We work with the model proposed by Silva and Torres [9] that explains the dynamics of HIV infection assuming homogenous mixing within communities. Here the entire population is divided into four compartments. These compartments are mutually exclusive in the sense that no person can belong to more than one compartment at any time point. The four compartments are: susceptible individuals (S), individuals infected with HIV but having no clinical symptoms of AIDS (I), individuals infected with AIDS and having ART treatment as they are supposed to belong to chronic stage (C), and individuals infected with HIV having clinical symptoms of AIDS (A). Note that a person belonging to I may transmit HIV to other individuals although he/she has no symptoms of AIDS. Under this situation, the model we consider is, d S d t=Λ - β ( I ( t ) + η C C ( t ) + η A A ( t ) ) S ( t ) N - μ S ( t ) (1) d I d t=β ( I ( t ) + η C C ( t ) + η A A ( t ) ) S ( t ) N - ( ρ + ϕ + μ ) I ( t ) + ω C ( t ) + α A ( t ) (2) d C d t=ϕ I ( t ) - ( ω + μ ) C ( t ) (3) d A d t=ρ I ( t ) - ( α + μ + d ) A ( t ) (4)

Here β represents the contact rate for HIV transmission, η_A (≥ 1) is the relative infectiousness of individuals with AIDS symptoms compared to those infected with HIV but no AIDS symptoms, and η_C (≤ 1) is the rate of partial restoration of immune function of HIV infected individuals who are correctly treated under ART. ρ and ϕ are the rates at which individuals in class I move to A and C respectively. α and ω are the respective rates at which individuals from class A and class C move to I. We assume that d is the HIV induced death rate while μ is the natural death rate in the population.

We consider the total population as N(t) = S(t) + I(t) + A(t) + C(t) implying that d N ( t ) d t = Λ - μ N ( t ) - d A ( t ). It can be easily shown that N ( t ) ≤ Λ μ.

We use the approach of next generation matrix [15] to calculate R₀. Note that the rate of new infections coming in the compartments I, C and A are β ( I + η C C + η A A ) S N, 0 and 0 respectively. On the other hand, new infections moving out of the compartments I, C, and A are (ρ + ϕ + μ)I − ωC − αA, (ω + μ)C − ϕI and (α + μ + d)A − ρI respectively. Thus following [9, 15], we have basic reproduction number (R₀) as: R 0 = S 0 β N ξ 2 ( ξ 1 + ρ η A ) + η C ϕ ξ 1 μ ( ξ 2 + ( ρ + ξ 1 ) + ϕ ξ 1 + ρ d ) + ρ ω d = S 0 N ζ γ ≈ ζ γ (5) where ζ = β(ξ₂(ξ₁ + ρη_A) + η_C ϕξ₁), γ = μ(ξ₂ + (ρ + ξ₁) + ϕξ₁ + ρd) + ρωd, ξ₁ = α + μ + d and ξ₂ = ω + μ.

Evaluation of CCS depends on the time to extinction of disease based on the given model. In order to find the time to extinction, we now construct the fully stochastic version of the model (1)—(4). Next, we write the Kolmogorov forward equation for this stochastic model. This would be the key step for further downstream analysis.

First, we note the nature of transitions and the respective transition rates from one compartment to another (Table 1).

Here, s = S/N, i = I/N, c = C/N, a = A/N. Then the Kolmogorov forward equation becomes, p s , i , c , a ′ ( t ) = λ 1 ( s - 1 , i , c , a ) p s - 1 , i , c , a + λ 2 ( s + 1 , i , c , a ) p s + 1 , i , c , a + λ 3 ( s + 1 , i - 1 , c , a ) p s + 1 , i - 1 , c , a + λ 4 ( s + 1 , i , c - 1 , a ) p s + 1 , i , c - 1 , a + λ 5 ( s + 1 , i , c , a - 1 ) p s + 1 , i , c , a - 1 + λ 6 ( s , i + 1 , c - 1 , a ) p s , i + 1 , c - 1 , a + λ 7 ( s , i + 1 , c , a - 1 ) p s , i + 1 , c , a - 1 + λ 8 ( s , i + 1 , c , a ) p s , i + 1 , c , a + λ 9 ( s , i - 1 , c + 1 , a ) p s , i - 1 , c + 1 , a + λ 10 ( s , i , c + 1 , a ) p s , i , c + 1 , a + λ 11 ( s , i - 1 , c , a + 1 ) p s , i - 1 , c , a + 1 + λ 12 ( s , i , c , a + 1 ) p s , i , c , a + 1 - κ ( s , i , c , a ) p s , i , c , a (6) where κ ( s , i , c , a ) = ∑ i = 1 12 λ i ( s , i , c , a ) and p_s,i,c,a(t) = P[S(t) = s, I(t) = i, C(t) = c, A(t) = a].

