted by 'x' if *li* ∈ Λ(*j*), or by '◊' if *li* ∈ Λ(*j*) and *li* is a ballast link; the generic entry (*fi*,Λ (*j*)

The considered scenario has been simulated with MatLab. In particular we have tested two simuation sets reported in subsection 4.2 and 4.3respectively. In subsection 4.2 we considered the Bottleneck Link Management by varying the weights of the bottleneck loads, while in subsection 4.3 we made comparisons between Load Balancing and Bottleneck Link

a)

(*fi*,*vj*) is denoted by 'x' if it is possible to route *fi* on path *vj*; the generic entry (*li*,Λ (*j*)

capacities in [Mbps]; the generic entry (*li*,*vj*) is denoted by 'x' if *li* ∈ Λ(*j*)

[kbps] 10 10 5.4 5.4 5.4 5.4 10 5.4 5.4 5.4 5.4 5.4 5.4 5.4 5.4 5.4 5.4 5.4 5.4

*v*1 x x x *v*2 x x x x *v*3 x x x x *v*4 x x x x x *v*5 x x x x x *v*6 x x x x *v*7 x x x x *v*8 x x x x

*l*1 *l*2 *l*3 *l*4 *l*5 *l*6 *l*7 *l*8 *l*9 *l*10 *l*11 *l*12 *l*13 *l*14 *l*15 *l*16 *l*17 *l*18 *l*19 *f*1 *f*2 *f*3 *f*4

*Remark.* Some methods are proposed in the literature to solve the above minimax optimization problem (see Warren *et al.*, 1967, Osborne and Wetson, 1969, Blander *et al.*, 1972, Blander and Charambous, 1972). The original minimax problem (11) is equivalent to the following:

$$\min\_{\mu, \mathbf{g} \in U} f(\mu, \mathbf{g}) \tag{14}$$

$$f(\boldsymbol{\mu}, \mathbf{g}) = \sum\_{\boldsymbol{\nu} \in \Omega} \left[ \mathcal{g}(\boldsymbol{\nu}) \right]^m \tag{15}$$

$$\begin{aligned} \mathcal{U} = \left\{ (u, g) \in \mathcal{R}^{V(F+1)} : u(f, v) \ge 0, \sum\_{v \in \Omega\_f} u(f, v) = 1, \\\\ \frac{c(l)}{c\_{\text{NOM}}(l)} \le g(v), \forall f \in \Phi, \forall v \in \mathfrak{Q}\_f \right\} \end{aligned} \tag{16}$$

where *g* is the vector of auxiliary variables *g*(*v*), *v* ∈ Ω. This is a nonlinear (linear if *m* = 1, quadratic if *m* = 2) programming problem that can be solved by well-established methods. We observe that the equivalence lies in the fact that, once (14) (15) (16) is solved, the optimal value assumed by *g*(*v*) coincides with *Gv* in (12), for *v* ∈ Ω, i.e., it represents the normalized bottleneck link load of path *v*.

The load balancing problem (7) (8), with constraints (5) (6) and the bottleneck load management problem (14) (15) (16) are easily seen to be convex. This allows standard minimization routines to be used for its solution, such as MatLab simulation tools.

*Remark*. The cost function (13) enlightens a further advantage of network decomposition. Indeed, in case the decomposition had not been performed, then (13) would describe an illposed optimal control problem whenever the worst bottleneck over the whole network happens to be an uncontrollable link. Similar considerations hold for cost function (11).

#### **4. Evaluation and comparison of optimal routing procedures**

#### **4.1 Network structure and decomposition**

The considered scenario is composed by 16 nodes and 19 links (see Fig. 1 a)). The traffic pattern involves 4 traffic flows of the same service class k, from 4 source nodes *ni*, *i* = 1,..,4, to 4 different destination nodes *nj*, *j* = 11,..,14. The traffic pattern is described by the set of traffic flows Φ = {*f*1,*f*2,*f*3,*f*4}, where each traffic flow is identified by the following triples: *f*1 = (*n*1*,n*11*,k*), *f*2 = (*n*2*,n*12*,k*), *f*3 = (*n*3*,n*13*,k*), *f*4 = (*n*4*,n*14*,k*). After performing the network decomposition as in Bruni *et al.*, 2010, we recognize three sub-networks (see, Fig. 1 b), c) and d)). The network topology is summarized in Table 1, where the network decomposition is reported as well.

( ) max *v v J u G* ∈Ω

*Remark.* Some methods are proposed in the literature to solve the above minimax optimization problem (see Warren *et al.*, 1967, Osborne and Wetson, 1969, Blander *et al.*, 1972, Blander and Charambous, 1972). The original minimax problem (11) is equivalent to

> min ( , ) *ug U*

(,) () [ ]*m v Jug gv* ∈Ω

=∈ ≥ =

where *g* is the vector of auxiliary variables *g*(*v*), *v* ∈ Ω. This is a nonlinear (linear if *m* = 1, quadratic if *m* = 2) programming problem that can be solved by well-established methods. We observe that the equivalence lies in the fact that, once (14) (15) (16) is solved, the optimal value assumed by *g*(*v*) coincides with *Gv* in (12), for *v* ∈ Ω, i.e., it represents the normalized

The load balancing problem (7) (8), with constraints (5) (6) and the bottleneck load management problem (14) (15) (16) are easily seen to be convex. This allows standard

*Remark*. The cost function (13) enlightens a further advantage of network decomposition. Indeed, in case the decomposition had not been performed, then (13) would describe an illposed optimal control problem whenever the worst bottleneck over the whole network happens to be an uncontrollable link. Similar considerations hold for cost function (11).

The considered scenario is composed by 16 nodes and 19 links (see Fig. 1 a)). The traffic pattern involves 4 traffic flows of the same service class k, from 4 source nodes *ni*, *i* = 1,..,4, to 4 different destination nodes *nj*, *j* = 11,..,14. The traffic pattern is described by the set of traffic flows Φ = {*f*1,*f*2,*f*3,*f*4}, where each traffic flow is identified by the following triples: *f*1 = (*n*1*,n*11*,k*), *f*2 = (*n*2*,n*12*,k*), *f*3 = (*n*3*,n*13*,k*), *f*4 = (*n*4*,n*14*,k*). After performing the network decomposition as in Bruni *et al.*, 2010, we recognize three sub-networks (see, Fig. 1 b), c) and d)). The network topology is summarized in Table 1, where the network decomposition is

minimization routines to be used for its solution, such as MatLab simulation tools.

**4. Evaluation and comparison of optimal routing procedures**

*U ug R uf v uf v*

≤ ∀ ∈Φ ∀ ∈Ω

( 1) ( , ) : ( , ) 0, ( , ) 1,

*f*

*v*

∈Ω

*f*

,

( ) ( ), , ( )

*c l gv f v c l*

*NOM*

**4.1 Network structure and decomposition**

bottleneck link load of path *v*.

reported as well.

*V F*

+

the following:

= (13)

*J u g* ∈ (14)

= (15)

(16)

Table 1. Network Topology and Decomposition; the first row shows the nominal link capacities in [Mbps]; the generic entry (*li*,*vj*) is denoted by 'x' if *li* ∈ Λ(*j*) *v*; the generic entry (*fi*,*vj*) is denoted by 'x' if it is possible to route *fi* on path *vj*; the generic entry (*li*,Λ (*j*)) is denoted by 'x' if *li* ∈ Λ(*j*), or by '◊' if *li* ∈ Λ(*j*) and *li* is a ballast link; the generic entry (*fi*,Λ (*j*) ) is denoted by 'x' if *fi* ∈ Φ(*j*).

The considered scenario has been simulated with MatLab. In particular we have tested two simuation sets reported in subsection 4.2 and 4.3respectively. In subsection 4.2 we considered the Bottleneck Link Management by varying the weights of the bottleneck loads, while in subsection 4.3 we made comparisons between Load Balancing and Bottleneck Link Management.

