Basic Reproduction Number (ℜ 0 )

The basic reproduction numberℜ₀is among the most important quantities in infectious disease epidemiology. It plays a fundamental role in the study of mathematical models of emerging infectious diseases in outbreak situations and provides information for the design of control strategies. In a population composed only of susceptible individuals, the average number of infections caused by an infected individual is defined asℜ₀.

Theℜ0is characterized mathematically by considering the transmission of infection

as a demographic process, in which producing offspring is considered as causing a new infection through transmission. Thus, in consecutive generations of infected individuals, we can consider the infection process. Consecutive generations increasing in size indicate a growing population, and the growth factor per generation indicates the growth potential.

This growth factor is the mathematical characterization of ℜ₀. Then we have that, if 0 < ℜ₀ < 1the infection will die out in the long run and ifℜ₀ > 1the infection will be able to spread in a population [45]. The higher theℜ₀the more difficult it is to control the epidemic. Theℜ₀can be affected by several factors, such as the duration of infectivity of the affected patients, the infectivity of the organism and the degree of contact between the susceptible and infected populations. To find theℜ₀in the ODE systems describing the behavior of epidemics we start with the equations describing the production of new infections and the state changes among infected individuals which is called the infected subsystem. We first linearize the infected subsystem over the infection-free steady state.

Linearization shows thatℜ₀characterizes the initial spread potential of an infected in a fully susceptible population and we assume that the change in the susceptible population is negligible during the first virus outbreak. In the next subsections, we show the relationship betweenℜ0and the local and global asymptotic stability at the infection-free equilibrium point [45].

The relationship betweenℜ₀and local stability in disease-free point

The basic reproduction number depends on how we define the infected and uninfected compartments, and not only on the structure of the model. We define the number of individuals in the compartments as𝑥 = (𝑥₁, ..., 𝑥_𝑛), where 𝑥_𝑖 ≥ 0, 𝑖 = 1, ..., 𝑛. Then, we ordered the compartments such that the first𝑚 (𝑚 < 𝑛) compartments correspond to infected individuals. Let𝑋_𝑠 be the set of all disease-free states, that is

𝑋_𝑠 = {𝑥 ≥ 0| 𝑥_𝑖 = 0, 𝑖 = 1, ..., 𝑚}.

𝑇_𝑖(𝑥) is the rate of new infections in compartment 𝑖, Σ⁺_𝑖(𝑥) the rate of transfer of individuals into compartment𝑖 by other means, and Σ⁻_𝑖(𝑥) the rate of transfer out of compartment𝑖. We assume that all functions are continuously differentiable at least twice in each variable. The disease transmission model is defined as:

̇𝑥_𝑖 = 𝑇_𝑖(𝑥) − Σ_𝑖(𝑥) = 𝑓_𝑖(𝑥), 𝑖 = 1, ..., 𝑛, (1.1) with non-negative initial conditions, whereΣ𝑖 = Σ⁻_𝑖 − Σ⁺_𝑖 and the following assumptions are satisfied:

(P1) If 𝑥 ≥ 0, then 𝑇_𝑖, Σ⁻_𝑖, Σ⁺_𝑖 ≥ 0for 𝑖 = 1, ..., 𝑚 (each function represents a directed transfer of individuals, and are non-negative).

(P2) if𝑥_𝑖 = 0thenΣ⁻_𝑖(𝑥) = 0(if a compartment is empty, there is no transfer of individuals out of the compartment by any means).

(P3) 𝑇_𝑖 = 0if𝑖 > 𝑚, (the incidence of infection for uninfected compartments is null).

(P4) If𝑥 ∈ 𝑋_𝑠 then𝑇_𝑖(𝑥) = 0 andΣ⁺_𝑖(𝑥) = 0 for𝑖 = 1, ..., 𝑚 (there is no inmigration of infectives, density independent).

1.1 | THEORICAL BACKGROUND

(P5) Consider a population near the disease-free equilibrium point𝑥₀. If we introduce some infected individuals it does not lead to an epidemic, i.e. the population remains close to𝑥₀, then according to the linearized system the population will return to𝑥₀.

̇𝑥 = 𝐷𝑓 (𝑥₀)(𝑥 − 𝑥0), (1.2)

where𝐷𝑓 (𝑥₀)is the Jacobian matrix,[

𝜕𝑓𝑖

𝜕𝑥_𝑗]_(𝑖,𝑗)evaluated at the𝑥₀. We have that some derivatives are one-sided, since𝑥₀is on the boundary of the domain. We focus our study to systems in which𝑥₀is stable in the absence of a new infection. That is, If𝑇 (𝑥)is set to zero, then all eigenvalues of𝐷𝑓 (𝑥0)have negative real parts.

The following lemma shows us a way to partition𝐷𝑓 (𝑥₀)by the above conditions.