Now, we find quasi-stationarity by conditioning on non-extinction of the disease. So we write, q s , i , c , a ( t ) = P [ S ( t ) = s , I ( t ) = i , C ( t ) = c , A ( t ) = a | ( I ( t ) , C ( t ) , A ( t ) ) ≠ ( 0 , 0 , 0 ) ] = p s , i , c , a ( t ) 1 - p • 000 ( t ) where p • 000 ( t ) = ∑ s = 0 ∞ P [ S ( t ) = s , I ( t ) = 0 , C ( t ) = 0 , A ( t ) = 0 ] = ∑ s = 0 ∞ p s , 0 , 0 , 0 ( t ).

Differentiating q_s,i,c,a(t) with respect to t, we have, q s , i , c , a ′ ( t ) = p s , i , c , a ′ ( t ) 1 - p • 000 ( t ) + p s , i , c , a ( t ) ( 1 - p • 000 ( t ) ) 2 . p • 000 ′ ( t ) . (7) Now, from Eq (6), we have, after simplification, p • 000 ′ ( t ) = ∑ s = 0 ∞ [ μ N p s - 1 , 0 , 0 , 0 ( t ) + μ ( s + 1 ) p s + 1 , 0 , 0 , 0 ( t ) + μ ( 0 + 1 ) p s , 1 , 0 , 0 ( t ) + μ ( 0 + 1 ) p s , 0 , 1 , 0 ( t ) + ( μ + d ) ( 0 + 1 ) p s , 0 , 0 , 1 ( t ) ] - ( μ N + μ s ) ∑ s = 0 ∞ p s , 0 , 0 , 0 ( t ) = μ p • 1 0 0 ( t ) + μ p • 0 1 0 ( t ) + ( d + μ ) p • 0 0 1 ( t ) = p • ( d , μ ) ( t ) ( s a y ) . (8) From Eqs (7) and (8) we have, q s , i , c , a ′ ( t ) = p s , i , c , a ′ ( t ) 1 - p • 0 0 0 ( t ) + p s , i , c , a ( t ) ( 1 - p • 0 0 0 ( t ) ) 2 . p • 0 0 0 ′ ( t ) = p s , i , c , a ′ ( t ) 1 - p • 0 0 0 ( t ) + p s , i , c , a ( t ) 1 - p • 0 0 0 ( t ) . p • ( d , μ ) ( t ) 1 - p • 0 0 0 ( t ) = p s , i , c , a ′ ( t ) 1 - p • 0 0 0 ( t ) + p s , i , c , a ( t ) 1 - p • 0 0 0 ( t ) . q • ( d , μ ) ( t ) , ( s a y ) (9) where q • ( d , μ ) ( t ) = p • ( d , μ ) ( t ) 1 - p • 0 0 0 ( t ) (10) Now, following Nåsell [6, 7] we have, q s , i , c , a ′ ( t ) = 0 ⇒ p s , i , c , a ′ ( t ) = - p s , i , c , a ( t ) ( 1 - p • 0 0 0 ( t ) ) . q • ( d , μ ) ( t ) ( 1 - p • 0 0 0 ( t ) ) = - q • ( d , μ ) ( t ) p s , i , c , a ( t ) ∴ p s , i , c , a ( t ) = c e - q • ( d , μ ) . t = q • ( d , μ ) e - q • ( d , μ ) . t (11)

Let τ_Q be the time to extinction when the initial distribution is equal to the quasi-stationary distribution. Therefore, we have, E ( τ Q ) = 1 q • ( d , μ ) . (12)

The disease-free equilibrium is obtained as: Σ 0 = ( S 0 , I 0 , C 0 , A 0 ) = ( Λ μ , 0 , 0 , 0 ). To find the other endemic equilibrium, if exists, we put N = N(0), x₁(t) = S(t)/N, x₂(t) = I(t)/N, x₃(t) = C(t)/N, and x₄(t) = A(t)/N. The equilibrium point is obtained by equating the first differentiation in Eqs (1)–(4) to zero [9, 10], i.e. x 1 ′ ( t ) = μ - β ( x 2 ( t ) + η C x 3 ( t ) + η A x 4 ( t ) ) x 1 - μ x 1 ( t ) = 0 (13) x 2 ′ ( t ) = β ( x 2 ( t ) + η C x 3 ( t ) + η A x 4 ( t ) ) x 1 - ( ρ + ϕ + μ ) x 2 ( t ) + ω x 3 ( t ) + α x 4 ( t ) = 0 (14) x 3 ′ ( t ) = ϕ x 2 ( t ) - ( ω + μ ) x 3 ( t ) = 0 (15) x 4 ′ ( t ) = ρ x 2 ( t ) - ( α + μ + d ) x 4 ( t ) = 0 (16)