Optimal Control Strategies for Multipath Routing:

0,00 0,10 0,20 0,30 0,40 0,50 0,60 0,70 0,80 0,90 1,00

> *m =* 1 *m =* 2 *m =* 3

0,00 0,10 0,20 0,30 0,40 0,50 0,60 0,70 0,80 0,90 1,00

> *m =* 1 *m =* 2 *m =* 3

0,00 0,10 0,20 0,30 0,40 0,50 0,60 0,70 0,80 0,90 1,00

> *m =* 1 *m =* 2 *m =* 3

*l* 1 *l*

2 *l*

5*l*

6 *l*

c)

Fig. 2. Sub-network 1: a) optimal control variables, b) bottleneck link loads, c) link loads.

*u*(*f*

1,*v*1) *u*(*f*

From Load Balancing to Bottleneck Link Management 413

u(1,1) u(1,2) u(4,7) u(4,8)

g(1) g(2) g(7) g(8)

*g*(*v*1) *g*(*v*2) *g*(*v*7) *g*(*v*8)

l1 l2 l5 l6 l7 l10 l11 l14 l16 l19 m = 1 0,50 0,00 0,00 0,93 0,54 0,93 0,00 0,00 0,93 0,93 m = 2 0,41 0,09 0,16 0,77 0,71 0,77 0,16 0,16 0,77 0,77 m = 3 0,41 0,09 0,17 0,75 0,73 0,75 0,17 0,17 0,75 0,75

10*l*

11*l*

14*l*

16*l* 19

7*l*

m = 1 0,93 0,54 0,54 0,93 m = 2 0,77 0,71 0,71 0,77 m = 3 0,75 0,73 0,73 0,75

b)

4,*v*7) *u*(*f*

4,*v*8 ) 8)

1,*v*2) *u*(*f*

m = 1 1,00 0,00 0,00 1,00 m = 2 0,83 0,17 0,17 0,83 m = 3 0,81 0,19 0,19 0,81

a)

d)

Fig. 1. a) Global Network, b) Sub-network 0 (Λ(0)), c) Sub-network 1 (Λ(1)), d) Sub-network 2 (Λ(2)).

#### **4.2 Optimal routing for different weights of bottleneck loads**

In this simulation set we consider that the bit rate of traffic flows *f*1, *f*3, *f*4 is equal to 5 Mbps whilst the bit rate of traffic flow *f*2 is equal to 5.4 Mbps. Fig. 2 and 3 show the dependence of the optimal solutions on index *m* of the Bottleneck Link Management problem.

b)

c)

d) Fig. 1. a) Global Network, b) Sub-network 0 (Λ(0)), c) Sub-network 1 (Λ(1)), d) Sub-network 2

In this simulation set we consider that the bit rate of traffic flows *f*1, *f*3, *f*4 is equal to 5 Mbps whilst the bit rate of traffic flow *f*2 is equal to 5.4 Mbps. Fig. 2 and 3 show the dependence of

the optimal solutions on index *m* of the Bottleneck Link Management problem.

**4.2 Optimal routing for different weights of bottleneck loads**

(Λ(2)).

Fig. 2. Sub-network 1: a) optimal control variables, b) bottleneck link loads, c) link loads.

Optimal Control Strategies for Multipath Routing:

balancing optimal solution.

network 2.

Rate

From Load Balancing to Bottleneck Link Management 415

In this simulation set we consider that all the traffic sources transmit with an increasing trend from 4.5 Mbps to 8.5 Mbps. Tables 2-5 show the network load as the sources bit rate increase, and compares the optimal bottleneck control solutions for *m* = 1, 2, 3 with the load

In Tables 2-5, we denote by bold characters the normalized link loads exceeding 1;

The bottleneck control for *m* ≥ 2 manages a higher network load than the load balancing approach. In fact, the tables show that the solutions of the bottleneck control problem are such that no link is overloaded until the flow rates exceed 5 Mbps, 6.5 Mbps and 6.5 Mbps for *m* = 1,2,3, respectively; on the other hand, the load balancing solutions are such that no link is overloaded until the flow rates exceed 5 Mbps. Similar results are obtained for sub-

[Mbps] 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5

*g*(*v*1) 0,74 0,83 0,93 **1,02 1,11 1,20 1,30 1,39 1,48 1,57**  *g*(*v*2) 0,40 0,45 0,50 0,55 0,60 0,65 0,70 0,75 0,80 0,85 *g*(*v*7) 0,40 0,45 0,50 0,55 0,60 0,65 0,70 0,75 0,80 0,85 *g*(*v*8) 0,74 0,83 0,93 **1,02 1,11 1,20 1,30 1,39 1,48 1,57**  *l1* 0,40 0,45 0,50 0,55 0,60 0,65 0,70 0,75 0,80 0,85 *l2* 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 *l5* 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 *l6* 0,74 0,83 0,93 **1,02 1,11 1,20 1,30 1,39 1,48 1,57**  *l7* 0,40 0,45 0,50 0,55 0,60 0,65 0,70 0,75 0,80 0,85 *l10* 0,74 0,83 0,93 **1,02 1,11 1,20 1,30 1,39 1,48 1,57**  *l11* 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 *l14* 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 *l16* 0,74 0,83 0,93 **1,02 1,11 1,20 1,30 1,39 1,48 1,57**  *l19* 0,74 0,83 0,93 **1,02 1,11 1,20 1,30 1,39 1,48 1,57**

Table 2. Sub-network 1: Optimal Solutions under bottleneck control, *m* =1.

**4.3 Comparisons between load balancing and bottleneck link management**

hereinafter, the corresponding links will be denoted as overloaded links.

*u*(*f*1,*v*1) 1,00 *u*(*f*1, *v*2) 0,00 *u*(*f*4, *v*7) 0,00 *u*(*f*4, *v*8) 1,00

Fig. 3. Sub-network 2: a) optimal control variables, b) bottleneck link loads, c) link loads.

u(f2,v3) u(f2,v4) u(f3,v5) u(f3,v6)

2,*v*4) *u*(*f*

g(v3) g(v4) g(v5) g(v6)

*g*(*v*3) *g*(*v*4) *g*(*v*5) *g*(*v*6)

l8 l9 l12 l13 l15 l17 l18 m = 1 0,46 0,46 0,00 0,46 0,46 1,00 0,46 m = 2 0,36 0,57 0,21 0,57 0,57 0,79 0,36 m = 3 0,31 0,62 0,31 0,62 0,62 0,69 0,31

13 *l*

15*l*

17*l* 18

12*l*

c)

Fig. 3. Sub-network 2: a) optimal control variables, b) bottleneck link loads, c) link loads.

9 *l*

m = 1 0,00 1,00 0,46 0,46 m = 2 0,21 0,79 0,57 0,57 m = 3 0,31 0,69 0,62 0,62

b)

3,*v*5) *u*(*f*

3,*v*6)

m = 1 0,00 1,00 0,50 0,50 m = 2 0,21 0,79 0,38 0,62 m = 3 0,31 0,69 0,33 0,67

a)

0,00 0,10 0,20 0,30 0,40 0,50 0,60 0,70 0,80 0,90 1,00

> *m =* 1 *m =* 2 *m =* 3

0,00 0,10 0,20 0,30 0,40 0,50 0,60 0,70 0,80 0,90 1,00

> *m =* 1 *m =* 2 *m =* 3

0,00 0,10 0,20 0,30 0,40 0,50 0,60 0,70 0,80 0,90 1,00

> *m =* 1 *m =* 2 *m =* 3

*l* 8 *l*

*u*(*f*

2,*v*3) *u*(*f*


*3.2. Total NO2 Load over the Area Using air Quality Zonal Modeling*

A total 60-day average NO2 tropospheric column was demonstrated through the air quality zonal modeling (Figure 6). Here, the whole country was divided into 145 grids (255 × 255 km2) where the spatial distribution of NO2 (Figure 6a) was displayed for the micro-level understanding of the air quality changes in different regions of the country. The area was also calculated under each grid category. The NO2 concentration ranges between (1.7e+15–3.2e+15) had 131 grids and the area was 5.5 million sq.km before the lockdown but the grids have increased to 191 and the area has also increased to 80 million sq.km. Conversely, the areas under the high NO2 concentration category (5.5e+15–9.8e+15) have

significantly decreased from 1.1 to 0.2 million sq.km. Moreover, there was a significant decrease in NO2 concentration in the areas that come under a very high concentration category where the NO2 tropospheric column ranges between 1.6e+16 and 2.6e+16; 13 grids were accounting for 0.55 million km2 before the lockdown but none of the areas were under this very high category after lockdown (Figure 6a).