Lemma 1.1.1. If𝑥0is the equilibrium disease-free point of system(1.1)and𝑓𝑖(𝑥)satisfies (P1)-(P5), then the derivatives𝐷𝑇 (𝑥₀)and𝐷Σ(𝑥₀)can be decomposed as

𝐷𝑇 (𝑥₀) = [

𝐓 0 0 0 ], 𝐷Σ(𝑥₀) =

[

𝚺 0

𝐽₃ 𝐽₄ ], where𝐓and𝚺are the𝑚 × 𝑚matrices defined by𝐓 =

[

𝜕𝑇𝑖

𝜕𝑥_𝑗(𝑥₀) ],𝚺 =

[

𝜕Σ𝑖

𝜕𝑥_𝑗(𝑥₀)

]with1 ≤ 𝑖, 𝑗 ≤ 𝑚. Further,𝐓is non-negative, 𝚺is a non-singular M-matrix¹and all eigenvalues of𝐽₄ have positive real part.

The(𝑗, 𝑘)entry of𝚺⁻¹is the average length of time this individual spends in compart-ment𝑗during his lifetime, we assumed that the population remains close to disease-free equilibrium and without reinfection. The(𝑖, 𝑗)entry of𝐓is the rate where individuals in𝑗 produce new infections in𝑖. Then, the entry(𝑖, 𝑘) in𝐓𝚺⁻¹ is the expected number of new infections in compartment𝑖 produced by the infected individual introduced in compartment𝑘. We define𝐓Σ⁻¹ the next-generation matrix for the model and

ℜ₀= 𝜌(−𝐓𝚺⁻¹) where𝜌(𝐴)denotes the spectral radius of a matrix𝐴.

The following lemmas are part of the proof of the Theorem (1.1.4). For more details of results (1.1.2)-(1.1.3) and Theorem (1.1.4) and its proofs, the readers could refer to [51].

Lemma 1.1.2. Let𝐻 be a non-singular M-matrix and suppose𝐵and𝐵𝐻⁻¹have the𝑍 sign pattern². Then𝐵is a non-singular M-matrix if and only if𝐵𝐻⁻¹is a non-singular𝑀−matrix.

Lemma 1.1.3. Let𝐻 be a non-singular𝑀−matrix and suppose𝐾 ≥ 0. Then,

1If𝐵is non-negative matrix, and𝑟 > 𝜌(𝐵)where𝜌(𝐵)is the spectral ratio of𝐵, then𝐴 = 𝑟 𝐼_𝑚−𝐵is non-singular M- matrix where𝐼_𝑚is the identity matrix. If𝑟 = 𝜌(𝐵), then𝐴is a singular M-matrix [51].

2A matrix𝐵 = [𝑏𝑖𝑗]has the𝑍sign pattern if𝑏𝑖𝑗 ≤ 0for all𝑖 ≠ 𝑗.

i. (𝐻 −𝐾 )is non-singular𝑀−matrix if and only if(𝐻 −𝐾 )𝐻⁻¹is a non-singular𝑀−matrix.

ii. (𝐻 − 𝐾 )is singular𝑀−matrix if and only if(𝐻 − 𝐾 )𝐻⁻¹ is a singular𝑀−matrix.

If all the eigenvalues of matrix 𝐷𝑓 (𝑥₀) have negative real parts then 𝑥₀ is locally asymptotically stable and unstable if any eigenvalue of𝐷𝑓 (𝑥₀)has positive real part. Using the Lemma (1.1.1) we can decompose the eigenvalues into two sets, the𝐓 − 𝚺eigenvalues and those of−𝐽₄ which all have negative real part. So the stability of the infection-free point depends on the𝐓 − 𝚺eigenvalues. The following theorem shows the relationship between theℜ0and the local stability in𝑥0.

Theorem 1.1.4. Consider the disease transmission model given by(1.1)and𝑓 (𝑥)satisfies (P1)-(P5). If𝑥₀is the equilibrium disease-free point of the model, then𝑥₀is locally asymptotically

stable ifℜ0< 1, but unstable ifℜ0> 1withℜ0= 𝜌(−𝐓𝚺⁻¹).

Proof. The (P1)-(P5) conditions are used in the Lemmas (1.1.2) and (1.1.3) that are applied in the proof.

Let𝐽₁ = 𝐓 − 𝚺. As𝚺is non-singular M-matrix and𝐓is non-negative,−𝐽₁= 𝚺 − 𝐓, has the𝑍 sign pattern. Thus,

𝑠(𝐽₁) < 0 ⇔ −𝐽₁is a non- singular matrix,

where𝑠(𝐽₁)denotes the maximum real part of all the eigenvalues of the matrix𝐽₁. Since 𝐓𝚺⁻¹ is non-negative−𝐽₁𝚺⁻¹ = 𝐼 − 𝐓𝚺⁻¹ also has the𝑍 sign pattern. For the application of Lemma (1.1.2) with𝐻 = 𝚺and𝐵 = −𝐽₁= 𝚺 − 𝐓, we have

−𝐽₁is a non-singular M-matrix⟺ 𝐼 − 𝐓𝚺⁻¹is a non-singular M-matrix.