For simplicity we use the notations: x_j(t) = x_j for j = 1, …, 4, ξ₁ = α + μ + d, ξ₂ = ω + μ, γ = ξ₁ ξ₂(ρ + ϕ + μ) − ωϕξ₁ − ραξ₂, ζ = β(ξ₁ ξ₂+ η_C ϕξ₁+ η_A ρξ₂).

Then solving Eqs (13)–(16), we have the endemic equilibrium as: x ^ 1= γ ζ = 1 R 0 (17) x ^ 2= ( ζ - γ ) ζ γ . μ ξ 1 ξ 2 = ( R 0 - 1 ) μ ζ . ξ 1 ξ 2 (18) x ^ 3= ( ζ - γ ) ζ γ . μ ϕ ξ 1 = ( R 0 - 1 ) μ ζ . ϕ ξ 1 (19) x ^ 4= ( ζ - γ ) ζ γ . μ ρ ξ 2 = ( R 0 - 1 ) μ ζ . ρ ξ 2 (20)

We consider a diffusion approximation to the stochastic version of our model (1)—(4). This would allow us to approximately evaluate the quasi-stationary distribution using a multivariate normal distribution when N is large and R₀ is greater than 1. First we find the multivariate normal distribution corresponding to the state variables. Let the changes in the scaled state variables x₁, x₂, x₃, and x₄ during the time interval (t, t + δt) be denoted by δx₁, δx₂, δx₃, and δx₄ respectively, where δx_i(t) = x_i(t + δt) − x_i(t), 1, 2, 3, 4.

Under the assumptions of the original process on the sequence of transitions, we evaluate the mean vector and covariance matrix for δx_i (i = 1, 2, 3, 4) during the time interval (t, t + δt) as follows. At first, we assume that we are in the state (S, I, C, A). So, the possible transitions from this state are:

The random variable δx₁ equals 1 N in cases (a), - 1 N in cases (b), (c), (d), (e), and 0 in other cases. Similarly, δx₂ equals 1 N in cases (c), (d), (e), (i), (k), - 1 N in cases (f), (g), (h), and 0 in other cases. δx₃ equals 1 N in cases (f), - 1 N in cases (i), (j), and 0 in other cases. δx₄ equals 1 N in cases (g), - 1 N in cases (k), (l), and 0 in other cases. Denoting x = (x₁, x₂, x₃, x₄)′ we have, E ( δ x ) = b ( x ) δ t + o ( δ t ) where b ( x ) = ( μ - β ( x 2 + η C x 3 + η A x 4 ) x 1 - μ x 1 β ( x 2 + η C x 3 + η A x 4 ) x 1 - ( ρ + ϕ + μ ) x 2 + ω x 3 + α x 4 ϕ x 2 - ( ω + μ ) x 3 ρ x 2 - ( α + μ + d ) x 4 ) (21)

Now to derive the covariance matrix we need to find the Jacobian matrix at point x, which is obtained from b(x) as, B ( x ) = ∂ b ( x ) ∂ x = ( - β ( x 2 + η C x 3 + η A x 4 ) - μ - β x 1 - β η C x 1 - β η A x 1 β ( x 2 + η C x 3 + η A x 4 ) β x 1 - ( ρ + ϕ + μ ) β η C x 1 + ω β η A x 1 + α 0 ϕ - ( ω + μ ) 0 0 ρ 0 - ( α + μ + d ) ) Then we approximate B(x) at equilibrium point x ^ = ( x ^ 1 , x ^ 2 , x ^ 3 , x ^ 4 ) ′ by B ( x ^ ). So, covariance matrix of δx = (δx₁, δ x₂, δx₃, δx₄)′ is,