**Figure 5.** Time series, area-average of NO2 tropospheric column (30% cloud screened) daily 0.25 deg. [OMI OMNO2d v003] 1/cm2 over 24 November 2019–22 March 2020, shape China. Source: OMI, Global Modeling and Assimilation Office, NASA, 2020.

**Figure 6.** (**a**) Time average map and zonal distribution of the NO2 tropospheric column, (**b**) declining rate (in total and percentage) of NO2, (**c**) trend analysis of the total NO2 tropospheric column of 145 random sites of China's mainland before and after pandemic. Source: OMI, Global Modeling and Assimilation Office, NASA.

The total column of NO2 decline rate in percentage was grouped into three categories, i.e., less than 50%, 51–70%, and more than 70%. The maximum decline rate (more than 70%) of the NO2 total column was measured in Guangzhou, Wuhan, and Chengdu city of China before and after the pandemic COVID-19 event (Figure 6b). Figure 6c reveals the trend of the total NO2 tropospheric column of 145 random sites before and after the pandemic where it is clearly visible that almost all the sample sites have experienced a low concentration of NO2 in the atmosphere.

In the graphs (Figure 7), steepness of the histogram revealed the degree of reduction in the NO2 concentration. Guangzhou city showed high steepness of NO2 reduction after the pandemic. The maximum (almost 60%) declining rate of NO2 was observed in Guangzhou, Wuhan, Chengdu, and Beijing; whereas the minimum (almost 40%) declining rate of NO2 total column in the atmosphere was found in Jinan, Shanghai, Anshan, Zhengzhou, Xinjiang, and Lanzhou.

**Figure 7.** Absolute trend line indicates 60-day average of NO2 tropospheric column (molecules/cm2), whereas the bar graph represents 10-day average concentration (X-axis) of NO2 in 15 major cities of China before (yellow) and after (purple) the COVID-19 lockdown period. Source: OMI, Global Modeling and Assimilation Office, NASA, 2020.


*Article* **Can Socioeconomic, Health, and Safety Data Explain the Spread of COVID-19 Outbreak on Brazilian Federative Units?**

#### **Diego Galvan 1,2,3,\* , Luciane E**ff**ting 4 , Hágata Cremasco 4 and Carlos Adam Conte-Junior 1,2,3**

Received: 19 October 2020; Accepted: 14 November 2020; Published: 30 November 2020

**Abstract:** Infinite factors can influence the spread of COVID-19. Evaluating factors related to the spread of the disease is essential to point out measures that take effect. In this study, the influence of 14 variables was assessed together by Artificial Neural Networks (ANN) of the type Self-Organizing Maps (SOM), to verify the relationship between numbers of cases and deaths from COVID-19 in Brazilian states for 110 days. The SOM analysis showed that the variables that presented a more significant relationship with the numbers of cases and deaths by COVID-19 were influenza vaccine applied, Intensive Care Unit (ICU), ventilators, physicians, nurses, and the Human Development Index (HDI). In general, Brazilian states with the highest rates of influenza vaccine applied, ICU beds, ventilators, physicians, and nurses, per 100,000 inhabitants, had the lowest number of cases and deaths from COVID-19, while the states with the lowest rates were most affected by the disease. According to the SOM analysis, other variables such as Personal Protective Equipment (PPE), tests, drugs, and Federal funds, did not have as significant effect as expected.

**Keywords:** coronavirus disease; artificial neural networks; SARS-CoV-2; ventilator; index development index; developing country

#### **1. Introduction**

In December 2019, cases of acute pneumonia in the city of Wuhan, China, called attention due to the speed of contagion [1]. The virus that caused this pneumonia in affected individuals is the SARS-CoV-2, the causative agent of the novel coronavirus (COVID-19). The number of cases quickly evolved into a pandemic, decreed by the World Health Organization (WHO) on 11 March 2020. Currently, a few months after the first cases, this disease has been plaguing the world [2].

At the moment, the pandemic epicenter is in Latin America, with Brazil, Argentina, Colombia, Peru, Mexico, and Chile being the most representative countries, concentrating more than 10 million cases. Brazil already has more than 5 million cases and 150,000 deaths, ranking third in the world in the number of cases and deaths by COVID-19, behind only to the United States and India [3].

Intense efforts by the scientific community are focused on finding drugs to treat [4] and develop a vaccine for the disease [5]. Other research tries to understand how the disease has spread around the world. Some factors can influence the spread of the virus, including the disregard of preventive measures adopted by the government by population [6], climatic [7–9], comorbidities [10,11], hospital structure, prepared professionals, personal protective equipment, and financial resources [4].

In order to monitor the pandemic, it is necessary to collect information on the most significant possible number of factors that can influence the disease's dissemination behavior and, with this, establish a relationship that allows to affirm which measures delay the spread of COVID-19. Statistical tools have been used to study the behavior of epidemics for years; this area treats, analyzes, and obtains information from large datasets that cannot be analyzed by traditional systems [12].

Artificial Neural Networks (ANNs) are tools that have been gaining scientific relevance to perform pattern recognition, classification, or prediction tasks. ANN is a non-linear computational model attempting to simulate human brain structure and decision-making [13]. Its architecture is inspired by biological neural networks and consists of simple processing units that store empirical knowledge through a learning process [14].

Among the most common types of neural networks are the Feed-Forward Neural Network (FFNN), the Convolutional Neural Network (CNN), and the Recursive Neural Network (RNN). FFNNs are the most common type of ANN in practical applications, consisting of one or more hidden layers of perceptrons (neurons) that require supervised training. The input data of the desired sample datasets and the output results are sent to the network several times until the error in the output is minimized [13].

Traditionally, several types of neural networks exist with various techniques such as Autoregression (AR) [15,16], Moving Average (MA) [17,18], Exponential Smoothing (ES) [19], Hybrid Methods (HM) [20–22], and Autoregressive Integrated Moving Average (ARIMA) [23]. They have been used to predict the dependent variable in a time series [15–23]. Among these techniques, unsupervised neural networks of the type Self-Organizing Maps model has mostly outperformed others in precision and accuracy [14].

Self-Organizing Maps (SOM) or Kohonen Map is a method for the analysis of multivariate data used for pattern recognition and classification, which may be taken as a non-linear generalization of principal component analysis [24]. The SOM algorithm consists of input nodes and a grid of computational connected nodes (neurons), which compete among themselves for activation as the one that most closely resembles the input vector. If the input data exhibit some similarity across the input classes, the neurons will organize themselves, showing similarity patterns in a grid [13,25].

SOM has already been used successfully for data mining in several areas of knowledge [24,26–30], like food science [26,28,29], fuels [27], monitoring chemical reactions [24,30], and other applications [25,31,32]. Recently, SOM was used to verify the spatial relationship of the COVID-19 spread in countries and states in Mexico [31]. Most published papers have the main theme of the modelling or prediction of the spread of COVID-19 [2,9,33–36]. In our literature searches, we did not evidence reports that simultaneously consider the historical data on the number of cases/deaths from COVID-19 with possible external factors that affect the spread of the virus through a spatial analysis by pattern recognition.

In a previous study, we demonstrated that the spread of COVID-19 varies according to Brazilian regions, states, and cities. However, it was not possible to allege the reason for this behavior. In this study, 14 possible factors that can affect the spread of COVID-19 were analyzed jointly by SOM. This analysis will verify which variables may be essential and will point out possible variables that had the most significant effect on each Brazilian state.

#### **2. Theoretical Foundations**

The Kohonen algorithm was proposed by Teuvo Kohonen in 1982 [14]. In SOM, there is no given output target, the objective of the algorithm is to find a set of neurons to represent the cluster, but with topological restrictions [31]. The learning process begins with the random initialization of the synaptic weights vector of each neuron. In sequence, three key steps are developed for the formation of the feature map: Competition, cooperation, and synaptic adaptation [24].

