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The authors have no competing interest related to this work.

After an epidemic outbreak, the infection persists in a community long enough to engulf the entire susceptible population. Local extinction of the disease could be possible if the susceptible population gets depleted. In large communities, the tendency of eventual damp down of recurrent epidemics is balanced by random variability. But, in small communities, the infection would die out when the number of susceptible falls below a certain threshold. Critical community size (CCS) is considered to be the mentioned threshold, at which the infection is as likely as not to die out after a major epidemic for small communities unless reintroduced from outside. The determination of CCS could aid in devising systematic control strategies to eradicate the infectious disease from small communities. In this article, we have come up with a simplified computation based approach to deduce the CCS of HIV disease dynamics. We consider a deterministic HIV model proposed by Silva and Torres, and following Nåsell, introduce stochasticity in the model through time-varying population sizes of different compartments. Besides, Metcalf’s group observed that the relative risk of extinction of some infections on islands is almost double that in the mainlands i.e. infections cease to exist at a significantly higher rate in islands compared to the mainlands. They attributed this phenomenon to the greater recolonization in the mainlands. Interestingly, the application of our method on demographic facts and figures of countries in the AIDS belt of Africa led us to expect that existing control measures and isolated locations would assist in temporary eradication of HIV infection much faster. For example, our method suggests that through systematic control strategies, after 7.36 years HIV epidemics will temporarily be eradicated from different communes of island nation Madagascar, where the population size falls below its CCS value, unless the disease is reintroduced from outside.

Periodicity in the recurrence of local epidemics has attracted mathematicians and epidemiologists for centuries. Soper [

Apart from the works by Andersson and Britton [

Technically, this paper determines the time to extinction of an epidemic outbreak theoretically and also develops a computation based approximate formulation of CCS using the diffusion approximation. The application of our method is free from any restriction on the number of variables in the system that defines the disease dynamics. To demonstrate our method we consider an HIV model by Silva, Torres, and Djordjevic [

In this paper, we have cited the HIV epidemic case of Madagascar, in Sub-Saharan Africa because of a special reason. In spite of less remarkable progress in the 90-90-90 HIV treatment target of the Joint United Nations Programme on HIV and AIDS (UNAIDS), Madagascar has shown a much lower prevalence and incidence rate of HIV infection, compared to some other African countries. It is interesting to note that, despite being in the part of Africa where HIV is in a state of epidemic, it has much lower rates of prevalence and incidence. Incidentally, Madagascar is situated in a geographically isolated location, separated by 60 km of water from the other eastern and southern countries that lie in the mainland of the African continent. This observation echoes a fact from Metcalf et al. [

We, therefore, applied our method on the demographic facts and figures of Madagascar to obtain the CCS value and mean time to extinction of the disease. Our calculations suggest that, unless the disease is reintroduced from outside, after 7.36 years HIV epidemics might fade out from different cities or communes of the island nation where the susceptible population size drops below 4585, the CCS value of Madagascar. This time period of 7.36 years could appear astonishing as HIV has become a long-standing epidemic situation in the countries of the Sub-Saharan AIDS belt. But there is a strong possibility that the idea of CCS would propel the controlling strategies of UNAIDS to eradicate the disease in the near future. This work suggests that the systematic spreading of control measures could lead to accelerated progress to achieve the 90-90-90 target of UNAIDS for HIV eradication. Such progress shall prevent the HIV epidemic in Madagascar unless the infection is reintroduced from outside. Moreover, countries like Uganda, Malawi (located in Eastern and Southern Africa) are separated from their neighboring countries by water bodies such as fresh water lakes. These countries have lower rates of prevalence and incidence compared to landlocked countries such as Eswatini (in Southern Africa). Notably, these rates are still much higher than that of Madagascar.

To date, CCS has largely been a part of mathematical theory only. But implementing CCS as controlling strategies of epidemics might prove beneficial in the future. Therefore, to utilize the potential that CCS holds, it is important to determine its value promptly for any disease dynamics. This motivated us to calculate CCS (approximately) using simplified computations. Our method will help in finding country-specific CCS and time to extinction of the disease, using real demographic facts and figures for HIV infection in different countries as parameter values. We emphasize the potential utility of CCS to guide public health practitioners to develop appropriate area-specific control strategies and interventions especially when they are limited, before or after an epidemic outbreak.

We work with the model proposed by Silva and Torres [

Here _{A} (≥ 1) is the relative infectiousness of individuals with AIDS symptoms compared to those infected with HIV but no AIDS symptoms, and _{C} (≤ 1) is the rate of partial restoration of immune function of HIV infected individuals who are correctly treated under ART.

We consider the total population as

We use the approach of next generation matrix [_{0}. Note that the rate of new infections coming in the compartments _{0}) as:
_{2}(_{1} + _{A}) + _{C} _{1}), _{2} + (_{1}) + _{1} + _{1} = _{2} =

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 (

Event | Transition | Transition rate |
---|---|---|

Immigration of susceptibles (S) | ( |
λ_{1}( |

Death of susceptibles (S) | ( |
λ_{2}( |

Susceptible (S) to Infected (I) | ( |
λ_{3}( |

Susceptible (S) to Infected under treatment (C) | ( |
λ_{4}(_{C} |

Susceptible (S) to Infected with AIDS (A) | ( |
λ_{5}(_{A} |

Infected (I) to Infected under treatment(C) | ( |
λ_{6}( |

Infected (I) to Infected with AIDS (A) | ( |
λ_{7}( |

Death of Infected (I) | ( |
λ_{8}( |

Infected under treatment (C) to infected (I) | ( |
λ_{9}( |

Death of Infected under treatment(C) | ( |
λ_{10}( |

Infected with AIDS symptoms (A) to Infected (I) | ( |
λ_{11}( |

Death of Infected with AIDS symptoms (A) | ( |
λ_{12}( |

Here, _{s,i,c,a}(

Now, we find quasi-stationarity by conditioning on non-extinction of the disease. So we write,

Differentiating _{s,i,c,a}(

Let _{Q} be the time to extinction when the initial distribution is equal to the quasi-stationary distribution. Therefore, we have,

The disease-free equilibrium is obtained as: _{1}(_{2}(_{3}(_{4}(

For simplicity we use the notations: _{j}(_{j} for _{1} = _{2} = _{1} _{2}(_{1} − _{2}, _{1} _{2}+ _{C} _{1}+ _{A} _{2}).