V ( δ x ) = 1 N S ( x ) δ t + o ( δ t ) where, S ( x ) = ( β ( x 2 + η C x 3 + η A x 4 ) x 1 - β x 1 x 2 - β η C x 1 x 3 - β η A x 1 x 4 μ ′ + μ x 1 - β x 1 x 2 β ( x 2 + η C x 3 + η A x 4 ) x 1 - ϕ x 2 - ω x 3 - ρ x 2 - α x 4 + ( ρ + ϕ + μ ) x 2 + ω x 3 + α x 4 - β η C x 1 x 3 - ϕ x 2 - ω x 3 ϕ x 2 + ( ω + μ ) x 3 0 - β η A x 1 x 4 - ρ x 2 - α x 4 0 ρ x 2 + ( α + μ + d ) x 4 ) We again approximate S(x) by S ( x ^ ), where x ^ is the equilibrium point. For large N, the process N ( x ( t ) - x ^ ) is approximated by a multivariate Ornstein-Uhlenbeck (O-U) process [16], with a local drift matrix B ( x ^ ) and local covariance matrix S ( x ^ ).

The stationary distribution of this O-U process approximates the quasi-stationary distribution. It is approximately normal with mean zero and covariance matrix Σ, where Σ is obtained by solving B ( x ^ ) Σ + Σ B ′ ( x ^ ) = - S ( x ^ ) . (22) Exact analytical solution for Σ is not straightforward [5]. However, since we are interested in calculating the CCS, we can easily solve the Eq (22) numerically using the parameter values and the equilibrium point.

Let σ_ij be the solution of the (i, j)th element of Σ, where i, j = 1, …, 4. The diffusion approximation guides us to consider appropriate mean and variance-covariance matrix for the joint distribution of x(t). Thus we have, N ( x ( t ) - x ^ ) ∼ N 4 ( 0 , Σ ) , with x ( t ) = ( x 1 ( t ) , x 2 ( t ) , x 3 ( t ) , x 4 ( t ) ) ′ , x ^ = ( x ^ 1 , x ^ 2 , x ^ 3 , x ^ 4 ) ′ , and Σ = ( σ 11 σ 12 σ 13 σ 14 σ 21 σ 22 σ 23 σ 24 σ 31 σ 32 σ 33 σ 34 σ 41 σ 42 σ 43 σ 44 ) (23)

The approximation for quasi-stationary distribution is obtained by using conditional truncated distributions of the above multivariate normal distribution. Thus in order to evaluate p_•000, p_•100, p_•010, and p_•001 we need to use a result from conditional truncated multivariate normal distribution, which is given below as theorem 1 (for proof see S1 Appendix). Even then, it is extremely difficult, if not impossible [5], to get an exact expression for q_•(d, μ). So, we evaluate an approximate expression of q_•(d, μ) using Result 1 (for proof see S1 Appendix).

Theorem 1 Let Y ∼ N_p(μ, Σ) and write y = ( y 1 ′ , y 2 ′ ) ′, μ = ( μ 1 ′ , μ 2 ′ ) ′, and Σ = ( Σ 11 Σ 12 Σ 21 Σ 22 ). Suppose instead of R p, Y is defined only on a truncated support c < y < d. Consider the partitions c = ( c 1 ′ , c 2 ′ ) ′ and d = ( d 1 ′ , d 2 ′ ) ′. Then, the conditional distribution of Y₁ given y₂ is given by f * ( y 1 | y 2 ) = f ( y 1 | y 2 ) ∫ c 1 d 1 f ( y 1 | y 2 ) d y 1 (24) where f(y₁|y₂) is the conditional probability density function of Y₁ given y₂ i.e. Y 1 | y 2 ∼ N q ( μ 1 + Σ 12 Σ 22 - 1 ( y 2 - μ 2 ) , Σ 11 . 2 = Σ 11 - Σ 12 Σ 22 - 1 Σ 21 )