The following is a brief description of the Kohonen algorithm, as described previously by Haykin [14]. The function is chosen to represent the topological neighborhood in the following equation.

$$h\_{\dot{\jmath}\dot{\iota}} = \exp(-d\_{\dot{\jmath}\dot{\iota}}^2/2\sigma^2),$$

where σ is the effective radius of the topological neighborhood, and *dj,i* is the lateral distance between the winning neuron *i* and the excited neuron *j*, defined through Euclidean distance. Over the training epochs, there is a reduction in the size of the neighborhood due to an exponential decay, described by the equation.

$$\sigma(n) = \sigma\_0 \exp(-n/\tau\_1) \quad n = 0, 1, 2, \dots$$

where σ0 is the effective radius in the initialization of the algorithm, τ1 is the time constant, with τ1 = 1000/log σ0 being recommended, and *n* is the number of training epochs.

During the adaptive process, the synaptic weight vector (**w***j* ) of the *j* neuron in the grid must be modified concerning the input vector **x**. The modification process is a modification of the Hebb learning postulate, described by the equation.

$$\mathbf{w}\_{j}(n+1) = \mathbf{w}\_{j}(n) + (\eta)h\_{j,i(x)}(\mathbf{x} - \mathbf{w}\_{j(n)})$$

where η(*n*) is the learning rate, which is variable and decreases during the training epochs, *n*. The learning rate decrease may be modelled by an exponential decay, as described in the equation. In this equation, η0 is the initial learning rate, and τ2 is another time constant; the recommended values are, respectively, 0.1 and 1000:

$$
\eta(n) = n\_0 \exp(-n/\tau\_2),
$$

#### **3. Methodology**

#### *3.1. Dataset*

The dataset used in the study by neural networks was obtained from websites of Brazilian government agencies or institutions, available for download. Before the SOM analysis, the variables that would constitute the study were selected. All variables related to socioeconomic, health, and safety factors that could explain the dissemination of COVID-19 in Brazil were included in our research. They were made available and updated daily by institutional sites of the Brazilian federal government.

Initially, data for all variables were converted and expressed as rates for each 100,000 inhabitants. For the conversion, we used data from the Brazilian population estimated in 2019 by the Instituto Brasileiro de Geografia e Estatística – IBGE (Brazilian Institute of Geography and Statistics); the public agency responsible for conducting censuses and organizing information related to the country's geosciences and social, demographic, and economic statistics [37]; available in https://www.ibge.gov.br/.

The numbers of cases and accumulated deaths by COVID-19 were analyzed for 16 epidemiological weeks. According to the international convention, each week starts on Sunday and ends on Saturday. The information represents the numbers recorded from 26 February to 13 June, 2020 (110 days), a period that corresponds from the 9th to the 24th epidemiological week of the year, first COVID-19 case record in Brazil. The data were obtained from the website of the Ministério da Saúde—MS (Ministry of Health) of Brazil, the government sector responsible for the administration and maintenance of public health in the country, updated daily with information about the COVID-19 pandemic in the country [38]; available in https://covid.saude.gov.br/.

The Federal Government of Brazil transfers financial resources, supplies, and equipment for structuring health services in the country to combat or treat patients affected by COVID-19. The MS frequently updates a panel of inputs considered essential, distributed to each Brazilian

state: Influenza vaccination—H1N1 and H3N2 (distributed and applied), drugs (chloroquine and oseltamivir), COVID-19 tests (rapid and reverse-transcriptase polymerase chain reaction—RT-PCR), hand sanitizer per liter, and Personal Protective Equipment—PPE (surgical masks, N95 mask, gowns, gloves, glasses, face shield, caps, and sneakers) [39]. These data were obtained on June 13, 2020, at https://covid-insumos.saude.gov.br/paineis/insumos/painel.php.

The destination of Federal funds transferred to each federative unit (Brazilian states) is monitored and presented on the Federal Government website [40], available in https://www.tesourotransparente. gov.br/visualizacao/painel-de-monitoramentos-dos-gastos-com-covid-19. The data used in this study represent the sum of all transfers made until 13 June 2020. For better visualization, the values were converted from the real (R\$) to the US dollar (US\$) by the quotation of R\$ 5.05 equals to US\$ 1.00, referring to 12 June 2020, provided by the Banco Central do Brasil—BCB (Central Bank of Brazil) [41], available in https://www.bcb.gov.br/.

Health data from public and private institutions, such as the number of physicians, nurses, ventilators, and Intensive Care Unit (ICU) beds for COVID-19 in each state, were obtained from the IBGE [42]; available in https://mapasinterativos.ibge.gov.br/covid/saude/.

Finally, the data referring to the Human Development Index (HDI) of each state for the year 2015 (last census registered in the country) were obtained from the Atlas do Desenvolvimento Humano do Brasil—ADHB (Human Development Atlas of Brazil), a public institution that provides information about human development in Brazil [43]; available in http://www.atlasbrasil.org.br/2013/pt/download/base/.

#### *3.2. Data Analysis by Artificial Neural Network*

The Kohonen Map routine developed was carried out in Matlab software (MathWorks, Natick, MA, USA) according to the algorithm previously described in Haykin [14] and Cremasco et al. [26]. The datasets were evaluated using three different approaches. In the first step, two SOM analyses were conducted to verify the distribution of the numbers of cases and deaths from COVID-19 in Brazilian regions and states. In the second step, the distribution of 14 variables for the Brazilian states was verified: ICU beds, ventilators, physicians, nurses, PPE, hand sanitizer, rapid test, PCR test, vaccines distributed, vaccines applied, chloroquine tablets, oseltamivir capsules, HDI, and federal funds distribution.

The SOM setup was a hexagonal topology of 8 × 8 and 4 × 4 for 27 Brazilian states and five regions, respectively, for the number of cases and deaths from COVID-19 (**first step**); 8 × 8 for 27 Brazilian states with 14 variables (**second step**) with 7000 training epochs to ensure convergence of the average quantization error. The initial neighborhood relationship was 3.5, decreasing to 0.07, with an initial learning rate of 0.1, decaying exponentially with the training epochs to 9.11 × 10−5 . It was used a computer Intel® Core™ i7–4790 CPU© 3.60 GHz, 32 GB RAM, and 250 GB HDD.

In order to obtain a better representation, the values collected after the SOM analysis were transposed through the color scale to the Brazilian cartographic map for each variable. This procedure was adopted to facilitate the interpretation of unfamiliar people with the Kohonen map. The original SOM output weight maps for the 14 variables generated are displayed in the Supplementary Material in Figures S1–S14.

#### *3.3. Spearman's Correlation Test*

The Spearman's rank correlation coefficient is adopted to determine the correlation between variables. It analyzes how well the association between two variables can be defined using a monotonic function. The correlation matrix was performed using the package "corrplot" in the Software R Core Team (Vienna, Austria) [44].

#### **4. Tests and Results**

Figure 1 shows the number of cases and deaths accumulated by Brazilian states and regions per 100,000 inhabitants. All states belonging to the South (S) and Central-West (CW) regions of the country Brazilian state.

representative states in the region.

had the lowest rates of cases and deaths by COVID-19 recorded, an average of 180 cases and 4 deaths per 100,000 inhabitants. Most of the Brazilian states with the highest rates of cases and deaths belong to the North (N) region, an average of 954 cases and 43 deaths per 100,000 inhabitants, mainly represented by Acre (AC), Amapá (AP), Amazonas (AM), Pará (PA), and Roraima (RR). SOM network classifies the input data as clusters that can be formed by one or more neurons. The definition of clusters is characterized by the presence of empty neurons between the groups. Nearby clusters share some similarity, that is, the greater the Euclidean distance, the greater the difference in behavior.

associated with a respective winning neuron, that is, the one that best represents in the analysis. The

*Int. J. Environ. Res. Public Health* **2020**, *17*, x 5 of 16

belong to the North (N) region, an average of 954 cases and 43 deaths per 100,000 inhabitants, mainly

Some states in the Brazilian Northeast (NE) and Southeast (SE) also had high rates of cases and deaths when compared to the other states in the South (S) and Central-West (CW) regions. The Northeast (NE) is the second region with the highest rate of cases and deaths, an average of 527 cases and 24 deaths per 100,000 inhabitants, the states in this region that were most aggravated by the disease were Ceará (CE), Maranhão (MA), and Pernambuco (PE). The Southeast (SE) is the third region with the highest rate of cases and deaths, with an average of 338 cases and 23 deaths by COVID-19, with Espírito Santo (ES), Rio de Janeiro (RJ), and São Paulo (SP) being the most

The neural network demonstrated that the spread of COVID-19 in Brazil has heterogeneous behavior; see Figure 1. It is essential to understand this behavior. The North of the country was more affected by COVID-19 than the South. The Northeast and Southeast differ in case rates but have similar death rates. In this sense, possible factors were evaluated that may explain the reason for this behavior and point out which measures are more relevant in the fight against COVID-19 in each

After the training phase of the SOM network, we generated the topological map of the 14

represented by Acre (AC), Amapá (AP), Amazonas (AM), Pará (PA), and Roraima (RR).