Then solving Eqs (

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 _{0} is greater than 1. First we find the multivariate normal distribution corresponding to the state variables. Let the changes in the scaled state variables _{1}, _{2}, _{3}, and _{4} during the time interval (_{1}, _{2}, _{3}, and _{4} respectively, where _{i}(_{i}(_{i}(

Under the assumptions of the original process on the sequence of transitions, we evaluate the mean vector and covariance matrix for _{i} (

_{C}

_{A}

The random variable _{1} equals _{2} equals _{3} equals _{4} equals _{1}, _{2}, _{3}, _{4})′ we have,

Now to derive the covariance matrix we need to find the Jacobian matrix at point _{1}, _{2}, _{3}, _{4})′ is,

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

Let _{ij} be the solution of the (

The approximation for quasi-stationary distribution is obtained by using conditional truncated distributions of the above multivariate normal distribution. Thus in order to evaluate _{•000}, _{•100}, _{•010}, and _{•001} we need to use a result from conditional truncated multivariate normal distribution, which is given below as theorem 1 (for proof see _{•}(_{•}(

_{p}(_{1} _{2} _{1}|_{2}) _{1} _{2}

Once we find

We determine the quasi-period of the oscillation about the critical point using linearisation method [

Now we can find the eigenvalues of the matrix in

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.

Our approach is not restricted to only two [

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 (_{0}) 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 (_{C} = 0.000105, and _{A} = 1.1, and

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 (_{0} = 14) and Bartlett [_{0} 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 (

Bartlett [

Shocking statistics (

When we look at the geographical locations of Uganda, Malawi, Eswatini, and Madagascar, interestingly we find that being at the heart of the Great Lakes region, Uganda is surrounded by three lakes among which one is a fresh water lake. These lakes, to some extent separate Uganda from the rest of the African countries. Malawi again is separated by a freshwater lake (Lake Malawi) that comprises of 25% of its area. But Eswatini is a landlocked country in Southern Africa. We observe that, even with quite impressive progress towards achieving the 90-90-90 target, countries that are more landlocked have higher rates of incidence and prevalence compared to other countries that are isolated to varying extent. Although being in the AIDS belt of Africa, Madagascar has shown remarkably lower rates of incidence and prevalence even with little success of the 90-90-90 target. It is possibly because of the isolated location of Madagascar that separates it from the mainland of the African continent to a great extent.

Metcalf et al. [

Madagascar has 111 large and small cities or districts with 39.6% of them have population size less than 20000, 30.6% with population size somewhere between 20000 − 30000, 16.2% has 30000 − 40000 and 5.4% have population more than 100000 (

We observe from our case study that in addition to the existing controlling measures of HIV epidemics in Madagascar, employing the idea of CCS might accelerate the eradication of the infection from smaller populations after the corresponding mean extinction time. Besides, CCS could be looked upon as an index of performance of epidemic controlling strategies in different countries.

Combating pandemic and epidemic outbreaks require efficient controlling strategies, as the available treatment facilities and/or vaccines become limited in such situations. But, the infection persists in the community long enough to engulf the entire susceptible population. Hence, in large communities, the infection recurs in the subsequent timeline. Interestingly, Bartlett observed that in smaller communities the infection dies out if the susceptible population drops below a threshold size (CCS). Later, Nåsell formulated CCS for infectious diseases having Susceptible-Infected model dynamics. But the analytical calculation of CCS for more complex disease dynamics or higher-order models are quite cumbersome. So, we propose a simplified computation based approach for rapid calculation of the approximate value of CCS, for HIV disease dynamics with four differential equations. Since our method is free from any restriction on the number of variables in the system, it could be applied to other diseases with higher-order model dynamics. We envisage CCS for devising control strategies during epidemic outbreaks such as COVID-19 [

So to sum up, through this paper, firstly, we have generalized the approach for CCS calculation for a system of differential equations having more than two variables. Our method for a system containing four variables may be extended to higher dimensions for any infectious disease dynamics. This is because our method is able to tackle multivariate normal distribution for calculating expected time to extinction as a function of CCS (Result 1). Secondly, we have developed a method to unleash the potential that CCS holds in devising control strategies by promptly determining its approximate numerical value from any disease dynamics using a simplified computation approach. In particular, we believe that our method will provide a suitable measure for accelerated extermination of epidemic infections by aiding policymakers in making decisions on control strategies. Thirdly, we hypothesize that, if we start treating spatially isolated regions or smaller communes and then progress to the district level, and after that to larger regions systematically, the annihilation of HIV infection could be accelerated. Although it is way difficult to reach out to smaller communes with treatments and other controlling measures (such as educating them about the infection etc.), CCS could be worth giving a trial, as this could utilize the natural force of damping down of epidemic oscillations to eradicate HIV infection from smaller populations.