Result 1 Let ϕ ( x ) = 1 2 π e - x 2 / 2 and Φ ( x ) = ∫ - ∞ x ϕ ( t ) d t f o r a n y x ∈ ( - ∞ , ∞ ) . Then, an approximate expression of q • ( d , μ ) is given as, q • ( d , μ ) = μ p • 100 + μ p • 010 + ( d + μ ) p • 001 1 - p • 000 (25) w h e r e p • 100 ≈ 1 2 N σ 22 * ϕ ( x ^ 2 * σ 22 * ) Φ ( x ^ 2 * σ 22 * ) . 1 2 N σ 33 * ϕ ( x ^ 3 * σ 33 * ) Φ ( x ^ 3 * σ 33 * ) . 1 2 N σ 44 ϕ ( x ^ 4 σ 44 ) Φ ( x ^ 4 σ 44 ) p • 010 ≈ 1 2 N σ 33 * * ϕ ( x ^ 3 * * σ 3 * * ) Φ ( x ^ 3 * * σ 33 * * ) . 1 2 N σ 22 * * ϕ ( x ^ 2 * * σ 2 * * ) Φ ( x ^ 2 * * σ 22 * * ) . 1 2 N σ 44 ϕ ( x ^ 4 σ 44 ) Φ ( x ^ 4 σ 44 ) p • 001 ≈ 1 2 N σ 44 * * * ϕ ( x ^ 4 * * * σ 44 * * * ) Φ ( x ^ 4 * * * σ 44 * * * ) . 1 2 N σ 22 * * * ϕ ( x ^ 2 * * * σ 22 * * * ) Φ ( x ^ 2 * * * σ 22 * * * ) . 1 2 N σ 33 ϕ ( x ^ 3 σ 33 ) Φ ( x ^ 3 σ 33 ) (26) where x ^ i * , σ i i * f o r i = 2 , 3, x ^ i * * , σ i i * * f o r i = 2 , 3, x ^ i * * * , σ i i * * * f o r i = 2 , 4 are obtained from the truncated conditional distribution of multivariate normal distribution as given in (23).

Once we find q • ( d , μ ), we have an expression for expected time to extinction E ^ ( τ Q ) using Eq (12). However, this involves N, which is unknown. Now, our aim is to find the above expression in another way so that we can equate them and solve for N. So, following [6], if we can find the quasi-period (T ^ 0), we may use the relation E ^ ( τ Q ) log 2 = T ^ 0 to get the value of N, which is the critical community size.

We determine the quasi-period of the oscillation about the critical point using linearisation method [17]. Note that for our model, the linearised system about the equilibrium point x ^ = ( x ^ 1 , x ^ 2 , x ^ 3 , x ^ 4 ) ′ can be written as: d x * d t = ( - β ( 1 + η C ϕ ξ 2 + η A ρ ξ 1 ) x ^ 2 - μ - β R 0 - β η C R 0 - β η A R 0 β ( 1 + η C ϕ ξ 2 + η A ρ ξ 1 ) x ^ 2 β R 0 - ( ρ + ϕ + μ ) β η C R 0 + ω β η A R 0 + α 0 ϕ - ( ω + μ ) 0 0 ρ 0 - ( α + μ + α ) ) x * (27) where x * = x - x ^.

Now we can find the eigenvalues of the matrix in Eq (27) and hence the angular frequency, provided there are imaginary roots. Using Eq (27) and putting the values of the parameters, we can also find the angular frequency (θ) numerically. Hence the quasi-period (T ^ 0, say) is obtained as T ^ 0 = 2 π θ, which is independent of N. Now noting the relation E ^ ( τ Q ) log 2 = T ^ 0 and using Result 1, we can solve for N.

Since we are dealing with a system consisting of four equations, it is not possible to obtain an explicit or compact expression for CCS (i.e. N) in terms of the model parameters. Hence we find an approximate value of CCS numerically.

Our approach is not restricted to only two [6] or three variables [5]. It generalizes the numerical calculation of CCS for any finite number of variables. Thus, our method for finding CCS is a general one and is applicable to any number of variables. Let n be the number of variables and m be the number of equations in the system of differential equations that explains the disease dynamics. An algorithm to calculate CCS for such a process is given below.

Note that CCS thus calculated is an approximate one as we have made some approximations at few stages. However, this algorithm is a general one and may be applied to any model involving a system of differential equations that explains the disease dynamics. However, the basic reproduction number (R₀) must be greater than 1 in order to calculate CCS using our method.

For HIV transmission, we can calculate the critical community size for any region or community. Note that this value of CCS and time to extinction are approximate as we have made some approximation while applying diffusion approximation using conditional truncated multivariate normal distribution. All calculations are based on the assumed model (1)-(4). We suppose that an individual in the susceptible compartment/class (S) either moves to the infected (I) compartment or infected under treatment (C) compartment or infected with AIDS (A) compartment; while an individual from infected (I) class may receive treatment and move to infected under treatment (C) class or get infected with AIDS and move to infected with AIDS (A) class; while an individual after treatment may have reduced viral load and thus move from class C to the infected (I) class; similarly, an individual infected with AIDS (A) may have reduced viral load and move to infected (I) class. Death may occur in each class while immigration occurs only in the susceptible class.