*Int. J. Environ. Res. Public Health* **2020**, *17*, x; doi: www.mdpi.com/journal/ijerph **Figure 1.** Geographical distribution of cases and deaths by COVID-19 in Brazilian states (above) and regions (below) per 100,000 inhabitants.

Some states in the Brazilian Northeast (NE) and Southeast (SE) also had high rates of cases and deaths when compared to the other states in the South (S) and Central-West (CW) regions. The Northeast (NE) is the second region with the highest rate of cases and deaths, an average of 527 cases and 24 deaths per 100,000 inhabitants, the states in this region that were most aggravated by the disease were Ceará (CE), Maranhão (MA), and Pernambuco (PE). The Southeast (SE) is the third region with the highest rate of cases and deaths, with an average of 338 cases and 23 deaths by COVID-19, with Espírito Santo (ES), Rio de Janeiro (RJ), and São Paulo (SP) being the most representative states in the region.

The neural network demonstrated that the spread of COVID-19 in Brazil has heterogeneous behavior; see Figure 1. It is essential to understand this behavior. The North of the country was more affected by COVID-19 than the South. The Northeast and Southeast differ in case rates but have similar death rates. In this sense, possible factors were evaluated that may explain the reason for this behavior and point out which measures are more relevant in the fight against COVID-19 in each Brazilian state.

After the training phase of the SOM network, we generated the topological map of the 14 variables evaluated together, which represents the distribution of each federative unit in Brazil, according to the winning neuron; see Figure 2. In the topological map, each federative unit is associated with a respective winning neuron, that is, the one that best represents in the analysis. The SOM network classifies the input data as clusters that can be formed by one or more neurons. The definition of

clusters is characterized by the presence of empty neurons between the groups. Nearby clusters share some similarity, that is, the greater the Euclidean distance, the greater the difference in behavior. **Figure 1.** Geographical distribution of cases and deaths by COVID-19 in Brazilian states (above) and regions (below) per 100,000 inhabitants.

*Int. J. Environ. Res. Public Health* **2020**, *17*, x 6 of 16

**Figure 2.** Distribution of Brazilian federative units according to the winning neuron. Where: Acre— AC; Alagoas—AL; Amapá—AP; Amazonas—AM; Bahia—BA; Ceará—CE; Distrito Federal—DF; Espírito Santo—ES; Goiás—GO; Maranhão—MA; Mato Grosso—MT; Mato Grosso do Sul—MS; Minas Gerais—MG; Pará—PA; Paraíba—PB; Paraná—PR; Pernambuco—PE; Piauí—PI; Roraima— RR; Rondônia—RO; Rio de Janeiro—RJ; Rio Grande do Norte—RN; Rio Grande do Sul—RS; Santa Catarina—SC; São Paulo—SP; Sergipe—SE; Tocantins—TO. **Figure 2.** Distribution of Brazilian federative units according to the winning neuron. Where: Acre—AC; Alagoas—AL; Amapá—AP; Amazonas—AM; Bahia—BA; Ceará—CE; Distrito Federal—DF; Espírito Santo—ES; Goiás—GO; Maranhão—MA; Mato Grosso—MT; Mato Grosso do Sul—MS; Minas Gerais—MG; Pará—PA; Paraíba—PB; Paraná—PR; Pernambuco—PE; Piauí—PI; Roraima—RR; Rondônia—RO; Rio de Janeiro—RJ; Rio Grande do Norte—RN; Rio Grande do Sul—RS; Santa Catarina—SC; São Paulo—SP; Sergipe—SE; Tocantins—TO.

We identified the formation of some clusters by evaluating the topological map. Among some of the clusters formed, we highlight the one formed in the lower-left corner of the map in Figure 2, represented by the states of the Southern region of the country, which contain the federative units: Paraná (PR), Rio Grande do Sul (RS), and Santa Catarina (SC). This cluster contains the federal units and the region of the country with the lowest rates of deaths and cases by COVID-19, as shown in Figure 1. Another cluster formed that we can highlight is composed of some states in the North region of the country, represented by the states of Acre (AC), Amapá (AP), and Roraima (RR), presented in the upper-right corner of the map. This cluster represents the region and some of the federative units with the highest rates of deaths and cases due to COVID-19 in the country. This same analysis can be done for the other formed clusters. We identified the formation of some clusters by evaluating the topological map. Among some of the clusters formed, we highlight the one formed in the lower-left corner of the map in Figure 2, represented by the states of the Southern region of the country, which contain the federative units: Paraná (PR), Rio Grande do Sul (RS), and Santa Catarina (SC). This cluster contains the federal units and the region of the country with the lowest rates of deaths and cases by COVID-19, as shown in Figure 1. Another cluster formed that we can highlight is composed of some states in the North region of the country, represented by the states of Acre (AC), Amapá (AP), and Roraima (RR), presented in the upper-right corner of the map. This cluster represents the region and some of the federative units with the highest rates of deaths and cases due to COVID-19 in the country. This same analysis can be done for the other formed clusters.

To verify the applicability of SOM analysis, we compared the results obtained with another unsupervised method. Thus, we evaluated the ability to group the data obtained using the Hierarchical Cluster Analysis (HCA) method. The results obtained with the HCA were very similar to those obtained with the SOM algorithm. The dataset presents different patterns according to the region in which the Brazilian states are located. In other words, most states in the North and Northeast regions formed a cluster, while most federative units in the Central-West, South, and Southeast formed another cluster. The graphic output generated by the HCA method is shown in Figure S15 of the Supplementary Material. To verify the applicability of SOM analysis, we compared the results obtained with another unsupervised method. Thus, we evaluated the ability to group the data obtained using the Hierarchical Cluster Analysis (HCA) method. The results obtained with the HCA were very similar to those obtained with the SOM algorithm. The dataset presents different patterns according to the region in which the Brazilian states are located. In other words, most states in the North and Northeast regions formed a cluster, while most federative units in the Central-West, South, and Southeast formed another cluster. The graphic output generated by the HCA method is shown in Figure S15 of the Supplementary Material.

In general, the topological map of the SOM network made it possible to state that the socioeconomic, health and safety data demonstrate that the spread of COVID-19 in the country varies according to the Brazilian federative units. However, only the topological map does not allow to state which of the 14 variables evaluated may be the main responsible for explaining the spread of COVID-19 in the country. Thus, we used weight maps in the following discussions. In order to obtain a better representation, the values collected from the weight maps were transposed through the color scale to the Brazilian cartographic map for each variable. However, when doing the transposition In general, the topological map of the SOM network made it possible to state that the socioeconomic, health and safety data demonstrate that the spread of COVID-19 in the country varies according to the Brazilian federative units. However, only the topological map does not allow to state which of the 14 variables evaluated may be the main responsible for explaining the spread of COVID-19 in the country. Thus, we used weight maps in the following discussions. In order to obtain a better representation, the values collected from the weight maps were transposed through the color scale to the Brazilian cartographic map for each variable. However, when doing the transposition procedure,

procedure, we lose the neighborhood relationship. In Figures S1–S14 of the Supplementary Material,

the original outputs of the weight maps for each variable in the SOM network are shown.

we lose the neighborhood relationship. In Figures S1–S14 of the Supplementary Material, the original outputs of the weight maps for each variable in the SOM network are shown. discussion, the weight maps were divided into four blocks and grouped according to similarities: (i) Available hospital infrastructure; (ii) inputs and tests available; (iii) drugs available; and (iv) financial resources,