Finally, since𝐓𝚺⁻¹is non-negative, all eigenvalues of𝐓𝚺⁻¹have magnitude less than or equal to𝜌(𝐓𝚺⁻¹). Then,

𝐼 − 𝐓𝚺⁻¹ is a non-singular M-matrix, ⟺ 𝜌(𝐓𝚺⁻¹) < 1.

Hence,

𝑠(𝐽1) < 0 ⇔ ℜ0< 1.

Analogously, we have that

𝑠(𝐽₁) = 0 ⟺ −𝐽₁is a singular M-matrix,

⟺ 𝐼 − 𝐓𝚺⁻¹is a singular M-matrix, for Lemma (1.1.3) with𝐻 = 𝚺and𝐾 = 𝐓

⟺ 𝜌(𝐓𝚺⁻¹) = 1.

The rest of the equivalences are obtained as in the non-singular case. Therefore,𝑠(𝐽₁) = 0 if and only ifℜ0= 1. Then, we have that𝑠(𝐽1) > 0if and only ifℜ0> 1.

1.1 | THEORICAL BACKGROUND

Global Stability for the Disease-Free Equilibrium

In this subsection, we list two conditions that if met, also guarantee the global asymp-totic stability of the disease-free point. First, we write the system as:

𝑑𝑥

𝑑𝑡 = 𝐹 (𝑥, 𝐼 ), 𝑑𝐼

𝑑𝑡 = 𝐺(𝑥, 𝐼 ), 𝐺(𝑥, 0) = 0, (1.3)

where𝑥 ∈ ℝ^𝑛−𝑚 denotes the number of uninfected individuals (components) and𝐼 ∈ ℝ^𝑚 denotes the number of infected individuals including latent, infectious, etc (components).

Let𝑋₀ = (𝑥^∗, 0)denotes the disease-free equilibrium point. The conditions (𝐻₁) and (𝐻₂) below must be satisfied to guarantee local asymptotic stability.

• (𝐻₁): For 𝑑𝑥

𝑑𝑡 = 𝐹 (𝑥, 0), 𝑥^∗is globally asymptotically stable (g.a.s) inΩ,

• (𝐻₂):𝐺(𝑥, 𝐼 ) = 𝐴𝐼 − 𝐺^∗(𝑥, 𝐼 ),𝐺^∗(𝑥, 𝐼 ) ≥ 0for(𝑥, 𝐼 ) ∈ Ω,

where 𝐴 = 𝐷_𝐼𝐺(𝑥^∗, 0) is an M-matrix (Jacobian of 𝐺 with the off-diagonal elements nonnegative) andΩis the invariant biologically feasible region of the system.

If the system (1.3) satisfies the above two conditions, then we have the following theorem:

Theorem 1.1.5. The fixed point𝑋₀ = (𝑥^∗, 0)is a globally asymptotic stable (g.a.s) equilibrium point of the system provided thatℜ₀< 1locally asymptotic stable (l.a.s) and that assumptions (𝐻1)and(𝐻2)are satisfied.

This theorem, its proof and examples of its application can be found in [38].

Proof. Let𝐼₀ = 𝐼 (0), we have that𝐼 (𝑡) ≥ 0if𝐼₀ > 0and that𝑒^𝐴𝑡 is a positive semigroup (since𝐴is an M- matrix). Hence, using the variation-of-constants formula, we have

0 ≤ 𝐼 (𝑡) = 𝑒^𝐴𝑡𝐼₀− ∫

𝑡 0

𝑒^{𝐴(𝑡−𝑠)}𝐺^∗(𝑥(𝑠), 𝐼 (𝑠))𝑑𝑠 ≤ 𝑒^𝐴𝑡𝐼₀. (1.4) Since𝐴is an M-matrix, A has a dominant eigenvalues𝑚(𝐴)with𝑚(𝐴) < 0forℜ₀< 1. Thus,

𝑡→∞lim||𝑒^𝐴𝑡|| = 0 ⟹ lim

𝑡→∞𝐼 (𝑡) = 0. (1.5)

Note that𝑥^∗is a g.a.s equilibrium of𝑑𝑥

𝑑𝑡 = 𝐹 (𝑥, 0), a limiting system of𝑑𝑥

𝑑𝑡 = 𝐹 (𝑥(𝑡), 𝐼 (𝑡)). Thus,

𝑡→∞lim𝑥(𝑡) = 𝑥^∗. (1.6)