It is clear that in order to calculate CCS for a region or community, we need to know the values of the parameters in the model. These parameter values are usually estimated based on different studies. As an illustration, we consider the values of the parameters based on the data available for Madagascar. However, the values of all parameters are not available in the literature. In such cases, we have assumed a few values considering the meaning and range of the parameters. Based on the available data for 2018 from the UNAIDS (www.unaids.org/en/regionscountries/countries/madagascar), we see that the number of people living with HIV who are on ART is 3500 whereas the number of adults and children living with HIV is 39000. The number of adults and children newly infected with HIV is 6100 and the number of deaths due to AIDS is 1700. Based on this information and following Nåsell [6] we have considered β = 0.156 × 365 = 56.94 and d = 0.044 × 365 = 16.06. Based on the available reports (http://cfs.hivci.org/country-factsheet.html) from the World Health Organisation (WHO), we have ω = 0.03 × 365 = 10.95, ϕ = 0.0035 × 365 = 1.2775, η_C = 0.000105, and μ = 0.015. Other parameters such as α = 0.001 × 365 = 0.365, η_A = 1.1, and ρ = 0.98 × 365 = 357.7 are based on available literature and intuitive meaning of the parameters. Given the poor socio-economic conditions of Madagascar, the 90-90-90 HIV treatment target initiative by UNAIDS has not performed remarkably well. At severe stages of HIV infection, available medication has very little effect. This information has led us to opt for such a high value of ρ and small value for α. Here, β is the contact rate (per year) of an HIV-infected person with the susceptible pool of individuals and ρ signifies the rate of progress (per year) of HIV-infected individuals to become AIDS patient. Similarly, we define other mentioned rates.

For this set of parameters, the CCS is obtained as 4585. So we can say that if the population in a commune drops below 4585, the infection will die out automatically. Obviously, the range of CCS for the HIV epidemic is far less than comparatively more infectious diseases such as measles that has much higher incidence rate. UNICEF reports that there were 244,607 cases of measles and 1,080 deaths in Madagascar from August 2018 to November 2019 (www.unicef.org/press-releases/measles-outbreaks-continue-unabated-five-countries-accounted-nearly-half-all-measles). Naturally, Nåsell [6] obtained CCS as 561, 000 (with R₀ = 14) and Bartlett [4] estimated CCS as 250, 000 to 300, 000 in England and Wales for measles. Note that, using the above parameters for Madagascar and assuming homogenous mixing within communities, we obtain R₀ as 4.059, which essentially means that in a completely susceptible population, a single infection may produce 4.059 secondary infections in the duration of infectiveness. Our calculation also indicates that temporary eradication or fade-out could happen after 7.365 years unless the infection is reintroduced from outside. This fade-out tends to happen in local spatial regions until the reintroduction of the disease occurs from other areas. However, this does not imply that a person already infected with HIV will be removed from the population within this period. In fact, she/he may continue to exist for more than 7.365 years and we assume that no new infection will be spread by her/him.

Next, we study the nature of CCS by varying one parameter and keeping others fixed at the above values. For this we consider the most important parameters (β, d, ϕ, ω, α, and ρ) in our model. Fig 1 indicates the relation of CCS with different parameters. Here, β is the contact rate for HIV transmission. An increase of β signifies an increment of infection circulation in the population. So the susceptible population will get infected at a much higher rate. In order to stop the infection, the population size should be very small. So if the number of the susceptible population falls below some threshold, the chance of the spread of infection will diminish. This indicates that smaller the population size, lesser would be the chance of spreading the infection. Thus, it is expected that if β increases, CCS will decrease (Fig 1A). d is the rate of death from class A. So as d increases, the spread of infection will decrease. For large values of d, a large susceptible population will remain unaffected. Hence the CCS value will be high for large d (Fig 1B). ϕ and ω respectively are the rates at which individuals in class I move to C and individuals in class C move to I. So, as ϕ increases, the number of individuals receiving treatment increases. It is expected that individuals under treatment will spread less infection than those who are infected but not treated. So, as more individuals go to class I from class C, the chance of spreading infection in the susceptible population decreases. But, it is known that the infection stays in a community long enough to engulf the entire susceptible population. Therefore, the infection might be eradicated if the susceptible population decreases. This would mean that, under the condition that ϕ increases, eradication of infection is expected to happen as the CCS value decreases (Fig 1C). On the contrary, as ω increases, CCS also increases (Fig 1D). Two other parameters viz. α and ρ represent the rates at which individuals in class A move to I and individuals in class I move to A respectively. For similar reasons, as α increases, CCS decreases (Fig 1E) but as ρ increases, CCS increases (Fig 1F).