*Int. J. Environ. Res. Public Health* **2020**, *17*, x 7 of 16

The 14 variables were evaluated jointly by ANN. However, for better results visualization and

The 14 variables were evaluated jointly by ANN. However, for better results visualization and discussion, the weight maps were divided into four blocks and grouped according to similarities: (i) Available hospital infrastructure; (ii) inputs and tests available; (iii) drugs available; and (iv) financial resources, The first set of variables is shown in Figure 3, composed of the rates of ICU beds, ventilators, physicians, and nurses per 100,000 inhabitants. In general, it is observed that the states that belong to the South, Southeast, and Central-West regions of Brazil had higher rates than the North and Northeast. There is evidence that these variables influence the rates of cases and deaths in each

The first set of variables is shown in Figure 3, composed of the rates of ICU beds, ventilators, physicians, and nurses per 100,000 inhabitants. In general, it is observed that the states that belong to the South, Southeast, and Central-West regions of Brazil had higher rates than the North and Northeast. There is evidence that these variables influence the rates of cases and deaths in each Brazilian state, and it can be said that the states with the lowest rates of ICU, respirators, physicians, and nurses, have the highest rates of cases and deaths from COVID-19. Brazilian state, and it can be said that the states with the lowest rates of ICU, respirators, physicians, and nurses, have the highest rates of cases and deaths from COVID-19. It is also evident that these were not the only relevant factors. The state of Rio de Janeiro (RJ), for example, has high rates of ICU beds, respirators, physicians, and nurses and despite this, it has a higher death rate than other states in the North region, such as Acre (AC), Amapá (AP), and Roraima (RR).

**Figure 3.** Geographical distribution of Intensive Care Unit (ICU) bed, ventilators, physicians and nurses in Brazilian states per 100,000 inhabitants. **Figure 3.** Geographical distribution of Intensive Care Unit (ICU) bed, ventilators, physicians and nurses in Brazilian states per 100,000 inhabitants.

The second set of variables presented in Figure 4 represents the inputs destined to each state by the Federal Government to combat and control the spread of COVID-19, composed by PPE, hand sanitizer, rapid test, and PCR test per 100,000 inhabitants. In general, the PPE and hand sanitizer rates It is also evident that these were not the only relevant factors. The state of Rio de Janeiro (RJ), for example, has high rates of ICU beds, respirators, physicians, and nurses and despite this, it has a higher death rate than other states in the North region, such as Acre (AC), Amapá (AP), and Roraima (RR).

distributed for each state did not have a direct relationship with the rates of cases or deaths by COVID-19 according to the SOM analysis. The second set of variables presented in Figure 4 represents the inputs destined to each state by the Federal Government to combat and control the spread of COVID-19, composed by PPE, hand sanitizer,

*Int. J. Environ. Res. Public Health* **2020**, *17*, x; doi: www.mdpi.com/journal/ijerph

COVID-19 in the country.

compared to other Brazilian states.

rapid test, and PCR test per 100,000 inhabitants. In general, the PPE and hand sanitizer rates distributed for each state did not have a direct relationship with the rates of cases or deaths by COVID-19 according to the SOM analysis. test is reliable but expensive and time-consuming [46]. It is necessary to adopt a balance between rapid tests and PCR; for example, if PCR test indices are higher, it will not be possible to follow the spread in a short period, while rapid tests, when misused, can generate mistaken information.

Rapid tests are cheaper, less reliable, and can result in false positives or negatives, while the PCR

*Int. J. Environ. Res. Public Health* **2020**, *17*, x 8 of 16

The states of Amapá (AP) and Rio de Janeiro (RJ), for example, had the highest rates of PPE distributed, and Amazonas (AM) had the highest rates of liters of hand sanitizer distributed per 100,000 inhabitants; however, these three states present higher rates of cases and deaths when

It is important to note that the data refer to the distributed quantity of inputs, and it is not possible to estimate which portion was used by hospitals or the population, or even say if they were used correctly. The cultural factor also interferes in these variables, mainly concerning the education

The disease tests in the country have been carried out homogeneously, as shown in Figure 4. This homogeneity is considered an important aspect and demonstrates that differences do not influence the data collected regarding the number of cases in the form of confirmation of the disease, which could question the reliability of the data. The states that had the highest test rates available were the Distrito Federal (DF), Rio Grande do Sul (RS), and Roraima (RR), which belong to the

and awareness of the local population about the use of these inputs [45].

**Figure 4.** Geographical distribution of Personal Protective Equipment (PPE), hand sanitizer, rapid tests, and PCR tests available to Brazilian states per 100,000 inhabitants. **Figure 4.** Geographical distribution of Personal Protective Equipment (PPE), hand sanitizer, rapid tests, and PCR tests available to Brazilian states per 100,000 inhabitants.

*Int. J. Environ. Res. Public Health* **2020**, *17*, x; doi: www.mdpi.com/journal/ijerph Previous measures as vaccines and drugs against diseases with similar symptoms, such as influenza H1N1 and H3N2, can facilitate the diagnosis of COVID-19, since they are relevant The states of Amapá (AP) and Rio de Janeiro (RJ), for example, had the highest rates of PPE distributed, and Amazonas (AM) had the highest rates of liters of hand sanitizer distributed per 100,000 inhabitants; however, these three states present higher rates of cases and deaths when compared to other Brazilian states.

It is important to note that the data refer to the distributed quantity of inputs, and it is not possible to estimate which portion was used by hospitals or the population, or even say if they were used correctly. The cultural factor also interferes in these variables, mainly concerning the education and awareness of the local population about the use of these inputs [45].

The disease tests in the country have been carried out homogeneously, as shown in Figure 4. This homogeneity is considered an important aspect and demonstrates that differences do not influence the data collected regarding the number of cases in the form of confirmation of the disease, which could question the reliability of the data. The states that had the highest test rates available were the Distrito Federal (DF), Rio Grande do Sul (RS), and Roraima (RR), which belong to the Central-West, South, and North regions, respectively, i.e., the most and least affected regions by COVID-19 in the country.

Rapid tests are cheaper, less reliable, and can result in false positives or negatives, while the PCR test is reliable but expensive and time-consuming [46]. It is necessary to adopt a balance between rapid tests and PCR; for example, if PCR test indices are higher, it will not be possible to follow the spread in a short period, while rapid tests, when misused, can generate mistaken information.

Previous measures as vaccines and drugs against diseases with similar symptoms, such as influenza H1N1 and H3N2, can facilitate the diagnosis of COVID-19, since they are relevant information during the patient's anamnesis. These procedures allow faster and more efficient identification of the disease, which makes the adopted treatments more assertive and precocious [47]. Another measure that focuses on significant discussions in the scientific community is the use of drugs against the coronavirus [48]. Among several drugs, the Brazilian Federal Government has invested and passed on chloroquine tablets to the states.

The third set of variables in Figure 5 consists of influenza vaccines distributed and applied, and drugs distributed by states per 100,000 inhabitants. The states of the North region, such as Amazonas (AM), Roraima (RR), Pará (PA), and Acre (AC), had a more significant disparity than the other states in the rates of vaccines applied and distributed. At first, this behavior could be related to health factors, shown in Figure 3. However, this disparity would also be observed in the Northeast region. Other intrinsic factors may include geographic and logistical factors that make distribution and access to the population a challenge to receive the vaccine. *Int. J. Environ. Res. Public Health* **2020**, *17*, x 10 of 16

**Figure 5.** Geographical distribution of influenza vaccines distributed, influenza vaccines applied, chloroquine tablets, and oseltamivir capsules available in Brazilian states per 100,000 inhabitants. **Figure 5.** Geographical distribution of influenza vaccines distributed, influenza vaccines applied, chloroquine tablets, and oseltamivir capsules available in Brazilian states per 100,000 inhabitants.

As far as we know, there is no drug with proven efficiency against SARS-CoV-2; some drugs have been evaluated to combat or alleviate the disease's symptoms [48]. Recent studies demonstrate that dexamethasone welcomes the preliminary treatment of critically ill patients with COVID-19 [49]. Recently, the Food and Drug Administration (FDA) revoked emergency use authorization for chloroquine and hydroxychloroquine in patients with COVID-19 [50].

**Figure 6.** Geographical distribution by Index Development Index (HDI) and Federal funds destined

to combat COVID-19 in Brazilian states per 100,000 inhabitants.

*Int. J. Environ. Res. Public Health* **2020**, *17*, x; doi: www.mdpi.com/journal/ijerph

Figure 5 shows the chloroquine rates distributed by the Federal Government per 100,000 inhabitants. Chloroquine tablets were more widely distributed in the states of the North region, which has higher rates of deaths and cases of COVID-19 in Brazil. The possibility of greater distribution of chloroquine in these states may have occurred as an attempt to treat the disease symptoms when there were no restrictions on its use.

*Int. J. Environ. Res. Public Health* **2020**, *17*, x 10 of 16

The oseltamivir rates distributed by the Federal Government per 100,000 inhabitants is shown in Figure 5. This drug is commonly used to treat symptoms caused by the influenza virus and is administered when there is still no confirmed diagnosis of COVID-19 [51]. Oseltamivir was more widely distributed in the Southern and Northern states of Brazil; however, it is not possible to establish a direct relationship with the rates of cases and deaths by COVID-19 using SOM analysis.

The last set of variables evaluated comprises the HDI and the distribution of federal funds destined to each state. According to Figure 6, the destination of Federal funds was more significant for some states in the North, the region which has the highest rates of cases and deaths due to COVID-19. More resources destined for this region can be guided by the scenario of cases and deaths registered. **Figure 5.** Geographical distribution of influenza vaccines distributed, influenza vaccines applied, chloroquine tablets, and oseltamivir capsules available in Brazilian states per 100,000 inhabitants.

**Figure 6.** Geographical distribution by Index Development Index (HDI) and Federal funds destined to combat COVID-19 in Brazilian states per 100,000 inhabitants. **Figure 6.** Geographical distribution by Index Development Index (HDI) and Federal funds destined tocombat COVID-19 in Brazilian states per 100,000 inhabitants.

The highest HDI values are in the Central-West, South, and Southeast regions, with the Federal District (DF), Santa Catarina (SC), and São Paulo (SP) being the highest values in the country. It is also worth mentioning that the most affected states by COVID-19 have some of the lowest HDIs in the country, showing problems that some of these states have in fundamental areas of the population service.

*Int. J. Environ. Res. Public Health* **2020**, *17*, x; doi: www.mdpi.com/journal/ijerph Figure 7 shows the Spearman's correlation test adopted to determine the correlation among variables. The test demonstrated that the influenza vaccine applied, ICU, ventilators, physicians, nurses, and HDI were the most correlated significant variables, all positive correlations, while chloroquine had a negative correlation with all these variables. The HDI is directly related to essential aspects of the population, such as family income, education level, health, among other factors; therefore, the correlation with the HDI was already expected, since it takes into account important aspects of health for its determination. Other variables evaluated showed minor correlations, while PPE did not correlate with any other variable.

*Int. J. Environ. Res. Public Health* **2020**, *17*, x 11 of 16

**Figure 7.** Spearman correlation coefficients for the 14 evaluated variables. Correlations with *p*-value > 0.05 are considered insignificant. **Figure 7.** Spearman correlation coefficients for the 14 evaluated variables. Correlations with *p*-value > 0.05 are considered insignificant.

#### **5. Discussion 5. Discussion**

Considering the importance of understanding the spread of the virus, many studies have been presented in recent months related to different aspects of the pandemic, with the application of computational intelligence tools to model and predict the spread of the disease [34]. Considering the importance of understanding the spread of the virus, many studies have been presented in recent months related to different aspects of the pandemic, with the application of computational intelligence tools to model and predict the spread of the disease [34].

In our study, we gave the data a differentiated approach from the reported works. It is not common to find simultaneous studies with historical data on the number of COVID-19 cases, related to external factors, that affect the spread of the virus through spatial analysis [31]. For this, we used unsupervised pattern recognition to analyze some variables, among many others, that may be related to the disease spread. In our study, we gave the data a differentiated approach from the reported works. It is not common to find simultaneous studies with historical data on the number of COVID-19 cases, related to external factors, that affect the spread of the virus through spatial analysis [31]. For this, we used unsupervised pattern recognition to analyze some variables, among many others, that may be related to the disease spread.

Brazil is one of the most affected countries by the pandemic and by the tremendous socioeconomic, territorial, climatic differences, among the country's federative units. We understand that it may be interesting to try to establish spread patterns of the virus in the country. Thus, once the relationship of these factors with the spread and lethality of the disease has been identified, we hope that our study can assist in the analysis of the data that have been generated by institutions and, consequently, assist in directing decision-making on practical combat actions. Brazil is one of the most affected countries by the pandemic and by the tremendous socioeconomic, territorial, climatic differences, among the country's federative units. We understand that it may be interesting to try to establish spread patterns of the virus in the country. Thus, once the relationship of these factors with the spread and lethality of the disease has been identified, we hope that our study can assist in the analysis of the data that have been generated by institutions and, consequently, assist in directing decision-making on practical combat actions.

It is important to note that our analysis does not aim to point out or compare which were the best actions adopted to contain the spread of COVID-19, so we approached each variable according to the significance indicated by the SOM analysis. It is important to note that our analysis does not aim to point out or compare which were the best actions adopted to contain the spread of COVID-19, so we approached each variable according to the significance indicated by the SOM analysis.

Based on SOM's clustering skills, we were able to spatially group similar federative units in terms of the number of cases and deaths by COVID-19 with data on the distribution of financial resources, equipment, health professionals and HDI, represented by a color scale. Thus, SOM's ability Based on SOM's clustering skills, we were able to spatially group similar federative units in terms of the number of cases and deaths by COVID-19 with data on the distribution of financial resources, equipment, health professionals and HDI, represented by a color scale. Thus, SOM's ability to cluster,

therefore, can benefit from similar strategies to deal with the virus spread.

to cluster, made it possible to cluster together with the federal units that are behaving similarly and,

made it possible to cluster together with the federal units that are behaving similarly and, therefore, can benefit from similar strategies to deal with the virus spread.

The SOM analysis allowed us to raise some hypotheses about the spread of the virus in Brazil. In general, the analysis indicated that the spread of the disease has a direct relationship with quite heterogeneous socioeconomic, health, and safety variations in the national territory; see topological map in Figure 2.

In Figure 2, we can see that the spread of the disease in Brazilian states has been differentiated and regionalized. Interestingly, the network separated the most and least regions affected by COVID-19. We found that there was a separation pattern, in which all the Southern federative units are located at the bottom left of the map (PR, RS, and SC), while the Southeast states (SP, RJ, MG, and ES) and the Central-West (MS, MT, DF, and GO) are in the upper left corner of the map. In the North and Northeast regions, the federative units were classified on the right side of the map.

For better visualization, we invite readers to check the original output weight maps of the variables in the Supplementary Material. Weight maps represent the overlap of the topological map (Figure 2) and allow us to evaluate the behavior of each variable for the segmentation of the federative units. Analyzing the weight maps for the 14 variables extracted from the SOM network, it was possible to show which were the main responsible for differentiating the states, regarding the number of cases and deaths by COVID-19 in Brazil by socioeconomic, health, and safety data.

According to the 14 weight maps, the HDI was one of the essential variables for the distribution obtained (Figure 6 and Figure S13). The analysis indicated that the federal units with the lowest HDI were considerably more affected by the pandemic and that they had greater difficulty in combating the spread of the virus. Meanwhile, the best-prepared Brazilian federations with higher HDI values were more capable of deal with the pandemic.

Possibly, the HDI was the main measure responsible for the separation due to its high correlation with the other variables. It is noteworthy that this index relates to health (life expectancy), education (adult literacy index and levels of education), and income (GDP—gross domestic product—per capita). Thus, it is possible to observe the correlation of the HDI in Figure 7, with other variables, such as ICU (Figure S1), ventilators (Figure S2), doctors (Figure S3), and nurses (Figure S4).

Another important variable that allowed us to assess the behavior of the spread of COVID-19 in Brazil was the influenza vaccine rates of doses distributed and applied, highly correlated according to the Spearman correlation test (Figure 7). According to Figure 6, Figures S9 and S10, we show that the federative units with the highest rates of vaccines applied had lower rates of cases and deaths from COVID-19, as pointed out by the SOM analysis. Although this procedure is not effective against SARS-CoV-2, this measure may have facilitated clinical diagnosis and made the treatment of affected patients by COVID-19 faster and more accurate, and may have reflected in the number of cases and deaths in these regions and federative units.

Other measures adopted in the country, such as the use of drugs as chloroquine and oseltamivir, did not allow to evaluate the spread of COVID-19, as shown in Figure 5 and the weight maps of the variables in Figures S11 and S12. The SOM analysis indicated that the federative units in the North and Northeast regions had higher rates of chloroquine tablets distribution but more cases and deaths from the disease were registered. On the other hand, oseltamivir capsules had higher distribution rates in the Southern and Northern states, representing the regions more and less affected by the disease, which means that this variable, the spread of COVID-19 was similar.

We evidence that PPE (Figure S5) and hand sanitizer (Figure S6) rates distributed for each state did not have a direct relationship with the rates of cases or deaths by COVID-19 when considering the SOM analysis (Figure 4). It is known that these measures directly influence the control of the COVID-19 dissemination [45]; in this sense, we show that the SOM analysis allowed us to verify that although these items have been distributed to federative units, the population has not followed the measures adopted or made use of these items properly. Figure 7 reinforces this hypothesis since the PPE did not correlate with any other variable. Thereby, government agencies must further encourage the use of available items.

Among the other less expressive variables evaluated, we show that the rapid test (Figure S8), PCR test (Figure S9), and distribution of Federal funds (Figure S14) did not present an obvious behavior that would allow explaining the spread of COVID-19 using the numbers of cases and deaths in the country, according to the SOM analysis indicated.

### **6. Conclusions**

Unsupervised Artificial Neural Networks of the Self-Organizing Map type have demonstrated that the spread of the coronavirus has heterogeneous behavior and varies among Brazilian regions and states. According to the analysis, among the 14 variables evaluated, the factors responsible for the highest relationship with the numbers of cases and deaths by COVID-19 in Brazilian states were: Rates of influenza vaccine applied, ICU beds, ventilators, physicians, nurses, and HDI, positively correlated according to the Spearman's correlation test. In general, the lowest rates of cases and deaths by COVID-19 were recorded in the Brazilian states with the highest rates of influenza vaccine applied, ICU beds, respirators, physicians, and nurses, per 100,000 inhabitants, which consequently has some of the highest HDI in the country.

As a future work, we intend to integrate the SOM algorithm to analyze the spatial and temporal aspects of the COVID-19 spread in a unified way to obtain a complete view and solution to the problem. In addition, to analyze the possibility of applying SOM in other diseases that have affected Brazil, caused by dengue virus, Chikungunya virus, Zika virus, and yellow fever virus.

**Supplementary Materials:** The following are available online at http://www.mdpi.com/1660-4601/17/23/8921/s1, Figure S1: Weight maps overlaid by topological maps for variable Intensive Care Unit (ICU) by Brazilian federative unit. Figure S2: Weight maps overlaid by topological maps for variable Ventilators by Brazilian federative unit. Figure S3: Weight maps overlaid by topological maps for variable Physicians by Brazilian federative unit. Figure S4: Weight maps overlaid by topological maps for variable Nurses by Brazilian federative unit. Figure S5: Weight maps overlaid by topological maps for variable Personal Protective Equipment by Brazilian federative unit. Figure S6: Weight maps overlaid by topological maps for variable Hand sanitizer by Brazilian federative unit. Figure S7: Weight maps overlaid by topological maps for variable Rapid test by Brazilian federative unit. Figure S8: Weight maps overlaid by topological maps for variable PCR test by Brazilian federative unit. Figure S9: Weight maps overlaid by topological maps for variable Influenza vaccines distributed by Brazilian federative unit. Figure S10: Weight maps overlaid by topological maps for variable Influenza vaccines applied by Brazilian federative unit. Figure S11: Weight maps overlaid by topological maps for variable Chloroquine tablets by Brazilian federative unit. Figure S12: Weight maps overlaid by topological maps for variable Oseltamivir capsules by Brazilian federative unit. Figure S13: Weight maps overlaid by topological maps for variable Human Development Index (HDI) by Brazilian federative unit. Figure S14: Weight maps overlaid by topological maps for variable Distribution of Federal funds by Brazilian federative unit. Figure S15: Dendrogram of an HCA for Brazilian Federative Units.

**Author Contributions:** D.G.: Conceptualization, formal analysis, investigation, methodology, software, validation, visualization, writing—review and editing. L.E.: Conceptualization, formal analysis, investigation, methodology, software, validation, visualization, writing—review and editing. H.C.: Conceptualization, formal analysis, investigation, methodology, software, validation, visualization, writing—review and editing. C.A.C.-J.: Methodology, writing—review and editing, supervision, project administration, funding acquisition. All authors have read and agreed to the published version of the manuscript.

**Funding:** The authors are thankful for the financial support provided by the Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro (FAPERJ), D.G.: [grant number E-26/200.062/2020, E-26/010.000.984/2019, and E-26/010.000148/2020]. http://www.faperj.br/. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

**Conflicts of Interest:** The authors have declared that no competing interests exist.

### **References**

1. Wells:, C.R.; Sah, P.; Moghadas, S.M.; Pandey, A.; Shoukat, A.; Wang, Y.; Wang, Z.; Meyers, L.A.; Singer, B.H.; Galvani, A.P. Impact of international travel and border control measures on the global spread of the novel 2019 coronavirus outbreak. *Proc. Natl. Acad. Sci. USA* **2020**, *117*, 7504–7509. [CrossRef] [PubMed]

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**1. Introduction**

The design and simulation of integrated circuits (IC) are assisted by compact MOSFET models, which started to be developed in the 1960s [1] for long-channel devices. The technological progress promoted the down-scaling of semiconductor devices, giving rise to short-channel effects and their interference in circuit performance; thereby, these shortchannel effects were incorporated into the existing long-channel based models to improve circuit-design efficiency.

Although BSIM [2,3] has been broadly used as the main MOSFET model to simulate MOS circuits in EDA tools, the complexity of its calculations and numerous parameters have opened a gap between circuit simulations and designs by hand [4,5], which has complicated the understandings of how the main MOSFET parameters relate to simulation results. Therefore, it is in designers' interest to have models founded on physics available in the simulator, such as those based on the inversion charge.

In the fast expanding ultra-low-voltage domain [6], some short-channel effects, such as velocity saturation, are not relevant; thus, a simplified MOSFET model can be satisfactory for circuit design. Targeting the increasing number of ultra-low-voltage designs [7–12], this work proposes a four-parameter model (4PM) based on the all-region advanced compact MOSFET model (ACM) [13].

In this work, the 4PM was carried out with the description language Verilog-A to easily simulate circuits in the commercial Cadence Virtuoso simulator, which implements BSIM 4.5 through a private propriety interface [14]. Hardware description languages

**Citation:** Adornes, C.M.; Alves Neto, D.G.; Schneider, M.C.; Galup-Montoro, C. Bridging the Gap between Design and Simulation of Low-Voltage CMOS Circuits. *J. Low Power Electron. Appl.* **2022**, *12*, 34. https://doi.org/10.3390/ jlpea12020034

Academic Editors: Orazio Aiello and Andrea Acquaviva

Received: 16 February 2022 Accepted: 5 May 2022 Published: 16 June 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

(HDLs), such as Verilog-A, are interchangeable with different simulators and assist designers in describing circuits and systems in a variety of behavioral modeling levels. We chose Verilog-A because it combines simplicity, functionality and portability [14].

The paper is structured as described in the following lines. Section 2 briefly introduces the four-parameter model (4PM). Section 3 describes the methods employed to extract the model's parameters and describes the extraction results of the parameters with temperature and process variations. Section 4 describes how to carry out the 4PM in Verilog-A for its inclusion in Cadence. Section 5 presents the results of the simulations carried out by using BSIM or the 4PM in Verilog-A. Four circuits designed according to the ACM model were simulated: a CMOS inverter, a ring oscillator, a self-biased current source and a common source amplifier. Conclusions are drawn in Section 6.